Abelian subgroup structure of groups of order 2^n

Summary on existence
Note that if there exists an abelian normal subgroup of a particular order, there exist abelian normal subgroups of all smaller orders, because any normal subgroup contains normal subgroups of all orders dividing its order.

Summary on congruence condition
Below are values of $$n$$ and $$k$$, the $$n$$ values in the rows and the $$k$$ values in the columns. A "Yes" indicates that in any group of order $$2^n$$, the number of abelian subgroups of order $$2^k$$ is either 0 or odd. A "No" indicates that there exists a group of order $$2^n$$ where the number of abelian subgroups of order $$2^k$$ is a nonzero even number.