Counting subgroups of a finite group containing a prescribed subgroup

TL;DR

This paper establishes an upper bound on the number of subgroups containing a given subgroup in a finite group, depending on the index of the subgroup, using a specific exponential formula.

Contribution

It provides a new explicit upper bound formula for counting subgroups containing a fixed subgroup in finite groups, improving understanding of subgroup structure.

Findings

Derived a specific exponential upper bound formula.

Bound depends on the index of the subgroup.

Enhances subgroup enumeration techniques.

Abstract

Let $R$ be a finite group, and let $T$ be a subgroup of $R$. We show that there are at most \[ 7.3722[R:T]^{\frac{\log_2[R:T]}{4}+1.8919} \] subgroups of $R$ containing $T$.