Patriarchal attitudes are bred in the family through the socialization process. The family, as a social institution, is a brewery for patriarchal practices by socializing the young to accept sexually differentiated roles.
Kate Millet, a radical feminist, points to ideological factors in her search for the roots of patriarchy. She attaches importance to socialization. Men are socialized to have a dominant temperament. This provides men with a higher social status which in turn lead to them filling social roles in which they can exercise mastery over women.
For two hundred years, patriarchy precluded women from having a legal or political identity and the legislation and attitudes supporting this provided the model for slavery. In the late 19th and early 20th centuries suffrage campaigners succeeded in securing some legal and political rights for women in the UK. By the middle of the 20th century, the emphasis had shifted from suffrage to social and economic equality in the public and private sphere and the women¡¦s movement that sprung up during the 1960s began to argue that women were oppressed by patriarchal structures.
Patriachy has probably the biggest part of gender inequality, yes. It's much more intricate than that; its continuance in this century is attributed to the reluctance of society to adapt to gender equality. This, along with racism and things of the like, will hopefully be eradicated in future generations. The most we can do is put down a solid foundation for future generations to build on.