The charges followed England's 1-0 victory in a European under-21 championship playoff in Krusevav in October.
"The sanction against the FSS has been imposed for the improper conduct of its supporters during and at the end of the match, as well as for the improper conduct of the Serbia players at the end of the game," European soccer's ruling body said in a statement.
England scored with the last kick of the game and fighting broke out between players and officials immediately after the final whistle.
Visiting full back Danny Rose, on loan at Sunderland, complained he was racially abused by Serbia fans before, during and after the game.
Serbia's assistant coach Predrag Katic and fitness coach Andreja Milunovic have been banned from football for two years.
Four Serbia players have been suspended for between two and four international matches. Goran Causic was banned for four games, Ognjen Mudrinski and Filip Malbasic for three matches and Nikola Ninkovic for two.
England defender Steven Caulker, who plays for Tottenham Hotspur, has been suspended for two international matches and Blackpool's Tom Ince for one.
($1 = 0.7641 euros) (Writing by Toby Davis; editing by Tony Jimenez)
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