Unfortunately many of the other answers here, including the accepted one are bordering on insecure apologetics, the fact is in Islam men are definitely "ranked" higher than women and this is clear to anyone who examines the Qur'an, the Sunnah and avoids drowning in the numerous pits of liberal reform that are so easy to fall into nowadays. Now I should be clear, Islam does grant women all the rights they need and deserve and women are never belittled, but the focus of this answer will be strictly to show that men "outrank" women, at least in this worldly life.
The Quranic verse 4/34
One of the clearest proofs of a Man's superiority over a woman is this verse of the Qur'an:
Men are in charge of women by [right of] what Allah has given one over the other and what they spend [for maintenance] from their wealth. So righteous women are devoutly obedient, guarding in [the husband's] absence what Allah would have them guard. But those [wives] from whom you fear arrogance - [first] advise them; [then if they persist], forsake them in bed; and [finally], strike them. But if they obey you [once more], seek no means against them. Indeed, Allah is ever Exalted and Grand.(4/34).
There are three clear takeaways from this verse.
(1) Men are in charge of women. If men and women are equal, then why would Allah declare one in charge of the other?
(2) Allah encourages women to be obedient. How can you be equal with somebody you are subservient to?
(3) Further, Allah gives men the authority to discipline their disobedient wives through various methods. Again, where is the equality? there is none, there never was and I think any such notion is born external to Islam.
A quick survey of various classical commentaries on this verse by the tafsir specialists of the Madhahib shows the same conclusion:
The two Jalals, Imams Mahalli and Suyuti two famous scholars, especially Al Suyuti for his expertise in the Qur'an sciences wrote:
Men are in charge of, they have authority over, women, disciplining them and keeping them in check, because of that with which God has preferred the one over the other, that is, because God has given them the advantage over women, in knowledge, reason, authority and otherwise, and because of what they expend, on them [the women], of their property
[ Quranx Ref: Tafsir al Jalalyn 4.34 4/34 ]
From the same madhab, Ibn Kathir the famous mufassir whose tafsir is probably the most widely used today wrote:
الرِّجَالُ قَوَّامُونَ عَلَى النِّسَآءِ
meaning, the man is responsible for the woman, and he is her maintainer, caretaker and leader who disciplines her if she deviates.
[ Quranx Ref: Tafsir Ibn Kathir 4.34 4/34) ]
The Reservation of Rulership for Men
Secondly, I highlight the Islamic reservation of leadership roles for men and strictly men as another evidence that the rank of a man is further up the figurative "totem pole" on the hierarchy (yes i said it and God forbid Jordan peterson reads this).
The Messenger of Allah said:
Never will succeed such a nation as makes a woman their ruler
[ Sunnah Ref: Sahih Bukhari 7099 ]
Every hadith in Sahih al Bukhari is accepted. I know there are some people who prefer Hadith books written closer to the time of the prophet, you lot should know that most of the earlier books generally contain weaker connections of the form
while Imam Bukhari preferred to find chains with a stronger link, and whenever he did use the weaker form, he ensured that the two narrators connected were stellar when it came to problems like tadlees. This quickly devolves into a debate of time vs connection strength.
As for the interpretation of this hadith, it is obvious that the Messenger of Allah PBUH does not consider it appropriate for women to rule a nation. There are some theories as to why the prophet said this. Some scholars explained that this is because of the verse 4/34 (mentioned earlier) as Allah put men in charge of women and if a woman were to manage a nation, then she would also be in charge of men and this would contradict that verse of the Qur'an.
It was on the basis of this Hadith and reasoning that some of the madhabs even forbade women from becoming judges and any positions of leadership, nevermind rulers. Regardless of whether one disagrees with that extrapolation of this statement of the Prophet, this is clearly a few metric tonnes of bomb dropped on feminist ideas of gender equality.
The Speech on Female Deficiencies
Another famous Hadith regarding this issue of gender equality is the speech the Messenger of Allah gave regarding the various deficiencies women have, in the intellect and in the religion:
The Messenger of Allah said:
"O women folk! You should give charity and be diligent in seeking Allah's forgiveness because I have seen (i.e., on the Night of the Ascension to the highest heavens) that dwellers of the Hell are women." A woman amongst them said: "Why is it that the majority of the dwellers of Hell are women?" The Prophet (ﷺ) replied, "You curse frequently and are ungrateful to your husbands. In spite of your lacking in wisdom and failing in religion, you are depriving the wisest of men of their intelligence." Upon this the woman asked: "What is the deficiency in our wisdom and in our religion?" He (ﷺ) replied, "Your lack of wisdom can be well judged from the fact that the evidence of two women is equal to that one man. You do not offer Salat (prayer) for some days and you do not fast (the whole of) Ramadan sometimes, it is a deficiency in religion.
[ Sunnah Ref: Riyad al Salihin 1879 ]
This Hadith is found in both Sahih al Bukhari and Sahih Muslim, giving it the status of
They agreed upon it
Meaning, Imam Bukhari and Muslim agreed upon it. This is important because although they both had the most stringent and rigorous of Hadith scrutiny, they differed in the emphasis. Imam Bukhari focused more on the reliability of each individual transmitter, while Imam Muslim was more careful with the transmission of the exact wording so their corroboration is a big deal although I couldn't find this hadith in earlier books. I may have missed it.
With that said, we move onto interpreting the text. The first thing to establish is that although the Prophet is speaking to a specific group of women, his speech here is clearly directed at women in general.
(1) - The fact that the prophet said most of the hell dwellers will be women.
If the speech was directed at only that specific group of women the prophet was talking to, then it would not make sense to mention this as surely that group alone could not fill the vastness of hell on their own.
(2) It is well known that the witness of a woman is half a man's, in general. Not just for that one group of women.
So this should conclusively bury the idea that men and women are equal, at least in this worldly life. Lastly, we should bear in mind the Quranic command of obedience to the rulings of Islam:
The only statement of the [true] believers when they are called to Allah and His Messenger to judge between them is that they say, "We hear and we obey." And those are the successful(24/51)