I want to upgrade to High Sierra or Mojave.

The model year of an iMac is available at Apple menu > About This Mac.

An early 2009 iMac can be upgraded to El Capitan 10.11.6 but no later version. See: How to download macOS - Apple Support

A late 2009 through mid 2011 iMac can be upgraded to High Sierra 10.13.6 but no later version. See: How to download macOS - Apple Support

A late 2012 through late 2013 iMac can be upgraded to Catalina 10.15.7 but no later. See: How to download macOS - Apple Support

A mid 2014 and late 2014 iMac can be upgraded to the latest version of Big Sur macOS 11 but no later. See: How to download macOS - Apple Support

A late 2015 iMac can be upgraded to the latest version of Monterey macOS 12. See:How to download macOS - Apple Support

A 2017 or later iMac can be upgraded to the current version of Ventura macOS 13. See: How to upgrade to macOS Ventura - Apple Support

Back up your data, check your applications for compatibility, click here, and then on the download link for High Sierra, or if your iMac is a 2012 or newer model, Mojave.

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