The books dive deep into this. Op answered it perfectly though. I have read every single book and and op nailed it all.

From the Sangheili culture we are exposed to, females typically serve an administrative role in their homes, wielding considerable authority in the matters that deal with home. Many assume Lekgolo to be asexual. Female San'Shyuum appear to live a civilian life. We do know that some serve in concubine roles. What we know about female Yanme'e is about the Queens, which hold considerable power over their species and in the Covenant. Kig-Yar have a matriarchal society, where only females can captain ships. We know nothing about female Unggoy, but given their species' role in the Covenant, I would assume they serve birthing roles. What little we know about female Jiralhanae (brothels) indicates they have a patriarchal society.

I'm pretty sure that the sangheili woman take care of the family/keep, and Kig-Yar/Jackals are a matriarch society. I'm not sure about unggoy and jirlhanaee.