The maximal subgroups of the Monster

The classification of the maximal subgroups of the Monster M is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether M contains maximal subgroups that are almost simple with socle PSL 2 ( 13 ) . However, this conclusion relies on reported claims, with unpublished proofs, that M has no maximal subgroups that are almost simple with socle PSL 2 ( 8 ) , PSL 2 ( 16 ) , or PSU 3 ( 4 ) . The aim of this paper is to settle all of these questions, and thereby complete the solution to the maximal subgroup problem for M , and for the sporadic simple groups as a whole. Specifically, we prove the existence of two new maximal subgroups of M , isomorphic to the automorphism groups of PSL 2 ( 13 ) and PSU 3 ( 4 ) , and we establish that M has no almost simple maximal subgroup with socle PSL 2 ( 8 ) or PSL 2 ( 16 ) . We also correct the claim that M has no almost simple maximal subgroup with socle PSU 3 ( 4 ) , and provide evidence that the maximal subgroup PSL 2 ( 59 ) (constructed in 2004) does not exist. Our proofs are supported by reproducible computations carried out using the publicly available Python package mmgroup for computing with M recently developed by M. Seysen. We provide explicit generators for our newly discovered maximal subgroups of M in mmgroup format.