things to remember with cyclic groups:
1) every subgroup is cyclic.
2) if G is cyclic of order n, and k divides n, G has a UNIQUE subgroup of order k (so it has one, and only one, for each divisor k of n).
in particular, if G = <a>, with |a| = n, and k|n, so that n = kd, then the subgroup of order k is <ad>.
for ABELIAN groups, ad is usually written da. since Z/36Z is generated by 1+36Z = , we need only consider <d> = <[d]> for each divisor d of 36.
the divisors of 36 are:
1,2,3,4,6,9,12,18, and 36.