Difference between conjugacy classes and subgroups? I am studying Group theory and Im not sure I understand the difference between a conjugacy class and subgroup?

A conjugacy class is a set of congugated elements, it is not necessarily a subgroup (to start with, the identity $e$ is not part of any conjugacy class, except $\{e\}$). Normal subgroups are a union of conjugacy classes, but the converse is not true. The cardinality of a conjugacy class is counted by the index of a subgroup: $|Cl_G(g)|=|G:C_G(g)|$, where $C_G(g)$ is the centralizer of $g$, the subgroup of all elements of $G$ that commute with $g$. The singletons among the conjugacy classes are hence the elements that are in the center $Z(G)$ of $G$.
