Subgroup properties are properties of subgroups of a group. These properties are assumed to satisfy only one condition : they must be invariant up to commuting isomorphism. That is, if ${\displaystyle G}$ and ${\displaystyle G'}$ are isomorphic groups, and ${\displaystyle H}$ is a subgroup of ${\displaystyle G}$ whose image under the isomorphism is ${\displaystyle H'}$ then ${\displaystyle H}$ has the property in ${\displaystyle G}$ if and only if ${\displaystyle H'}$ has the property in ${\displaystyle G'}$.