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](https://en.wikipedia.org/wiki/Sporadic_group) a diagram describing the weaker property of "which G\_i is a *subquotient* of which G\_j."



Embeddings of the sporadic simple groups
See Table 2, p. 362 in:


* Wilson, Robert A.
Is the Suzuki group Sz(8) a subgroup of the Monster? Bull. Lond. Math. Soc. 48 (2016), no. 2, 355-364. <https://doi.org/10.1112/blms/bdw012>


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:


* Griess, Robert L., Jr. The friendly giant. Invent. Math. 69 (1982), no. 1, 1-102. <https://doi.org/10.1007/BF01389186>
* Wilson, Robert A. Is J1 a subgroup of the Monster? Bull. London Math. Soc. 18 (1986), no. 4, 349-350. <https://doi.org/10.1112/blms/18.4.349>


