The subgroups of $SL_2(Z)$ can be divided in two types. There are the congruence subgroup, i.e. subgroups that contain a principal congruence subgroup $\Gamma(N)$, and the non-congruence subgroups. Often, the form a subgroup is given in does not allow an easy and direct determination of the type. In this talk I will present a method to check for a subgroup given by permutations if it is a congruence subgroup or not.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/246