Solvability of a group based on its number of subgroups

Abstract

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by Betz and Nash. We also provide an application of our results in studying comaximal subgroup graph of a group. Finally, we conclude with some open issues.