Subgroups Original Question: Let G be an abelian group of order 540. What is the largest possible number of subgroups of order 3 such a group G can have? My attempt: The fundamental theorem of group theory says that any group G of order 540 can be expressed as G1 X G2 where G1 is a group of order 27 and G2 is a group of order 20. If I'm searching for every group of order 3, then I guess I'm actually searching for every element of order 3. (number of subgroups of order 3= number of elements of order 3 divides by 2). Since an element of order 3 can only be in G1, I need to be searching for elements of order 3 in a group of order 27. This will require classifying all groups of order 27, but I don't know how to classify these groups. Help? 
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