Aside from women, Black people, gay people, and Catholics are special targets of the ruling elite. Black people are shipped off to a colony in the Midwest, suggesting an apartheid approach to race. Offred and the other handmaids see the dead bodies of gay men and Catholics dangling from a wall they pass on the way to the market.
These groups are targeted because the elite in Gilead considers them evil and corrupting influences who bring chaos and false ideas into the pure Christian state they are trying to establish. Atwood is illustrating the ideas of certain forms of Christianity that hold that Black people carry the mark of Cain on their bodies in the form of dark skin and so are cursed by God. This would justify exiling them. Likewise, gay sex is seemingly condemned by verses in both the Old and New Testaments. The Catholic Church has sometimes been characterized as the Great Satan by fundamentalist groups and the Pope as the anti-Christ. These interpretations of the Bible would allow Gilead to feel justified in acting on its prejudices.
As for women, as the novel shows, fear of women's independence and equality, especially the female desire for control over their reproduction and sexual functions, fuels much of the fundamentalist backlash against the liberal state. The oppression of women, however, especially the handmaids, is particularly fueled by a sudden drop in fertility due to pollution and environmental issues. In order to have children at all, there is intense pressure to control the bodies of fertile women so that they are directed solely towards producing more children. All of this oppression is fueled by interpretations of the Bible that say women have a second-class, subordinate status to men.
These hierarchies matter so much to the ruling elite because of their fear of chaos and disorder as socially destructive forces—and particularly from their fear of the sharply declined birth rate. The ruling regime represents a repressive, authoritarian mindset that believes that people have to be forced by those "above" them to do what someone else has decided is best for them. In reality, this system benefits a small elite of older white men.