Groups with finitely many isomorphic classes of non-abelian subgroups

We study groups in which the non-abelian subgroups fall into finitely many isomorphic classes. We prove that a locally generalized radical group with this property is abelian-by-finite and reduced minimax. The same conclusion holds for locally generalized coradical groups. Here a generalized radical group is a group with an ascending series whose factors are either locally nilpotent or locally finite, and a generalized coradical group is a group with a descending series whose factors are either locally nilpotent or locally finite.