= Rational conjugation of finite-index modular subgroups
For any <finite-index subgroup> $\Gamma\leq SL_2(\mathbb Z)$ and rational positive-determinant $\gamma$, the intersection $SL_2(\mathbb Z)\cap\gamma^{-1}\Gamma\gamma$ has finite index. Multiply $\gamma$ by a positive rational scalar to obtain an integral <matrix> $A$ with positive integer <determinant> $D$. For $h=I+DB\in\Gamma(D)$, $AhA^{-1}=I+AB\operatorname{adj}(A)$ is integral, so $\Gamma(D)$ lies in $SL_2(\mathbb Z)\cap\gamma^{-1}SL_2(\mathbb Z)\gamma$. This intersection has finite index. Its further intersection with $\gamma^{-1}\Gamma\gamma$ has relative index at most $[SL_2(\mathbb Z):\Gamma]$. No assumption that $\Gamma$ itself is a <congruence subgroup> is needed.
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