Normal-monotone subgroup-defining function

Definition with symbols
A subgroup-defining function $$f$$ is said to be normal-monotone if whenever $$N \triangleleft G$$ is a normal subgroup, then $$f(N) \le f(G)$$.

Stronger properties

 * Verbal subgroup-defining function
 * Monotone subgroup-defining function

Weaker properties

 * Characteristic-monotone subgroup-defining function
 * Direct factor-monotone subgroup-defining function

Frattini subgroup
The Frattini subgroup is not a monotone subgroup-defining function, but it is normal-monotone. The crucial fact used here is the modular property of groups.