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  "body": "\n## Summary\n\nThe [[subgroups|0X1JXT47GA]] of a [[group|G5S1M9]] $G$, ordered by inclusion, form a complete lattice.\nThe meet of a family of subgroups is their intersection, and the join is the [[subgroup generated by|0RP7VWQ0YN]] their union.\nThe least element is the trivial subgroup $\\{e\\}$ and the greatest element is $G$.\n\n## Remarks\n\nFor two subgroups $H$ and $K$, the meet is $H \\cap K$ and the join is $\\langle H \\cup K \\rangle$.\nThe lattice of subgroups of $(\\mathbb{Z}, +)$ is the set of the subgroups $n\\mathbb{Z}$, and $m\\mathbb{Z} \\subseteq n\\mathbb{Z}$ exactly when $n$ divides $m$.\nThe lattice of normal subgroups of a group is a modular lattice.\n\n## Explanations\n\n### L1 human\n\nYou can draw the subgroups of a small group as a diagram, with a line from each subgroup up to each next bigger subgroup.\nThis diagram shows the lattice of subgroups.\nTwo subgroups always have a largest common subgroup below them and a smallest subgroup above them.\n",
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