In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of a direct product and reveals beautiful connections among several elementary topics from group theory. We decompose the proof of Goursat's Theorem and its corollaries into a sequence of exercises which, when considered as a problem set for undergraduates, may be suitable for inclusion either in a mathematics capstone course, or even in the group theory section of a modern algebra final exam. On the other hand, the exercises may be strategically assigned to parallel the natural flow of the group theory presentation in abstract algebra so that by the end of the semester the theorem can be stated and proved without much difficulty.