Classification of Normal Subgroups of Hecke Group H6 in Terms of Parabolic Class Number

In [3], Greenberg showed that $n≤6t3$ so that $μ = nt≤6t4$ for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to $n≤6t2$ always and $n≤t2$ if Γ/N is not abelian. Newman, [5], obtained another generalisation of these results. Hecke groups are generalisations of the modular group. We particularly deal with one of the most important cases, $q = 6.$