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This is essentially a reference request. According to the classification of the finite simple groups, there are 26 (or arguably 27) sporadic groups. Let us denote these by G_1 , ... , G_{26} (resp. G_{27}).

Question: Which G_i can be embedded into which G_j?

I was unable to find the answer to this question in the literature, but I suspect that the experts will know where to look. Note that it is not too difficult to find a diagram describing the weaker property of "which G_i is a subquotient of which G_j."

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  • $\begingroup$ this (subquotient) property is weaker but its negation is stronger, and this is information provided by the diagram... $\endgroup$
    – YCor
    Commented Jan 2, 2022 at 10:33
  • $\begingroup$ @YCor : thanks for spotting that. I should have copy-pasted the link en.wikipedia.org/wiki/Sporadic_group instead. $\endgroup$
    – user203598
    Commented Jan 2, 2022 at 16:18
  • $\begingroup$ @YCor : done, thanks. $\endgroup$
    – user203598
    Commented Jan 2, 2022 at 16:28

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See Table 2, p. 362 in:

Which lists all simple groups that are contained in a sporadic group.

It seems the question you ask about (which sporadics can be embedded into another sporadic) was settled in these papers:

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    $\begingroup$ spin : Thank you! That is a wonderfully precise (and quick!) answer to my question =). $\endgroup$
    – user203598
    Commented Jan 2, 2022 at 16:22

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