Here's a computational approach. In fact, there are $13$ isomorphism classes of groups of order $60$. These can be accessed by the GAP AllSmallGroups
function.
Here's the input code:
for G in AllSmallGroups(60) do Conj:=ConjugacyClassesSubgroups(LatticeSubgroups(G)); SubgroupSizes:=Set(Conj,C->Size(C[1])); Print(StructureDescription(G)," ",IsAbelian(G)," ",SubgroupSizes,"\n"); od;
And here is the output:
C5 x (C3 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C3 x (C5 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C15 : C4 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C60 true [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] A5 false [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ] C3 x (C5 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C15 : C4 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] S3 x D10 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C5 x A4 false [ 1, 2, 3, 4, 5, 10, 12, 15, 20, 60 ] C6 x D10 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C10 x S3 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] D60 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ] C30 x C2 true [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
Each row identifies one of the $13$ groups of order $60$, writes whether or not it's abelian, and the list of subgroup sizes of that group.
The notation as to which group is which should be largely self-explanatory, except that ":" represents a semi-direct product.
- "$G$ is abelian" is true for $2$ out of $13$ isomorphism classes of groups of order $60$.
- "$G$ has a subgroup of order $30$" is true for $11$ out of $13$. (The two counter-examples are $A_5$ and $C_5 \times A_4$.)
- "$G$ has a subgroup of order $2$, $3$, and $5$" is true for $13$ out of $13$.
- "$G$ has a subgroup of order $6$, $10$, and $15$" is true for $11$ out of $13$. (The two counter-examples are $A_5$, which has no subgroup of order $15$, and $C_5 \times A_4$, which has no subgroup of order $6$.)
Disclaimer: I don't mean to undermine the importance of understanding the theoretical methods for answering these questions.