How can I know if the Garmin GTX 345 transponder supports modes A and C?
It's on the spec page: Garmin GTX 345:


Physical dimensions (unit, rack, connectors); for panel mounts, depth is measured from face of instrument panel): 1.65” x 6.30” x
10.07” (42 x 160 x 256 mm)
Display type: Digital
Weight (unit, rack, connectors): 3.1/3.2 lbs (1.41/1.45 kg)
Voltage range: 14/28 VDC (18/20 W Max)
Transmit power: 200 W minimum
Temperature: -40° C to +70° C
Operating altitude: To 55,000 ft (16,800 m)
Cooling input: Not required
Environmental compliance (TSO Approved): DO-160G
Software compliance (TSO Approved): DO-178 Levels B, C, D, E
Hardware compliance (TSO Approved): DO-254 Level C
TSO compliance (Approved): TSO-C88b (w/opt. alt. encoder; TSO-C112e (Class 1, Level 2els; TSO-C154c (Class A1S); TSO-C157a (Class 1);
TSO-C166b (Class A1S); TSO-C195a (Class C1,C2,C3,C4)
Mount type: Panel
Transponder type: Mode A/C, S and ES
Squawk code selection: Push-button
Traffic Information Services (TIS) alerts compatible: Yes


I have one of these units, and I can confirm that it supports A/C, S, ES, and ADS-B.