Subgroup structure of groups of order 32

Numerical information on counts of subgroups by order
We note the following:

Here is the GAP code to generate this information:

gap> AllAndNormalSubgroupCounts(2,5); [ [ 1, [ [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ] ], [ 2, [ [ 7, 7 ], [ 19, 7 ], [ 19, 7 ], [ 3, 3 ] ] ], [ 3, [ [ 3, 3 ], [ 7, 7 ], [ 7, 7 ], [ 3, 3 ] ] ]  , [ 4, [ [ 3, 3 ], [ 7, 3 ], [ 7, 7 ], [ 3, 3 ] ] ], [ 5, [ [ 7, 3 ], [ 11, 5 ], [ 11, 3 ], [ 3, 3 ] ] ],  [ 6, [ [ 11, 1 ], [ 23, 3 ], [ 11, 3 ], [ 3, 3 ] ] ], [ 7, [ [ 11, 1 ], [ 15, 3 ], [ 11, 3 ], [ 3, 3 ] ] ],  [ 8, [ [ 3, 1 ], [ 7, 3 ], [ 11, 3 ], [ 3, 3 ] ] ], [ 9, [ [ 11, 3 ], [ 19, 3 ], [ 11, 3 ], [ 3, 3 ] ] ],  [ 10, [ [ 3, 3 ], [ 11, 3 ], [ 11, 3 ], [ 3, 3 ] ] ], [ 11, [ [ 7, 1 ], [ 11, 3 ], [ 11, 3 ], [ 3, 3 ] ] ],  [ 12, [ [ 3, 3 ], [ 7, 5 ], [ 7, 3 ], [ 3, 3 ] ] ], [ 13, [ [ 3, 3 ], [ 11, 3 ], [ 7, 3 ], [ 3, 3 ] ] ],  [ 14, [ [ 3, 3 ], [ 11, 3 ], [ 7, 3 ], [ 3, 3 ] ] ], [ 15, [ [ 3, 1 ], [ 3, 3 ], [ 7, 3 ], [ 3, 3 ] ] ],  [ 16, [ [ 3, 3 ], [ 3, 3 ], [ 3, 3 ], [ 3, 3 ] ] ], [ 17, [ [ 3, 1 ], [ 3, 3 ], [ 3, 3 ], [ 3, 3 ] ] ],  [ 18, [ [ 17, 1 ], [ 9, 1 ], [ 5, 1 ], [ 3, 3 ] ] ], [ 19, [ [ 9, 1 ], [ 9, 1 ], [ 5, 1 ], [ 3, 3 ] ] ],  [ 20, [ [ 1, 1 ], [ 9, 1 ], [ 5, 1 ], [ 3, 3 ] ] ], [ 21, [ [ 7, 7 ], [ 19, 19 ], [ 19, 19 ], [ 7, 7 ] ] ],  [ 22, [ [ 15, 7 ], [ 43, 11 ], [ 27, 11 ], [ 7, 7 ] ] ], [ 23, [ [ 7, 7 ], [ 19, 11 ], [ 19, 11 ], [ 7, 7 ] ] ],  [ 24, [ [ 7, 3 ], [ 19, 7 ], [ 11, 11 ], [ 7, 7 ] ] ], [ 25, [ [ 11, 3 ], [ 23, 9 ], [ 19, 11 ], [ 7, 7 ] ] ],  [ 26, [ [ 3, 3 ], [ 15, 9 ], [ 11, 11 ], [ 7, 7 ] ] ], [ 27, [ [ 19, 3 ], [ 47, 7 ], [ 31, 7 ], [ 7, 7 ] ] ],  [ 28, [ [ 15, 3 ], [ 27, 5 ], [ 23, 7 ], [ 7, 7 ] ] ], [ 29, [ [ 7, 3 ], [ 19, 5 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 30, [ [ 11, 3 ], [ 23, 3 ], [ 15, 7 ], [ 7, 7 ] ] ], [ 31, [ [ 11, 3 ], [ 23, 3 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 32, [ [ 3, 3 ], [ 15, 3 ], [ 7, 7 ], [ 7, 7 ] ] ], [ 33, [ [ 7, 3 ], [ 19, 1 ], [ 7, 7 ], [ 7, 7 ] ] ],  [ 34, [ [ 19, 3 ], [ 31, 7 ], [ 31, 7 ], [ 7, 7 ] ] ], [ 35, [ [ 3, 3 ], [ 15, 7 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 36, [ [ 7, 7 ], [ 11, 11 ], [ 11, 11 ], [ 7, 7 ] ] ], [ 37, [ [ 7, 3 ], [ 11, 7 ], [ 11, 11 ], [ 7, 7 ] ] ],  [ 38, [ [ 7, 1 ], [ 7, 7 ], [ 11, 11 ], [ 7, 7 ] ] ], [ 39, [ [ 19, 3 ], [ 27, 3 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 40, [ [ 11, 3 ], [ 19, 3 ], [ 15, 7 ], [ 7, 7 ] ] ], [ 41, [ [ 3, 3 ], [ 11, 3 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 42, [ [ 11, 1 ], [ 11, 3 ], [ 15, 7 ], [ 7, 7 ] ] ], [ 43, [ [ 15, 1 ], [ 19, 3 ], [ 15, 7 ], [ 7, 7 ] ] ],  [ 44, [ [ 7, 1 ], [ 11, 3 ], [ 15, 7 ], [ 7, 7 ] ] ], [ 45, [ [ 15, 15 ], [ 43, 43 ], [ 43, 43 ], [ 15, 15 ] ] ],  [ 46, [ [ 23, 7 ], [ 67, 19 ], [ 51, 35 ], [ 15, 15 ] ] ], [ 47, [ [ 7, 7 ], [ 19, 19 ], [ 35, 35 ], [ 15, 15 ] ] ],  [ 48, [ [ 15, 3 ], [ 27, 15 ], [ 35, 35 ], [ 15, 15 ] ] ], [ 49, [ [ 19, 1 ], [ 39, 15 ], [ 35, 35 ], [ 15, 15 ] ] ],  [ 50, [ [ 11, 1 ], [ 15, 15 ], [ 35, 35 ], [ 15, 15 ] ] ], [ 51, [ [ 31, 31 ], [ 155, 155 ], [ 155, 155 ], [ 31, 31 ] ] ] ]

Counts of abelian subgroups and abelian normal subgroups
Two additional points:


 * For the abelian groups: note that abelian implies every subgroup is normal and also that subgroup lattice and quotient lattice of finite abelian group are isomorphic. Thus, when the whole group is abelian, we have: number of abelian subgroups of order 2 = number of abelian normal subgroups of order 2 = number of abelian subgroups of order 16 = number of abelian normal subgroups of order 16. Separately, we have number of abelian subgroups of order 4 = number of abelian normal subgroups of order 4 = number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8.
 * The "number of abelian normal subgroups" columns depend only on the Hall-Senior genus, i.e., two groups with the same Hall-Senior genus have the same "number of abelian normal subgroups" of each order. The Hall-Senior genus is the part of the Hall-Senior symbol excluding the very final subscript, so for instance $$32\Gamma_2a_1$$ and $$32\Gamma_2a_2$$ both belong to the Hall-Senior genus $$32\Gamma_2a$$ and hence have the same number of abelian normal subgroups of each order.

Here is the GAP code to generate this information:

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