  What are simplest ways to construct a finite group from its atomic
constituents? To understand part-whole relations between finite simple groups
and the global structure of finite groups, we axiomatize complexity measures on
finite groups. From the Jordan-H\"older theorem and Frobenius-Lagrange
embedding in an iterated wreath product, any finite group $G$ can be
constructed from a unique collection of simple groups, its Jordan-H\"older
factors, each with well-defined multiplicities through iterated extension. What
is the least number of levels needed in such a hierarchical construction if a
level is allowed to include several of these atomic pieces? To answer this
question rigorously, we give a natural set of hierarchical complexity axioms
for finite groups, and prove these axioms are satisfied by a unique maximal
complexity function $\mathbf{cx}$. We prove this function is the same as the
minimal number of "spans of gems" (direct products of simple groups) in a
subnormal series with all factors of this type. This hierarchical complexity is
thus effectively computable, and bounded below by all other complexity measures
satisfying the axioms, including generalizations of derived length and Fitting
height.
  For solvable groups, the unique maximal group complexity measure satisfying
the axioms agrees with the restriction of the one for all finite groups, and in
addition satisfies an embedding axiom. In both cases, the complexity of a group
is bounded above and below by various natural functions. In particular,
hierarchical complexity is sharply bounded above by socle length, with a
canonical decomposition. Examples illustrate applications of the bounds and
axiomatic methods in determining complexity of groups. We show also that
minimal decompositions need not be unique in terms of what components occur nor
their ordering. The complexity axioms are also shown to be independent.
