C7 chord inversions
C7/E, C7/G and C7/Bb are the first, second and third inversions of the C7. This means that the bass tone is shifting from C to E, G or B flat.
C7/E
C7/G
C7/Bb
Theory and information
All these chords consist of the notes C, E, G, Bb. The difference is that the order of notes shifts. E is the bass note in the first inversion, G is the bass note in the second inversion and Bb is the bass note in the third inversion.
Comments
C7/E (032310) is a C dominant 7th with E in the bass and an omitted 5th. In this chord, the first string can be omitted. Concerning the pictured C7/G, the index finger should touch the 5th string slightly. It's also possible to place the thumb on the sixth string and use 332310 instead.
Alternative fingerings
C7/E can also be played as 012010, similar with C7/Bb and with only one high C note. In a similar fashion, C7/G can alternatively be played as 312010. A fingerpicking option for C7/Bb is X1X010.
Movable shapes
Versions with movable barre shapes:
C7/G: 335353
C7/Bb: X12313
Alternative chord names
C7/Bb is identical with C7/A# and C/Bb.
Try in a chord progression
Dm7/G - C7/E - F