  A subgroup of a group $G$ is called algebraic if it can be expressed as a
finite union of solution sets to systems of equations. We prove that a
non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is
algebraic if and only if there exists a finite subgroup $K$ of $G$ such that
$C_G(K) \leq H \leq N_G(K)$. We provide some applications of this result to
free products, torsion-free relatively hyperbolic groups, and ascending chains
of algebraic subgroups in acylindrically hyperbolic groups.
