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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.\nThen the map $K \\mapsto K/N$ is a bijection from the set of [[subgroups|0X1JXT47GA]] $K$ of $G$ with $N \\subseteq K$ to the set of all subgroups of $G/N$. Here $G/N$ is the [[quotient group|0KYX2N69J0]].\nLet $K$ and $L$ be subgroups of $G$ that contain $N$. Then $K \\subseteq L$ if and only if $K/N \\subseteq L/N$. Also, $K$ is normal in $G$ if and only if $K/N$ is normal in $G/N$.\n\n## Remarks\n\nThe inverse map sends a subgroup $\\bar{K}$ of $G/N$ to its preimage $\\{g \\in G : gN \\in \\bar{K}\\}$.\nThe bijection is an isomorphism of [[lattices of subgroups|0B8BW4YWTF]], and it keeps indices: $[L : K] = [L/N : K/N]$ when $N \\subseteq K \\subseteq L$.\nFor example, the subgroups of $\\mathbb{Z}/12\\mathbb{Z}$ correspond to the subgroups $d\\mathbb{Z}$ of $\\mathbb{Z}$ with $d$ dividing $12$.\n\n## Explanations\n\n### L1 human\n\nThe subgroups of a quotient group $G/N$ are the same as the subgroups of $G$ that contain $N$.\nPassing to the quotient does not create new subgroups or lose old ones above $N$.\nSo the picture of the subgroups above $N$ in $G$ is the full picture of the subgroups of $G/N$.\n",
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    "name": "Correspondence theorem",
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        "locator": "Theorem D (groups)",
        "url": "https://en.wikipedia.org/wiki/Isomorphism_theorems"
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  "name": "Correspondence theorem",
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