Finite groups are classified by the number of conjugacy classes of nontrivial non-self-normalizing subgroups for small counts, with bounds on nilpotency class and derived length in general.
Herzog, On Finite Simple Groups of Order Divisible by Three Primes On ly, J
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On non self-normalizing subgroups
Finite groups are classified by the number of conjugacy classes of nontrivial non-self-normalizing subgroups for small counts, with bounds on nilpotency class and derived length in general.