Characteristic not implies powering-invariant in nilpotent group
Normal not implies left-transitively fixed-depth subnormal
Normal not implies right-transitively fixed-depth subnormal
Normal not implies strongly potentially characteristic
Normal not implies normal-potentially characteristic
Subnormal not implies conjugate-permutable
Order-conjugate and Hall not implies order-dominating
Left-transitively WNSCDIN not implies normal
Left-transitively homomorph-containing not implies subhomomorph-containing
Strictly characteristic not implies fully invariant
Locally inner automorphism-balanced not implies central factor
Conjugacy-closed normal not implies central factor
Potentially characteristic not implies characteristic-potentially characteristic
Potentially characteristic not implies normal-potentially characteristic
Potentially characteristic not implies normal-extensible automorphism-invariant
No common composition factor with quotient group not implies complemented
Automorph-permutable not implies permutable
Conjugacy-closed and Hall not implies retract
No nontrivial homomorphism to quotient group not implies complemented normal
Subisomorph-containing not implies homomorph-containing
Cocentral not implies amalgam-characteristic
Characteristic not implies strictly characteristic
Order-conjugate and Hall not implies order-dominated
Subgroup of abelian group not implies abelian-extensible automorphism-invariant
Quotient group need not be isomorphic to any subgroup
Normal not implies amalgam-characteristic
Characteristic not implies potentially fully invariant
Normal not implies potentially fully invariant
Characteristic not implies amalgam-characteristic
Characteristic not implies fully invariant
Iterated agemo subgroup not implies agemo subgroup
Image-closed characteristic not implies fully invariant
Abnormal normalizer not implies pronormal
Fusion system-relatively weakly closed not implies isomorph-normal
2-subnormal not implies automorph-permutable
Finite direct power-closed characteristic not implies fully invariant
Right-transitively homomorph-containing not implies subhomomorph-containing
Complemented central factor not implies direct factor
Finite not implies divisibility-closed in abelian group
Characteristic not implies quasiautomorphism-invariant
Homomorph-containing not implies no nontrivial homomorphism to quotient group
Transitively normal not implies central factor
Local powering-invariant not implies local divisibility-closed
Normal not implies normal-potentially relatively characteristic
Characteristic not implies injective endomorphism-invariant
Normal-potentially characteristic not implies characteristic
Sylow not implies local divisibility-closed
Characteristic-isomorph-free not implies normal-isomorph-free in finite
Endomorphism kernel not implies complemented normal
Fully invariant subgroup of abelian group not implies divisibility-closed
Powering-invariance does not satisfy lower central series condition in nilpotent group
Subgroup of abelian group not implies abelian-potentially characteristic
Normal not implies image-potentially fully invariant
Characteristic subgroup of abelian group not implies local powering-invariant
Powering-invariant not implies local powering-invariant
Lattice-complemented not implies permutably complemented
Verbal subgroup of abelian group not implies local powering-invariant
Order-conjugate not implies order-dominating
Central factor not implies direct factor
Normal-isomorph-free not implies isomorph-free in finite
Image-closed fully invariant not implies verbal
3-subnormal not implies finite-automorph-join-closed subnormal
Characteristic subgroup of abelian group not implies divisibility-closed
Direct factor not implies characteristic
Contranormal not implies self-normalizing
Complemented characteristic not implies left-transitively complemented normal
Divisibility-closed not implies local divisibility-closed
Existentially bound-word not implies verbal
Complemented normal not implies direct factor
P-automorphism-invariant not implies characteristic
Characteristic direct factor not implies fully invariant
Powering-invariant not implies divisibility-closed
Endomorph-dominating not implies automorph-conjugate
Finite index not implies local powering-invariant
1-closed transversal not implies permutably complemented
Order-conjugate not implies order-dominated
Characteristic not implies characteristic-isomorph-free in finite
Subhomomorph-containing not implies variety-containing
Subgroup of abelian group not implies powering-invariant
Characteristic not implies elementarily characteristic
Powering-invariant and normal not implies quotient-powering-invariant
Base of a wreath product not implies elliptic
Conjugacy-closed not implies weak subset-conjugacy-closed
Self-normalizing not implies contranormal
No nontrivial homomorphism from quotient group not implies characteristic
Left-transitively WNSCDIN not implies characteristic
Series-equivalent characteristic subgroups may be distinct
Pure subgroup of torsion-free abelian group not implies direct factor
Direct factor not implies amalgam-characteristic
Abelian-quotient not implies kernel of a bihomomorphism
Characteristic not implies direct factor
Quasiautomorphism-invariant not implies 1-automorphism-invariant
Transitively normal not implies conjugacy-closed normal
Intermediately characteristic not implies isomorph-containing in group of prime power order
Complemented normal not implies local powering-invariant
4-subnormal not implies finite-conjugate-join-closed subnormal
Pronormal not implies join with any distinct conjugate is the whole group