The class of acylindrically hyperbolic groups consists of groups that admit a particular nice type of non-elementary action on a hyperbolic space, called an acylindrical action. This class contains many interesting groups such as non-exceptional mapping class groups, Out(Fn) forn > 1, and right-angled Artin and Coxeter groups, among many others. Such groups admit uncountably many different acylindrical actions on hyperbolic spaces, and one can ask how these actions relate to each other. Inthis talk, I will describe how to put a partial order on the set of acylindrical actions of a given group on hyperbolic spaces, which roughly corresponds to how much information about the group different actions provide. This partial order organizes these actions into a poset. I will givesome structural properties of this poset and, in particular, discuss for which (classes of) groups the poset contains a largest element.