I'm trying to prove the following, but I don't really know where to start...
If $\theta : G \rightarrow H$ is a surjective groupmorfism with $ker(\theta) = N$. If we define$ S := \{U|N\leq U \leq G\}$ and $T := \{V |V\leq H\}$
Then $\theta$ and $\theta^{-1} $ induce inverse bijections between $S $ and $T$. Even more these bijections keep inclusions, indices, normal subgroups and quotient groups.
Could anybody help me with how to prove this?