Abelian subgroup structure of groups of order 32

Abelian normal subgroups of order 16
Note that index two implies normal, so the abelian subgroups of order 16 are precisely the same as the abelian normal subgroups of order 16.

For an abelian group of order 32 and rank $$r$$ (i.e., $$r$$ is the minimum size of generating set), the number of abelian normal subgroups of order 16 is $$2^r - 1$$, which could be 1, 3, 7, 15, or 31.
 * Existence is not guaranteed!: It is possible for there to be no abelian normal subgroup of order 16.
 * Count: The congruence condition on number of abelian subgroups of prime index tells us that the number of abelian normal subgroups of order 16, if nonzero, is odd. However, we can be more specific. In a non-abelian group of order 32, the number of abelian normal subgroups is 0, 1, or 3. The "3" case arises if and only if the inner automorphism group is a Klein four-group, which occurs only for the Hall-Senior family (isoclinism class) $$\Gamma_2$$.

We now construct a table derived from the above, that lists the total number of abelian normal subgroups of order eight and exponent bounded by some specific number (2, 4, 8, or 16). Note that the exponent dividing 2 count may be a nonzero even number (specifically, it can be 2), but the exponent dividing 4, exponent dividing 8, and exponent dividing 16 counts are all either zero or odd. As noted earlier, for non-abelian groups, these counts are either 0, 1, or 3.

Abelian normal subgroups of order 8

 * Existence: There exist abelian normal subgroups of order 8 by the existence of abelian normal subgroups of small prime power order. In fact, any subgroup that is maximal among abelian normal subgroups must have orer at least 8.
 * Count: The congruence condition on number of abelian subgroups of prime-cube order tells us that the number of abelian normal subgroups of order 8 is odd.

Abelian characteristic subgroups
We make cases based on the nilpotency class, isoclinism class (i.e., Hall-Senior family) and other aspects of the group structure:


 * For an abelian group of order 32, the whole group is the unique subgroup of itself that is maximal among abelian characteristic subgroups.