Classifying Groups With a Small Number of Subgroups

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 nonabelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 in the On-Line Encyclopedia of Integer Sequences introduced by Slattery.