Suppose $G$ is a group with an equivalence relation defined on it; let $[g]$ denote the equivalence class of $g \in G$. In reading some group theory texts I have sometimes seen the following property: we have some subgroup $H$ of $G$ such that the following holds:
If $g \in H$, then $[g] \subseteq H$.
In other words, if an element of the group can be found in the subgroup $H$, then it brings along its entire equivalence class into $H$ as well. (And the converse trivially holds because $g \in [g]$.)
Are there any fairly general conditions on the equivalence relation and/or on the subgroup $H$ (or the group $G$) in order for this property to hold? Is there a name for this sort of property? What are some important/common examples of this phenomenon?