In , Greenberg showed that n <= 6t(3) so that mu - nt <= 6t(4) for a normal subgroup N of level n and index mu having t parabolic classes in the modular group Gamma. Accola, , improved these to n <= 6t(2) always and n <= t(2) if Gamma/N is not abelian. In this work we generalise these results to Hecke groups. We get results between three parameters of a normal subgroup, i.e. the index mu, the level n and the parabolic class number t. We deal with the case q = 4, and then obtain the generalisation to other q. Two main problems here are the calculation of the number of normal subgroups and the determination of the bounds on the level n for a given t.