An abelian group of a certain order I want to find all the abelian groups of order less than or equal to $40$ up to isomorphism. Here is what I came up with:
$\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{5}$ which is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{10}$.
$\mathbb{Z}_{2} \times \mathbb{Z}_{4} \times \mathbb{Z}_{5}$ which is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{20}$ or $\mathbb{Z}_{10} \times \mathbb{Z}_{4}$.
$\mathbb{Z}_{8} \times \mathbb{Z}_{5}$ which is isomorphic to $\mathbb{Z}_{40}$. 
 A: You found all abelian groups, up to isomorphism, of order equal to $40$. There are MANY more abelian groups with orders less than $40$.
It wouldn't be difficult to find all the abelian groups, up to isomorphism, of order less than or equal to $40$; it would just take a bit of time. 
You can start simply with knowing the trivial group is abelian and cyclic. And of course, there are all the abelian and cyclic groups of prime order less than $40$ for which there is one and only one group per prime $p:\; \mathbb Z_p,\;$ for $p$ equal to each of $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37$.
Then it's simply breaking all the composite numbers less than 40 (since we've covered $40$) into their corresponding prime factors, and applying the Fundamental Theorem of Finitely Generated Abelian Groups, just as you did for abelian groups of order $40$.
A: The $68$ non-isomorphic abelian groups of order at most $40$ can be found quickly using GAP using the code:
S:=Concatenation(List([1..40],n->AllSmallGroups(n,IsAbelian)));
for G in S do
  Print(StructureDescription(G),"\n");
od;

which outputs:
1
C2
C3
C4
C2 x C2
C5
C6
C7
C8
C4 x C2
C2 x C2 x C2
C9
C3 x C3
C10
C11
C12
C6 x C2
C13
C14
C15
C16
C4 x C4
C8 x C2
C4 x C2 x C2
C2 x C2 x C2 x C2
C17
C18
C6 x C3
C19
C20
C10 x C2
C21
C22
C23
C24
C12 x C2
C6 x C2 x C2
C25
C5 x C5
C26
C27
C9 x C3
C3 x C3 x C3
C28
C14 x C2
C29
C30
C31
C32
C8 x C4
C16 x C2
C4 x C4 x C2
C8 x C2 x C2
C4 x C2 x C2 x C2
C2 x C2 x C2 x C2 x C2
C33
C34
C35
C36
C18 x C2
C12 x C3
C6 x C6
C37
C38
C39
C40
C20 x C2
C10 x C2 x C2

