No.
patriarchy noun pa·tri·ar·chy ˈpā-trē-ˌär-kē plural patriarchies 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 of descent and inheritance in the male line broadly : control by men of a disproportionately large share of power 2 : a society or institution organized according to the principles or practices of patriarchy For 20 years the country was ruled as a patriarchy.
You actually mean
gender role
A culturally and socially determined set of expected behaviors, attitudes, and characteristics based on concepts of masculinity and femininity. A gender role should not be confused with gender identity, which refers to an individual’s internal sense of being masculine, feminine, on a spectrum between the two, a gender unrelated to that binary, or no gender at all.
So… Republican