There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.
An Overview of the Foundations of the Hypergroup Theory
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On the borderline of fields and hyperfields, part II -- Enumeration and classification of the hyperfields of order 7
There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.