Subgroups of cyclic groups
Finite cyclic groups
Finite cyclic groups
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For every finite group *G* of order *n*, the following statements are equivalent:
* *G* is cyclic.
* For every divisor *d* of *n*, *G* has at most one subgroup of order *d*.
If either (and thus both) are true, it follows that there exists exactly one subgroup of order *d*, for any divisor of *n*. 
This statement is known by various names such as **characterization by subgroups**. (See also [cyclic group](/wiki/Cyclic_group "Cyclic group") for some characterization.)

There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the [Klein group](/wiki/Klein_group "Klein group") is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of the characterization.


Can a group exist where every subgroup has at most one subgroup of order d?