iPhone 5c won't update beyond v. 10

These are the iOS options based on iPhone Model

• iOS 13.5.1 —> 6S / 6S Plus, SE, 7 / 7 Plus, 8 / 8 Plus, X, XR, XS /XS Max, 11,11 Pro /11 Pro Max
• iOS 12.4.7 —> Final release supported on iPhone 5S, iPhone 6/6 Plus, iPad Air, iPad Mini 2, iPad Mini 3 and iPod Touch (6th generation)
• iOS 10.3.4 —> Final release supported on iPhone 5 and iPad (4th generation) Wi-Fi + Cellular
• iOS 10.3.3 —> Final release supported on iPhone 5C and iPad (4th generation) Wi-Fi
• iOS 9.3.6 —> Final release supported on iPhone 4S, iPad 2 Wi-Fi + Cellular, iPad (3rd generation) Wi-Fi + Cellular, and iPad Mini (1st generation) Wi-Fi + Cellular
• iOS 9.3.5 —> Final release supported on iPod Touch (5th generation), iPad 2 Wi-Fi, iPad (3rd generation) Wi-Fi, and iPad Mini (1st generation) Wi-Fi
• iOS 7.1.2 —> Final release supported on iPhone 4.
• iOS 6.1.5 —> Final release supported on iPhone 3GS, and iPod Touch (4th generation)

You may not have profile unless you have installed you based on certain requirement of some apps.

Is this your corporate iPhone? Do you have any profile installed by your corporate / organisation / enterprise / School Management then please contact the system administrator of your corporate / organisation / enterprise / School Management

If you have installed any profile on your own --> Install or remove configuration profiles on iPhone - Apple ...

The iPhone 5c is too old and now outdated. It's hardware isn't powerful enough to upgrade past 10.3.4.