Maximal subgroups of sporadic groups

TL;DR

This paper surveys the extensive 50-year research on classifying maximal subgroups of sporadic simple groups and their automorphism groups, highlighting progress and remaining challenges.

Contribution

It provides a comprehensive overview of methods and results in the classification of maximal subgroups of sporadic simple groups and their automorphisms.

Findings

Most sporadic groups' maximal subgroups are classified

Only a few cases in the Monster remain unresolved

The survey summarizes 50 years of research progress

Abstract

A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups.