Theory of Computing
-------------------
Title     : The Threshold for Subgroup Profiles to Agree is Logarithmic
Authors   : James B. Wilson
Volume    : 15
Number    : 19
Pages     : 1-25
URL       : https://theoryofcomputing.org/articles/v015a019

Abstract
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For primes $p> 2$ and $e> 3$ 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 deterministic polynomial-time 
isomorphism test.