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We refine Soundararajan's result to show that if $4 \\nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D \\equiv A \\pmod{B}$ down to $-X$ such that the ideal class group of $\\mathbb{Q} (\\sqrt{D})$ contains an element of order $g$ is bounded below by $X^{\\frac{1}{2} + \\epsilon(g) - \\epsilon}$, where the exponent is the same as in Soundararajan's theorem. 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Let $A,B,g \\ge 3$ be positive integers such that $\\gcd(A,B)$ is square-free. We refine Soundararajan's result to show that if $4 \\nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D \\equiv A \\pmod{B}$ down to $-X$ such that the ideal class group of $\\mathbb{Q} (\\sqrt{D})$ contains an element of order $g$ is bounded below by $X^{\\frac{1}{2} + \\epsilon(g) - \\epsilon}$, where the exponent is the same as in Soundararajan's theorem. 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