  We introduce the new notion of quotient-saturation as a measure of the
immensity of the quotient structure of a group. We present a sufficient
condition for a finitely presented group to be quotient-saturated, and use it
to deduce that non-elementary finitely presented subgroups of a hyperbolic
group (in particular, non-elementary hyperbolic groups themselves) are
quotient-saturated. Finally, we elaborate on the previous results to extend the
scope of this property to finitely presented acylindrically hyperbolic groups.
