Let
$$\Gamma(N)=\left\{ \pmatrix{a & b \\ c & d} \in \text{SL}_2(\mathbb{Z}) : \pmatrix{a & b \\ c & d} \equiv \pmatrix{1 & 0 \\ 0 & 1} \mod N \right\}$$
$$\Gamma_0(N)=\left\{ \pmatrix{a & b \\ c & d} \in \text{SL}_2(\mathbb{Z}) : \pmatrix{a & b \\ c & d} \equiv \pmatrix{* & * \\ 0 & *} \mod N \right\}$$
$$\Gamma_1(N)=\left\{ \pmatrix{a & b \\ c & d} \in \text{SL}_2(\mathbb{Z}) : \pmatrix{a & b \\ c & d} \equiv \pmatrix{1 & * \\ 0 & 1}\mod N \right\}.$$
I want to check whether these subgroups of $\text{SL}_2(\mathbb{Z})$ are normal subgroups. The first one is easy, as I can find a surjective homomorphism, such that $\Gamma(N)$ is in the kernel for all $N\geq1$.
For the other groups, what is the smartest and shortest way to (dis)prove it?