Huh? I'm missing a logic here

The given information goes -> An exclusive New York City club is trying very hard to pick its members from the social elite. An exact selection mechanism has been agreed upon that will ensure the entrance of only the most suave members from the group A, B, C, D, E, F, G and H. Being "in" or "out" of the club is determined by the following rules:

If A is in, then G is out.
If H is out, then B is in.
If D is out, then E is out.
If H is in, then C is in.
If B is out, then G and D are out.

Question: If B is out, then who must be in?
Possible answers:
1) A 2) C 3) D 4) E 5) F

The answer according to the answer key is 2) C because if H is in than C is in.

I see the rule that if H is out then B is in. I see one that says that if B is out then G & D are out (and by extension of another rule so is E). I see squat that says that if G is out then A must be in (so they could both be out). I see nothing that says that if B is out then H is in (thus no reason for C to be in by entention of H being in). I don't see anything that says that the opposite of the rules applies. Since the question asks about a "must" it can't be F for which there is no rule. What did I miss?
If 2) C is the answer than why couldn't it also be 1) A?