Classifying groups with a small number of subgroups

TL;DR

This paper establishes lower bounds on the number of subgroups in finite groups based on their prime factorization, classifies groups with few subgroups, and extends an integer sequence related to subgroup counts.

Contribution

It introduces new lower bounds for subgroup counts and classifies all abelian and certain non-abelian groups with a small number of subgroups, extending existing sequences.

Findings

Classified all abelian groups with ≤22 subgroups.

Classified all non-abelian groups with ≤19 subgroups.

Extended the integer sequence A274847 related to subgroup counts.

Abstract

We provide lower bounds on the number of subgroups of a group $G$ as a function of the primes and exponents appearing in the prime factorization of $|G|$. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and all non-abelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 \cite{OEIS} introduced by Slattery in \cite{Slattery}.