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Liverpool assistant coach Sipke Hulshoff was banned for two games and fined £7,000, while Liverpool and Everton were fined £65,000 and £50,000 respectively for failing to control their players.
James Tarkowski's 98th-minute equaliser sparked chaotic scenes, with Everton fans entering the field to celebrate.
Liverpool thought Tarkowski's goal, which was checked by the video assistant referee (VAR), should have been ruled out for a push on defender Ibrahima Konate in the build-up.
Slot exchanged words with one of the assistant referees before shaking hands with Oliver, who immediately showed him a red card.
Hulshoff was also sent off.
Slot, who was unable to speak to the media after the match because of his red card, later voiced his frustration over the amount of stoppage time played, after five minutes were initially signalled.
In a news conference two days after the game, Slot said "emotions got the better of me".
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