Max-sensitive subgroup

Symbol-free definition
A subgroup of a group is termed max-sensitive if its intersection with any maximal subgroup of the whole group is either equal to it or a maximal subgroup of it.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed max-sensitive in $$G$$ if, for any maximal subgroup $$M$$ of $$G$$, the group $$H$$ &cap; $$M$$ is either the whole of $$H$$ or a maximal subgroup of $$H$$.

In terms of the subgroup intersection restriction formalism
The property of being max-sensitive is the balanced subgroup property in the subgroup intersection restriction formalism corresponding to the subgroup property of being a maximal subgroup.

Metaproperties
The subgroup property of being max-sensitive is transitive, on account of being a balanced subgroup property with respect to a restriction formalism.

The subgroup property of being max-sensitive is identity-true, that is, every group is max-sensitive as a subgroup of itself.

It is also trivially true, that is, the trivial subgroup is always a max-sensitive subgroup.