Some groups with computable Chermak-Delgado lattices

Let $G$ be a finite group and let $H \leq G$. We refer to $\ord {H} \ord {C_G(H)}$ as the {\it Chermak-Delgado measure of $H$} with respect to $G$. Originally described by A. Chermak and A. Delgado, the collection of all subgroups of $G$ with maximal Chermak-Delgado measure, denoted $\CD {G}$, is a sublattice of the lattice of all subgroups of $G$. In this paper we note that if $H \in \CD G$ then $H$ is subnormal in $G$ and prove if $K$ is a second finite group then $\CD {G \times K} = \CD G \times \CD K$. We additionally describe the $\CD {G \wr C_p}$ where $G$ has a non-trivial center and $p$ is an odd prime and determine conditions for a wreath product to be a member of its own Chermak-Delgado lattice. We also examine the behavior of centrally large subgroups, a subset of the Chermak-Delgado lattice. DOI: 10.1017/S000497271200019