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I have heard that the following is a conjecture due to Thompson: The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group.

My question is: did Thompson really conjecture this? If so, is there any literature on the subject?

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I've seen this conjecture attributed to Wall (1961), for example: On a conjecture of G.E. Wall. This is a recent article (journal version appeared in 2007), and it gives a bunch of references. The conjecture remains open. Here is a very recent article which does not attack the conjecture itself, but uses it as an inspiration for a different conjecture.

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    @Stefanos Yes, and this holds in greater generality: if $H$ is a normal subgroup of $G$, and $G_1,G_2$ are subgroups containing $H$, then $G_1/H=G_2/H$ if and only if $G_1=G_2$. Indeed, if there exists $a\in G_1\setminus G_2$, then the coset $aH$ is contained in $G_1/H$ but not in $G_2/H$.2012-06-21