Maximal subgroup

<h2 id="existence-of-maximal-subgroup">Existence of maximal subgroup</h2>
<p>Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finite <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> under inclusion.  There are, however, infinite <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> that contain no maximal subgroups, for example the <a href="/facts/Pr%C3%BCfer_group/QtzBWXFv">Prüfer group</a>.
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<h2 id="maximal-normal-subgroup">Maximal normal subgroup</h2>
<p>Similarly, a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> <i>N</i> of <i>G</i> is said to be a maximal normal subgroup (or maximal proper normal subgroup) of <i>G</i> if <i>N</i> < <i>G</i> and there is no normal subgroup <i>K</i> of <i>G</i> such that <i>N</i> < <i>K</i> < <i>G</i>.  We have the following theorem:
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Theorem:   A normal subgroup <i>N</i> of a group <i>G</i> is a maximal normal subgroup if and only if the <a href="/facts/Quotient_group/HlwpHtjb">quotient</a> <i>G</i>/<i>N</i> is <a href="/facts/Simple_group/e5hGF7nT">simple</a>.
<h2 id="hasse-diagrams">Hasse diagrams</h2>
<p>These <a href="/facts/Hasse_diagram/EqeJbUH0">Hasse diagrams</a> show the <a href="/facts/Lattice_of_subgroups/hdIChdWA">lattices of subgroups</a> of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> S4, the <a href="/facts/Dihedral_group/1PUlRh4M">dihedral group</a> D4, and C23, the third <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct power</a> of the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> C2.
The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram.
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Maximal subgroup open-in-new

Maximal subgroup