Classifying groups with a small number of subgroups

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}.

12 pages, 3 tables; V2 Fixes an error (which has no effect on the final results) and also adds a few citations/details suggested by early readers; V3 fixes minor typographical errors and clarifies the explanation of Slattery's previous work