The club have vowed to take action against the group, after complaints about the behaviour of the “so-called” supporters, at the Cheaper Insurance Direct Stadium, where Dumbarton beat Livingston 1-0.
Following the match, Livingston announced the allegations on their website.
The club said there had been complaints about fans sneaking alcohol into the stadiumn and smashing bottles in the stand as well as shouting racial abuse at Darren Cole.
They said anyone identified as being party to such abuse would be dealt with in accordance with the club’s zero tolerance stance on racism.
A statement on the club’s official website said: “Livingston FC has received several complaints this evening regarding the behaviour of a small group of so-called supporters who attended today’s game in Dumbarton.
“We understand these individuals allegedly arrived at the game drunk, and illegally secreted bottles of alcohol into the stadium.
“It’s further alleged that one or more of these bottles was later smashed inside the stand, and one individual launched an assault upon another.
“We have also been told that this same group shouted racist abuse at Darren Cole, among a list of other expletives aimed at our club captain.”
They added: “We are now fully investigating this issue and shall advise further in due course.”
The club have said they expect their supporters to act as ambassadors, “especially” when the team are playing away from home.
The club said the “overwhelming majority” behave at matches – but that all it took “is one rotten apple to taint the reputation of the club and all our decent supporters”.
“We fully accept that football generates passion in the stands, and from time to time frustrations boil over into verbal comments,” the club added.
“This is part of the game – but only within reason.”
The club said it operates a zero tolerance approach to racist abuse, and those found to have indulged in such acts will be dealt with accordingly.
“Violence and other antisocial behaviour is similarly not tolerated, and we will cooperate fully with Police Scotland should this ever be required,” the club added.
Following Saturday’s events, fans have taken to social media to express their outrage.
Blair Cockburn tweeted: “Utterly embarrassed to see ‘so-called’ Livi fans kicking off with each other, and racist shouts against Darren Cole. Horrible.”
Another fan, Andy Brown, from Livingston, wrote on the club’s Facebook page: “Livingston has a great record over the years for having a friendly, sporting support both home and away.
“Let’s not let a few idiots tarnish the good reputation of the club.”
Robert Wilson, the chairman of Livingston Football Club, was not available for comment.