Just so we are clear- there are classes of amplificationandthere are types of amplifiers.
The class of operation has mostly to do with how the output devices in the amp are turned on and how long they are on. In this regard both solid state and tube amps can be class A, class AB, class B or D (yes, you can build a tube amp that is class D). Class C is not used for audio- its typically used for radio frequency transmitters.
A *type* of amplifier has to do with its topology (circuit design) and often what sort of devices it uses in the output section. For example an SET is a type of amp using entirely single-ended circuitry and a triode for a power tube. An OTL is a tube amp that does not use an output transformer. There are single-ended solid state amps (ala First Watt amps from Nelson Pass) and most solid state amps are output transformerless, so the OTL moniker is usually reserved for tube amps. These days all class D amps are solid state, but they can have half-bridge and full bridge configurations.