Upper join-closed subgroup property

Definition with symbols
A subgroup property $$p$$ is said to be upper join-closed if given $$H \le G$$ and $$K_i, i \in I$$ are intermediate subgroups of $$G$$ containing $$H$$ (indexed by a nonempty set $$I$$) and $$H$$ satisfies $$p$$ in each $$K_i$$, we have that $$H$$ satisfies $$p$$ in the defining ingredient::join of subgroups $$\langle K_i \rangle_{i \in I}$$.

Stronger metaproperties

 * Lower-intersection upper-join closed subgroup property
 * LU-join closed subgroup property
 * Upward-closed subgroup property
 * Izable subgroup property

Weaker metaproperties

 * Stronger than::Finite-upper join-closed subgroup property
 * Stronger than::Permuting-upper join-closed subgroup property

Related notions
Given a subgroup property $$p$$ that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup $$H$$ of $$G$$ associate a unique largest subgroup $$M$$ containing $$H$$ for which $$H$$ satisfies $$p$$ in $$M$$.

Such a subgroup property is termed an izable subgroup property and the $$M$$ that we get is termed the izing subgroup of $$H$$ for that subgroup property.

Normality
Normality is an upper join-closed subgroup property, viz, if $$H \le G$$ and $$K_1, K_2$$ are intermediate subgroups such that $$H \triangleleft K_1$$ and $$H \triangleleft K_2$$, then $$H \triangleleft $$.

Central factor
The property of being a central factor is also upper join-closed, in fact, it is izable.