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The goal of the paper is to give fine-grained hardness results for the Subgraph Isomorphism (SI) problem for fixed size induced patterns $H$, based on the $k$-Clique hypothesis that the current best algorithms for Clique are optimal. Our first main result is that for any pattern graph $H$ that is a {\em core}, the SI problem for $H$ is at least as hard as $t$-Clique, where $t$ is the size of the largest clique minor of $H$. This improves (for cores) the previous known results [Dalirrooyfard-Vassilevska W. STOC'20] that the SI for $H$ is at least as hard as $k$-clique where $k$ is the size of the largest clique {\em subgraph} in $H$, or the chromatic number of $H$ (under the Hadwiger conjecture). For detecting \emph{any} graph pattern $H$, we further remove the dependency of the result of [Dalirrooyfard-Vassilevska W. STOC'20] on the Hadwiger conjecture at the cost of a sub-polynomial decrease in the lower bound. The result for cores allows us to prove that the SI problem for induced $k$-Path and $k$-Cycle is harder than previously known. Previously [Floderus et al. Theor. CS 2015] had shown that $k$-Path and $k$-Cycle are at least as hard to detect as a $\lfloor k/2\rfloor$-Clique. We show that they are in fact at least as hard as $3k/4-O(1)$-Clique, improving the conditional lower bound exponent by a factor of $3/2$. Finally, we provide a new conditional lower bound for detecting induced $4$-cycles: $n^{2-o(1)}$ time is necessary even in graphs with $n$ nodes and $O(n^{1.5})$ edges.