Patriarchy: 1 – Social organization marked by the supremacy of the father in the clan or family, the legal dependence of wives and children, and the reckoning descent and inheritance in the male line. 2 – A society organized according to the principles of patriarchy.¹
Matriarchy: 1 – A system of social organization in which descent and inheritance are traced through the female line. 2 – A family, group, or state governed by a matriarch.¹ (Although, I am of the mind that the word Matriarchy should be defined exactly the same as Patriarchy, except with the genders reversed.)
Demogarchy²: 1 – A system of social organization in which descent and inheritance are traced by which child is chosen to be best suited to fill the role of family leader, regardless of gender or place in birth order.
¹ From my thirty-six year old Webster’s Dictionary.
² A word I made up.