Direct-product-closed subgroup property

Definition with symbols
A subgroup property $$p$$ is said to be direct-product-closed if whenever $$H_1 \le G_1$$ satisfies $$p$$, and $$H_2 \le G_2$$ satisfies $$p$$, then $$H_1 \times H_2$$ satisfies $$p$$ as a subgroup of $$G_1 \times G_2$$.

Stronger metaproperties

 * Intersection-closed subgroup property that also satisfies inverse image condition

Weaker metaproperties

 * Direct-power-closed subgroup property

Related metaproperties

 * Subgroup properties satisfying image condition
 * Subgroup properties satisfying inverse image condition
 * Subgroup properties satisfying transfer condition
 * Subgroup properties satisfying intermediate subgroup condition