  We investigate the relationship between various isomorphism invariants for
finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to
characterize, compare and quantify the effectiveness and complexity of
invariants for group isomorphism.
  It turns out that a surprising number of invariants and characteristic
subgroups that are classic to group theory can be detected and identified by a
low dimensional Weisfeiler-Leman algorithm. These include the center, the inner
automorphism group, the commutator subgroup and the derived series, the abelian
radical, the solvable radical, the Fitting group and $\pi$-radicals. A low
dimensional WL algorithm additionally determines the isomorphism type of the
socle as well as the factors in the derived series and the upper and lower
central series.
  We also analyze the behavior of the WL algorithm for group extensions and
prove that a low dimensional WL algorithm determines the isomorphism types of
the composition factors of a group.
  Finally we develop a new tool to define a canonical maximal central
decomposition for groups. This allows us to show that the Weisfeiler-Leman
dimension of a group is at most one larger than the dimensions of its direct
indecomposable factors. In other words the Weisfeiler-Leman dimension increases
by at most 1 when taking direct products.
