Most states have counties also. In Arkansas where I live, there are 75. It goes city/county/state/country. Louisiana has parishes instead of counties to help convolute things more if that's helpful lol.I agree with Larry, let’s complicate this, first up I live in England, UK (the real one not the Noble and Cooley Xanadu) and over here there is no such thing as a geographic state, the queen won’t allow it.
The term you are reaching for is County. So the naming headers should run City/County/Country. British people will get confused if you ask them for information on their state. My current state is antagonistic.
If you really wanna get crazy, each county has a "seat", a city that serves as the county's base of govt. Each city/town/village/etc. also has it's own govt. So each breakdown of land gets it's own govt breakdown also. Not sure how it works in the UK. It can get quite confusing and sticky here. But that's for another day.