Other way around. John Quincy Adams, Henry Clay, and his friends left to form their own party because they had been outmaneuvered in their control for the Democratic-Republican Party.
Somewhat like how the Southern Democrats left for the Republican Party in the late 1960s/early 1970s after they had been outmaneuvered by the pro-civil rights Democrats. Doesn't mean the current Democrats aren't the same party as the Democrats of 1940s and 1950s.
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