A comment from @mentalidade_logica mentioned the last matriarchal society on earth, the Mosuo in China (see https://en.wikipedia.org/wiki/Mosuo).

I checked it out why it works and well it's interesting.

Basically the father figure (in sense of raising kids) is taken by the brother of the mother (i.e. uncle) and biological father has no importance. Co-habitation is banned, random guys can only sleep the night (if given permission). The female can switch guys as fast as she likes, but always 1 at a time (serial monogamy). There is no financial support required from the father and mostly is unknown who the father is. There are no private rooms, except for girls in fertile age. Each girl past puberty is given such room.

Fascinating.