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  "body": "\n## Definition\n\nLet $\\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].\nLet $H$ be a [[subgroup|0X1JXT47GA]] of $\\mathbb{M}$ and let $N$ be a normal subgroup of $H$.\nA group isomorphic to such a [[quotient group|0KYX2N69J0]] $H/N$ is a subquotient of $\\mathbb{M}$.\nA *pariah group* is a [[sporadic group|PZ49Z4CBFZ]] that is not a subquotient of $\\mathbb{M}$.\n\n## Remarks\n\nThere 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 $\\mathrm{Ly}$.\nExample: the [[Janko group|PSEC8D7CX0]] $J_1$ is a pariah.\nNon-example: the [[Conway group|366D4ETGQH]] $\\mathrm{Co}_1$ is not a pariah.\nThe 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 $\\mathbb{M}$.\n\n## Explanations\n\n### L1 human\n\nA pariah is a sporadic simple group that cannot be found inside the Monster, even as a quotient of a subgroup.\n",
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