Almost normal subgroups: Is there any notion which is weaker than normal subgroup?

Consider the set of all conjugates of a subgroup $H\leq G$, defined by $C=\{gHg^{-1}\mid g\in G\}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.

A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 \lhd H_1 \lhd \cdots \lhd H_{n-1} \lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.

Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.

What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.