The name of the culture that exists within the dominant culture is .

A subculture.

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Subculture

Explanation:

  Subculture is a group of people or movement with distinct behavioral characteristics that differentiate them from a broader culture of which they are part. They are small "groups" or a culture not fully developed, groups that cultivate and preserve the same ideas in terms of aesthetics, religion, occupational, political, sexual, or a combination of these factors. The subculture can stand out due to the age of its members, or by their ethnicity, class and/or gender.