TITLE: Is $J_1$ a subquotient of the monster group?
QUESTION [8 upvotes]: Edit:  I was able to make a 3D diagram of the happy family if anyone is interested!
https://www.youtube.com/watch?v=_4IjnIcECoQ
I'm working on a twitter thread about the monster group, because I saw an interview with John Conway and he was very interested in the monster group.  
Here is a link to the wikipedia article on the monster:
https://en.wikipedia.org/wiki/Monster_group
I'm very interested in the diagram of the sporadic groups.  (Also if anyone would like to provide some basic understanding of the sporadic groups that would be helpful.) I'm trying to make a 3D version of the diagram with wire... 
When I look at the diagram 
I can understand that the sporadic groups that are maximal are the ones that are circled. And $M$ is the monster, with $B$ the baby monster. And it looks like $J_1$ is a subquotient of the maximal group $O'N$.  I'm not very familiar with subquotients, so I am wondering if $J_1$ is a subquotient of $M$?  That is, for my purposes, do I need to include $O'N$ and $J_1$ in the diagram of the monster?

REPLY [15 votes]: No $J_1$ is not involved in (i.e. is not a subquotient of) the Monster.
The six sporadic simple groups listed here as "Pariahs" are precisely those that are not involved in the Monster, namely $J_1$,   O'N,    $J_3$,     Ru,     $J_4$,     Ly.
There is a lot of information about the Monster and its subgroups in Section 5.8 of Robert Wilson's book, "The Finite Simple Groups''.
Note that, apart from a few uncertainties about a few possible groups ${\rm PSL}_2(q)$ for some small values of $q$, the maximal subgroups of the Monster are now all known, so locating which other simple groups are involved in it is moderately straightforward, using the ATLAS of Finite Simple Groups or the weblink above.
In fact Robert Wilson proved in a 1986 paper "Is $J_1$ a subgroup of the Monster" in Bull. London Math. Soc. 18, 349-150 that $J_1$ is not a subgroup, which was much more difficult at the time, because much less was known about the maximal subgroups of the Monster.