The lawsuit was filed on Sunday by the Union of Jewish Students of France (UEJF) and SOS-Racisme, the organisations said in a press release.
They referred to a survey carried out between March 31 and May 10 by their members and those of a third association, SOS Homophobie.
In this "first mass test of social networks," the groups uncovered 586 instances of content that was "racist, anti-Semitic, denied the Holocaust, homophobic (or) defended terrorism or crimes against humanity," the joint statement said.
"It's a mystery whether moderating teams in social media are actually working," said Sacha Reingewirtz, president of the UEJF.
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"These platforms seem more shocked about content with bare breasts, which is swiftly censored, than about incitement to hatred," Sopo said.
"Our legal step aims at getting the authorities to apply the law so that these organisation submit to it in full."
The ruling may be the first step in a full judgement in the case, which can take months or years.