Every fingering that fits a hand
x000
open · no 5
5400
easy · no 5
5430
easy
5450
easy · no 5
545x
easy · no 5
What is true of it
C6 is C, E, G and A — 1, 3, 5 and 6 counted from C.
These are also the notes of Am7/C. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is x000: 0 fingers and nothing above the third 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
C6/E
x453
C6/G
0000
C6/A
2000
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.