Subrack lattices of finite solvable and metacyclic groups

A group $G$ with conjugation operation is a rack. We call such racks \emph{group racks}. In this paper we study finite group racks via their subrack lattices. Heckenberger, Shareshian, and Welker proved that the isomorphism type of the subrack lattice of a finite group determines whether the group is solvable. Our first result shows that if $G$ is a finite solvable group and $H$ is a finite group whose subrack lattice is isomorphic to that of $G$, then $H$ is solvable and the derived length of $H$ has the same derived length as $G$. Our second result is that if $G$ is a finite metacyclic group and $H$ is a group whose subrack lattice is isomorphic to that of $G$, then $H/Z(H)$ is metacyclic. As a further application of our analysis of finite metacyclic groups, we answer a question of Heckenberger, Shareshian, and Welker in the affirmative by constructing two finite groups with isomorphic subrack lattices that are not isomorphic as racks.