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Homomorph-containing subgroup

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===Important classes of examples===
[[Normal Sylow subgroup]]s, [[normal Hall subgroup]]s, as well as subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}} See also the section [[#Stronger properties]] in this page.
===Examples in small finite groups===
| [[Stronger than::homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}}
|}
 
==Facts==
 
* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}}
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