It is reflexive since G is a subgraph of G for any graph G.
It is not irreflexive since it is reflexive.
It is not symmetric, since if H is some proper subgraph of G, then G will not be a subgraph of H.
It is not antisymmetric since if A is a subgraph of B, it is still possible for B to be a subgraph of A in the case that A is the same graph as B.
It is indeed transitive. If C is a subgraph of B and B is a subgraph of A, then it must be the case that C is a subgraph of A.