Thompson has proved that, up to conjugation, there are only finitely subgroups of containing a (non-trivial) principal congruence subgroup of the modular group of a fixed genus. (The genus of a (discrete) subgroup of is the genus of the Riemann surface , where is hyperbolic 2-space.)
This paper provides a detailed version of Thompson’s theorem for the two lowest genus values. They compile a complete list of all congruence subgroups of of genus 0 and 1. The proofs involve computations using Magma.
The results answer a question of Conway and Norton by listing all the so-called “moonshine groups”, i.e. the genus zero subgroups of which contain (i) a subgroup of the type and (ii) translations only defined by rational integers.
The authors discuss the overlap between their results and those of K. S. Chua and M. L. Lang in “Congruence subgroups associated to the Monster” [Exp. Math. 13, No. 3, 343–360 (2004), see Zbl 1099.11018 above].