  For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$
that have equal multisets of isomorphism types of proper subgroups and proper
quotient groups, isomorphic character tables, and power maps. This obstructs
recent speculation concerning a path towards efficient isomorphism tests for
general finite groups. These groups have a special purpose polylogarithmic-time
isomorphism test.
