Number of Distinct Conjugate Subsets is Index of Normalizer

Theorem
Let $G$ be a group.

Let $S$ be a subset of $G$.

Let $\map {N_G} S$ be the normalizer of $S$ in $G$.

Let $\index G {\map {N_G} S}$ be the index of $\map {N_G} S$ in $G$.

The number of distinct subsets of $G$ which are conjugates of $S \subseteq G$ is $\index G {\map {N_G} S}$.