Finding items that are contained in two lists as substrings

I have two lists:

var files = new List<string> {"ax_11118.txt", "ax_422226.txt", "ax_4346436.txt", "678678678.txt"};
var codes = new List<string> { "1111", "1234", "5555" };

I need to find the items in files which contain this format: "ax_code". The result in this case should be : { "ax_11118.txt" }

I know how to make it with foreach, but I was thinking if there is a cleaner way.