The Stabilizer subgroup of a set $H$ under the Conjugation action (vid)

Normalizer of $H$ ($N(H)$) is equal to $G$ iff $H$ is a Normal subgroup.

$|H|$ divides $|N(H)|$ ($|N(H)|$ divides $|G|$). both because they are subgroups (Lagrange's theorem). Let $c$ be the number by which $|H|$ divides $|N(H)|$, i.e. the number of different conjugate subgroups to $H$ (that is subgroups we get by applying Conjugation action to $H$). This is the number of elements in the orbit of $H$ under this action. Then by the Orbit-stabilizer theorem, $|G| = |N(H)|c$. ?? See here why. But don't see why it is related to orbit-stabilizer... He may have made a mistake..