1-completed subgroup

Symbol-free definition
A subgroup of a group is said to be 1-completed if there is a single element outside the subgroup such that that element, along with the subgroup, generates the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be 1-completed if there is an element $$x$$ in $$G$$ such that the subgroup generated by $$H$$ and $$x$$ is the whole of $$G$$.

Stronger properties

 * Maximal subgroup
 * Cyclic-quotient subgroup

Weaker properties

 * Finitely completed subgroup
 * Abelian-completed subgroup