The most common versions use different formulations:

 - SVC: uses the hinge-loss in the cost function

 - SVR: uses the epsilon-insensitive loss in the cost function, a modification of the hinge allowing errors in both directions, which also adds an additional hyperparameter to be tuned

So, from an algorithmic point of view I'm inclined to say neither is a subset of the other.

Thanks Foued Saâdaoui and Bruno Hebling Vieira.

Support vector machines (SVMs) are a set of supervised learning methods used for classification, regression and outliers detection. For more information see the following link...

https://scikit-learn.org/stable/modules/svm.html