On groups with few subgroups not in the Chermak-Delgado lattice

TL;DR

This paper classifies finite groups with fewer than five subgroups outside their Chermak-Delgado lattice, showing such groups are nilpotent except for S_3, and extends previous classifications.

Contribution

It extends prior work by classifying all finite groups with less than five subgroups outside the Chermak-Delgado lattice and proves such groups are mostly nilpotent.

Findings

Groups with fewer than five subgroups outside the Chermak-Delgado lattice are nilpotent.

The only non-nilpotent group with five or fewer subgroups outside the lattice is S_3.

Classification extends previous results on groups with at most two subgroups outside the lattice.

Abstract

We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasol\u{a} and T\u{a}rn\u{a}uceanu asked how many subgroups are not in the Chermak-Delgado lattice and classified all groups with two or less subgroups not in the Chermak-Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak-Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak--Delgado lattice is nilpotent. In this vein we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak-Delgado lattice is S_3.