Does there exist Fuchsian groups, which is not conjugated in $SL(2, \mathbb{R})$ to a subgroup of $SL(2, \mathbb{Z})$, but still contains a congruence subgroup?



Non congruence subgroups containing congruence subgroups.
How about the normalizer of $\Gamma\_0(N)$ in $PSL\_2(\mathbb{R})$ for $N > 1$, which is the subgroup generated by $\Gamma\_0(N)$ and $\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}$?


(I wouldn't call these "non-congruence subgroups" as you do in your title: that technical term is usually parsed as "(non-congruence) subgroup". This isn't a subgroup of $PSL\_2(\mathbb{Z})$ at all, but it certainly contains a principal congruence subgroup; perhaps we should call it a "congruence non-subgroup"?)


