# Group having subgroups of all orders dividing the group order

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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## Definition

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 $d$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of prime power order the order is a prime power prime power order implies subgroups of all orders dividing the group order (all the other examples, such as finite abelian groups of other orders) |FULL LIST, MORE INFO
finite abelian group finite and abelian: any two elements commute |FULL LIST, MORE INFO
finite nilpotent group finite and nilpotent; also equivalent to saying that there is a normal subgroup of every order dividing the group order Finite supersolvable group|FULL LIST, MORE INFO
finite supersolvable group finite and supersolvable; it has a normal series where all the quotient groups are cyclic groups finite supersolvable implies subgroups of all orders dividing the group order subgroups of all orders dividing the group order not implies supersolvable |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite solvable group finite and a solvable group Hall's theorem finite solvable not implies subgroups of all orders dividing the group order |FULL LIST, MORE INFO