An explicit upper bound on the number of subgroups of a finite group

TL;DR

This paper establishes a new explicit upper bound on the number of subgroups in a finite group, improving understanding of subgroup enumeration in group theory.

Contribution

It provides the first explicit bound of this form, linking subgroup count to group order with a precise inequality.

Findings

Finite groups of order r have at most 7.3722 * r^{(log_2 r)/4 + 1.5315} subgroups.

The bound is explicit and improves previous asymptotic estimates.

The result advances the quantitative understanding of subgroup structure in finite groups.

Abstract

In this paper we prove that a finite group of order $r$ has at most $$ 7.3722\cdot r^{\frac{\log_2r}{4}+1.5315}$$ subgroups.