In which of the following pairs of sets is $H$ a subgroup of $G$? Select ALL that apply.

A

$H$ is the set of odd integers together with $0$, and $G$ is the group of all integers under addition.

B

$H$ is the set of all integer powers of $2$, and $G$ is the group of all nonzero rational numbers under multiplication.

C

$H$ is the set of all $2\times 2$ matrices with integer coefficients, and $G$ is the group of all $2\times 2$ matrices with real coefficients under addition of matrices.

D

$H$ is the set all composite natural numbers (i.e., positive non-prime integers), and $G$ is the group of all nonzero rational numbers under multiplication.