The incident, at the Otkrytie Arena in Moscow on Dec. 4, came ten days after Turkish forces shot down a Russian war plane near the Turkish-Syrian border.
"Following videos that appeared in the press, we saw five Turkish flags, which were burnt by Spartak supporters," head of the RFU's disciplinary committee Artur Grigoryants told journalists.
"According to Russian law, it is forbidden to use sporting arenas as a place for manifesting non-sporting statements.
"If this incident happens again, the club will face much harsher sanctions."
Relations between the two countries have cooled since the Russian aircraft was shot down on Nov. 24. Russia says one of the pilots, who ejected from the plane, was subsequently killed by militants in Syria, and a Russian marine also died during the rescue mission.
Spartak won the match 1-0 and are currently fourth in the Russian Premier League.
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