C Chord Form -1 First, to B
The C chord form, like the G, have options down one fret. For the G, these are F♯/G♭ type chords. For C, these are B types.
Some thought could even be given to F & B-flat, respectively; but this looks a lot like E & A form territory for these chords: the boundary, where C/A forms become one thing – points of possibilities around the root.
Play each of these chords, making sense of where the mods are located. Any of these fully closed (no opens) can be moved to 12 positions, and maintain their quality (chord type).
And, check any of the chords with open strings, position by position. Some of them are sweet sounding harmonies. Example: move the first B7 shape to P6. In this position, the chord is E7 (the open B is the 5th of the chord).