Upward-closure operator

Symbol-free definition
Let $$p$$ be a subgroup property. Then the upward closure of $$p$$ is defined as the property of being a subgroup such that all subgroups containing it satisfy property $$p$$ in the whole group.

Definition with symbols
Let $$p$$ be a subgroup property. Then, the upward closure of $$p$$ is defined as the following subgroup property $$q$$: A subgroup $$H \le G$$ satisfies property $$q$$ in $$G$$, if for every subgroup $$K$$ with $$ H \le K \le G$$, $$K$$ satisfies $$p$$ in $$G$$.

Properties
Applying the upward closure operator twice is the same as applying it once. In other words, the properties that are fixed under the upward closure operator are precisely the same as the properties that can be obtained as images of the upward closure operator. A property that is fixed under the upward closure operator is termed an upward-closed subgroup property.

If $$p \le q$$ (both are subgroup properties) then the $$UC(p) \le UC(q)$$ where $$UC$$ denotes the upward closure.