Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of $\Gamma$ is a congruence subgroup of level $N$ if it contains the principal congruence subgroup of level $N$.

Are there methods for establishing upper bounds on the possible levels of $G$, say given information about $\Gamma/G$? Establishing *lower* bounds is trivial: the minimal $N$ such that $T^N \in G$ (if such an $N$ exists) is the *minimal* possible level of $G$.

For example, suppose $\Gamma/G \cong (\mathbb{Z}_3 \times \mathbb{Z}_9) \rtimes S_3$ or more generally of the form (finite abelian group)-semidirect product-(symmetric group). I'm generally thinking about the case where $G$ is the kernel of some unitary representation with finite image.