Every fingering that fits a hand
x001
open · no 5
5767
moderate
546x
easy · no 5
5463
hard · no 5
5467
hard · no 5
What is true of it
C7 is C, E, G and B♭ — 1, 3, 5 and b7 counted from C.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is x001: one finger, held at the 1st fret.
1 of them use open strings, which is what makes a shape quiet to change to.
It leaves out the 5. 4 strings cannot carry 4 notes, and the 5 is the one this chord can lose without changing what it is.
1 string stays silent.
Where its notes are on the neck
The brass dots are C, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
C7/E
9a8a
C7/G
0001
C7/B♭
3003
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.