Subgroup contained in finitely many intermediate subgroups

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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Definition

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 $H\leq K\leq G$ is finite.

Relation with other properties

Stronger properties

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

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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If $H\leq K\leq G$ 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 $G$ .

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If $H\leq K\leq G$ 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 $K$ .

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
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If $H\leq K\leq G$ 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 $G$ .