The purpose of subsequence matching is to find a query sequence from a long data sequence. Due to the abundance of applications, many solutions have been proposed. Virtually all previous solutions use the Euclidean measure as the basis for measuring distance between sequences. Recent studies, however, suggest that the Euclidean distance often fails to produce proper results due to the irregularity in the data, which is not so uncommon in our problem domain. Addressing this problem, some non-Euclidean measures, such as Dynamic Time Warping (DTW) and Longest Common Subsequence (LCS), have been proposed. However, most of the previous work in this direction focused on the whole sequence matching problem where query and data sequences are the same length. In this paper, we propose a novel subsequence matching framework using a non-Euclidean measure, in particular, LCS, and a new index query scheme. The proposed framework is based on the Dual Match framework where data sequences are divided into a series of disjoint equi-length subsequences and then indexed in an R-tree. We introduced similarity bound for index matching with LCS. The proposed query matching scheme reduces significant numbers of false positives in the match result. Furthermore, we developed an algorithm to skip expensive LCS computations through observing the warping paths. We validated our framework through extensive experiments using 48 different time series datasets. The results of the experiments suggest that our approach significantly improves the subsequence matching performance in various metrics.