,"Quick phrase"
"1-automorphism-invariant subgroup",
AEP-subgroup,
"Abelian subgroup",
"Ascendant subgroup",
"Automorph-conjugate subgroup",
"Base of a wreath product",
"C-closed subgroup",
CEP-subgroup,
"Centrally closed subgroup",
"Characteristic central factor",
"Characteristic direct factor",
"Characteristic subgroup","invariant under all automorphisms,automorphism-invariant,strongly normal,normal under outer automorphisms"
"Characteristic subgroup of group of prime power order",
"Characteristic transitively normal subgroup",
"Characteristically complemented characteristic subgroup",
"Characteristically complemented normal subgroup",
"Characteristically complemented subgroup",
"Cocentral subgroup",
"Commutator-verbal subgroup",
"Completely distinguished subgroup",
"Completely divisibility-closed subgroup",
Component,
"Composition subgroup",
"Conjugacy-closed characteristic subgroup",
"Conjugacy-closed normal subgroup",
"Conjugacy-closed subgroup",
"Conjugate-dense subgroup",
"Conjugate-large subgroup",
"Contranormal subgroup",
"Cyclonormal subgroup",
"Descendant subgroup",
"Direct factor","factor in internal direct product,normal with normal complement,has centralizing complement"
"Divisibility-closed subgroup",
"Finite direct power-closed characteristic subgroup",
"Free factor",
"Fully normalized subgroup",
"Hall direct factor",
"Hall subgroup",
"Hereditarily normal subgroup",
"IA-automorphism-balanced subgroup",
"Image-closed characteristic subgroup",
"Image-closed fully invariant subgroup",
"Improper subgroup",
"Inert subgroup",
"Injective endomorphism-invariant subgroup","invariant under all injective endomorphisms,injective endomorphism-invariant"
"Isomorph-containing subgroup","contains all isomorphic subgroups,weakly closed in any ambient group"
"Isomorph-free subgroup","no other isomorphic subgroups,no isomorphic copies,only subgroup of its isomorphism type"
"Large subgroup",
"Left-transitively 2-subnormal subgroup",
"Left-transitively WNSCDIN-subgroup",