Summary: This publication studies the concept of subsystems and introduces two important classes of systems. One class contains systems with the property of complete subsystems and the other contains systems with the property of completely trivial subsystems. It discusses the following questions in great detail and uses the concepts of linearly ordered sets, partially ordered sets, ordered groups and ordered algebras: existence, heredity, infinity, cardinality, proper classes, proper subclasses, categories, isomorphic categories, etc.