What class is the fish in?

Fish are a complex group of animals that are not all simply lumped into once class. There are four basic groups; Bony Fish, Lobe finned fish, Cartelagenous fish and jawless fish, but they are all separate classes, all they have in common is the fact that they are in the phylum Chordata. This strange system stems from the fact that the Word fish is a common name for swimming animals that have a certain look to them, but doesnt nessesarily represent any real genetic similarities.