,"Difficulty level","Fact about"
"Characteristic implies normal",1,"Characteristic subgroup; 1,Normal subgroup; 1"
"Characteristic not implies fully invariant",2,"Characteristic subgroup; 1,Fully invariant subgroup; 1"
"Characteristicity does not satisfy image condition",2,"Characteristic subgroup; 1,Image condition; 2"
"Characteristicity does not satisfy intermediate subgroup condition",2,"Characteristic subgroup; 1,Intermediate subgroup condition; 2"
"Characteristicity does not satisfy lower central series condition",2,"Characteristic subgroup; 1,Lower central series condition; 2"
"Characteristicity is centralizer-closed",1,"Characteristic subgroup; 1,Centralizer-closed subgroup property; 2"
"Characteristicity is commutator-closed",1,"Characteristic subgroup; 1,Commutator-closed subgroup property; 2"
"Characteristicity is not finite direct power-closed",3,"Characteristic subgroup; 1,Finite direct power-closed subgroup property; 2"
"Characteristicity is not finite-relative-intersection-closed",2,"Characteristic subgroup; 1,Finite-relative-intersection-closed subgroup property; 2"
"Characteristicity is not upper join-closed",3,"Characteristic subgroup; 1,Upper join-closed subgroup property; 2"
"Characteristicity is quotient-transitive",1,"Characteristic subgroup; 1,Quotient-transitive subgroup; 2"
"Characteristicity is strongly intersection-closed",1,"Characteristic subgroup; 1,Strongly intersection-closed subgroup property; 2"
"Characteristicity is strongly join-closed",1,"Characteristic subgroup; 1,Strongly join-closed subgroup property; 2"
"Characteristicity is transitive",1,"Characteristic subgroup; 1,Transitive subgroup property; 2"
"Characteristicity satisfies partition difference condition",1,"Characteristic subgroup; 1,Partition difference condition; 2"
"Every group is characteristic in itself",,"Characteristic subgroup; 1,Identity-true subgroup property; 2"
"No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined",,"Subnormal subgroup; 1,Conditionally lattice-determined subgroup property; 2,Normal subgroup; 1,Characteristic subgroup; 1,Fully invariant subgroup; 1,Normal Hall subgroup; 1,Normal Sylow subgroup; 1,Complemented normal subgroup; 1,Sylow retract; 1,Hall retract; 1,Retract; 1"
"Normal not implies characteristic",2,"Normal subgroup; 1,Characteristic subgroup; 1"
"Trivial subgroup is characteristic",,"Characteristic subgroup; 1,Trivially true subgroup property; 2"