So sandwich is the parent category, and hot dogs are a type of sandwich? Are burgers, too?

Oh no I accidentally started researching, there is an actual British Sandwich Association that defines sandwich as "any form of bread with a filling, generally assembled cold". The USDA, however, has different definitions for open and closed sandwiches and it depends on the percentage amounts of bread and meat... I guess if you put cheese on your bread it's not a sandwich at all!