Any finite group whose comaximal subgroup graph has independence number at most 51 is solvable, while thresholds 14 and 6 similarly force supersolvability and nilpotency, with a few listed exceptions.
Cameron, Graphs defined on groups, International Journal of Group Theory, Vol
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On Independence Number of Comaximal Subgroup Graph
Any finite group whose comaximal subgroup graph has independence number at most 51 is solvable, while thresholds 14 and 6 similarly force supersolvability and nilpotency, with a few listed exceptions.