[Kadia]

I believe it's not strange for the clans or branches to be Matriarch as Kuki, one of Kudou's branch family did designate his daughter as a head of family. This is where i got it mixed up with Fujibayashi clan.

Spoiler for Koki's family:

The old gentleman who called Retsu "sensei" was the former master of the Kuki
Clan, Kuki Mamoru. He was over sixty years in age, but considering Retsu was
about to turn ninety, he was not yet at retirement age. He had turned over the clan
to his eldest daughter in order to work as Retsu’s hands.