Subgroup contained in finitely many intermediate subgroups

Symbol-free definition
A subgroup of a group is said to be contained in finitely many intermediate subgroups if the number of subgroups of the whole group containing that subgroup is finite.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be contained in finitely many intermediate subgroups if the number of subgroups $$K$$ of $$G$$ such that $$H \le K \le G$$ is finite.

Stronger properties

 * Weaker than::Subgroup of finite index
 * Weaker than::Subgroup of finite double coset index

Facts

 * If a normal subgroup is contained in only finitely many intermediate subgroups, it has finite index. This follows from the fourth isomorphism theorem and the fact that a group having finitely many subgroups is finite.

Metaproperties
If $$H \le K \le G$$ are groups such that $$K$$ is contained in only finitely many intermediate subgroups of $$G$$ and $$H$$ is contained in only finitely many intermediate subgroups of $$K$$, it may well happen that $$H$$ is contained in infinitely many intermediate subgroups of $$G$$.

If $$H \le K \le G$$ are groups such that $$H$$ is contained in only finitely many intermediate subgroups of $$G$$, then $$H$$ is also contained in only finitely many intermediate subgroups of $$K$$.

If $$H \le K \le G$$ are groups such that $$H$$ is contained in only finitely many intermediate subgroups of $$G$$, then $$K$$ is also contained in only finitely many intermediate subgroups of $$G$$.