On Leinster groups of order pqrs

Abstract

A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. Let $p<q<r<s$ be primes. We prove that if $G$ is a Leinster group of order $p^{2}qr$, then $G\cong Q_{20}\times C_{19}$ or $Q_{28}\times C_{13}$. We also prove that no group of order $pqrs$ is Leinster.