Contra operator

Symbol-free definition
Let $$p$$ be a strongly intersection-closed subgroup property. In other words, an arbitrary intersection of subgroups with property $$p$$ also has property $$p$$, and further, any group has property $$p$$ as a subgroup of itself.

Then the contra-property to $$p$$, viz the property obtained by applying the contra operator to $$p$$ is defined as the property of being a subgroup such that there is no proper subgroup containing it that satisfies $$p$$.

Definition with symbols
Let $$p$$ be a strongly intersection-closed subgroup property. In other words, an arbitrary intersection of subgroups with property $$p$$ also has property $$p$$, and further, any group has property $$p$$ as a subgroup of itself.

Then the contra-property to $$p$$, viz the property obtained by applying the contra operator to $$p$$ is defined as the following property $$q$$: a subgroup $$H$$ satisfies $$q$$ in $$G$$ if there is no proper subgroup $$K$$ of $$G$$ containing $$H$$, for which $$K$$ satisfies $$p$$ in $$G$$.

Facts
If $$p$$ is a trim subgroup property then any $$p$$-simple group has the property that every nontrivial subgroup satisfies contra-$$p$$.

In particular, if $$p$$ is a simple-complete subgroup property, then every nontrivial subgroup is potentially contra-$$p$$.