  We introduce a new graph invariant of finite groups that provides a complete
characterization of the splitting types of unramified prime ideals in normal
number field extensions entirely in terms of the Galois group. In particular,
each connected component corresponds to a division (Abteilung) of the group. We
compute the divisions of the alternating group, and compile a list of
characteristics of groups that the invariant reveals. We conjecture that the
invariant distinguishes finite groups. Our exposition borrows elements from
graph theory, group theory, and algebraic number theory.
