Size of conjugacy class of subgroups equals index of normalizer

Statement
Let $$G$$ be a group and $$H$$ be a subgroup of $$G$$. Let $$\mathcal{C}$$ denote the conjugacy class of subgroups of $$H$$ in $$G$$, i.e., the set of all subgroups of $$G$$ that are fact about::conjugate subgroups to $$H$$ (note that $$H \in \mathcal{C}$$). Then, there is a bijection:

$$G/N_G(H) \leftrightarrow \mathcal{C}$$

where $$N_G(H)$$ is the fact about::normalizer of $$H$$ in $$G$$ and $$G/N_G(H)$$ denotes the coset space. In particular:

$$[G:N_G(H)] = |\mathcal{C}|$$

In other words, the size of the conjugacy class of subgroups $$\mathcal{C}$$ equals the index of $$N_G(H)$$ in $$G$$.

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Fundamental theorem of group actions