Let $G$ be a (possibly infinite) group and $H \unlhd G$. The normality of $H$ in $G$ naturally introduces the $H$-valued map $(g,h) \mapsto g^{-1}hg$, which turns out to be a $G$-action on $H$. Given $h \in H$, the orbit by $h$ is the set $O_h:=\lbrace g^{-1}hg \mid g \in G\rbrace \subseteq H$. The set $O:=\lbrace O_h \mid h \in H\rbrace$ is a partition of $H$.
Can we state anything on the cardinality of $O$?