Consider the cyclic group (Z/15Z,0,+). It has order 15. It is generated by each of
the 8 = 
(15) invertible elements.
The other possible orders of elements of (Z/15Z,0,+) are
1,
3,
5.
There is a unique element of order (1) = 1, namely 1.
(1) = 1, namely 1.
The (3) = 2 elements of order 3 are 5 and 10.
(3) = 2 elements of order 3 are 5 and 10.
The (5) = 4 elements of order 5 are 3, 6, 9, and 12.
(5) = 4 elements of order 5 are 3, 6, 9, and 12.