update to 10.3.4 on iphone 5

If you have 10.3.3 installed and you want to update to 10.3.4 and you can't, try to update with iTunes.

Get help with iOS updates

https://support.apple.com/en-us/HT209571

Get help with over-the-air iOS updates

https://support.apple.com/en-us/HT201435

Get help with iOS update and restore errors

https://support.apple.com/en-us/HT204770

See the chart below to determine whether you can upgrade your device and to what you can upgrade. If you do not have a Software Update option present on your iDevice, then you are trying to upgrade to iOS 5 or higher. You will have to connect your device to your computer and open iTunes to update.

iPhone, iPod Touch, and iPad iOS Compatibility Chart

     Device                                       iOS Version

iPhone 1                                       iOS 3.1.3

iPhone 3G                                    iOS 4.2.1

iPhone 3GS                                  iOS 6.1.x

iPhone 4                                       iOS 7.1.x

iPhone 4S                                     iOS 9.x.x

iPhone 5                                       iOS 10.x.x

iPhone 5c                                     iOS 10.x.x

iPhone 5s                                     iOS 12.x.x

iPhone 5se                                   iOS 12.x.x

iPhone 6                                       iOS 12.x.x

iPhone 6 Plus                              iOS 12.x.x

iPhone 7                                      iOS 12.x.x

iPhone 7 Plus                              iOS 12.x.x

iPhone 8                                  iOS 12.x.x

iPhone 8 Plus                              iOS 12.x.x

iPhone X                                   iOS 12.x.x

iPhone Xr                                   iOS 12.x.x

iPhone Xs                                   iOS 12.x.x

iPhone Xs Max                           iOS 12.x.x

=====================================

As you can see you have the last version of iOS available for your model.