Are The Six Duchies patriarchal?

I’ve seen it argued on here before that the Six Duchies are not patriarchal and I remember thinking ‘yes they are’ but not knowing exactly why I thought they were.

Now I am re-reading I think I can finally put my finger on it. Whenever The Six Duchies are talked about, the leaders are always referred to as ‘dukes’. It seems that they are all male led.

However what is confusing is that we do see female soldiers and stuff and from my understanding the king or queen is chosen according to the order of birth rather than sex.

So I’m a bit confused. Are The Six Duchies patriarchal or not?

(I’ve put ‘Spoilers all’ since I’ve read all the books and this may lead onto discussion of books beyond Farseer)