TITLE: How many non isomorphic groups of order 30 are there?
QUESTION [5 upvotes]: Let $|G|=30$.  I have prove that there is the only subgroup of order $15$, which I'll denote $H$. Now I do know how to classify the group.  After thinking, I made the following steps.
1) Possible order of subgroup $K$ of $G$ of order 2 are 1, 3, 5, 15.
Case 1. if $G$ contain only one element of order 2, then $G \cong Z_{30}$.
Now I cannot solve for the next steps.
Please give me any hints or any other method.

REPLY [5 votes]: There is a general description of groups with cyclic Sylow subgroups:
Marshall Hall, The Theory of Groups -  Theorem 9.4.3.