Equivalences of Acting on Collections of Subgroups by Conjugation

In general, suppose $G$ acts on its collection of subgroups by conjugation and $H\le G$ is a specific Subgroup. Then:

• The subgroup $H$ is fixed under this action exactly when $H⊴G$
• The stabilizer $G_H$ is the Normalizer $N_G(H)$ of $H$ in $G$
• The number of subgroups conjugate to $H$ equals $|G:N_G(H)|$.