The chords shown below are already compatible with the C Major scale. These are the diatonic chords.
I ii iii IV V vi viidim
C major = C , Dm , Em , F , G , Am , Bdim
But given all these different chords that work in the key, a smaller subset of highly common progressions exists. For example, some such common progressions are:
Why would one limit themselves to these common chord progressions?