Pariah groups

Remarks

There are exactly six pariah groups: $J_1$, $J_3$, $J_4$, the O'Nan group $\mathrm{O'N}$, the Rudvalis group $\mathrm{Ru}$ and the Lyons group $\mathrm{Ly}$. Example: the Janko group $J_1$ is a pariah. Non-example: the Conway group $\mathrm{Co}_1$ is not a pariah. The Monster contains an involution whose centraliser $H$ has a normal subgroup $N$ of order $2^{25}$ with $H/N \cong \mathrm{Co}_1$, so $\mathrm{Co}_1$ is a subquotient of $\mathbb{M}$.