Group having subgroups of all orders dividing the group order

Symbol-free definition
A group having subgroups of all orders dividing the group order is a group with the property that for any natural number dividing the order of the group, there is a subgroup whose order is that natural number.

Definition with symbols
Suppose $$G$$ is a finite group of order $$n$$. We say that $$G$$ is a group having subgroups of all orders dividing the group order if, for any positive divisor $$d$$ of $$n$$, there exists a subgroup $$H$$ of $$G$$ of order $$d$$.

Stronger properties

 * Weaker than::Group of prime power order:
 * Weaker than::Finite nilpotent group
 * Weaker than::Finite supersolvable group:

Weaker properties

 * Stronger than::Finite solvable group: This follows from Hall's theorem.

Incomparable properties

 * Group having a Sylow tower:

Facts

 * Every finite solvable group can be embedded inside a finite group having subgroups of all orders dividing the group order.