Coded Caching is a promising solution to reduce the peak traffic in broadcast networks by prefetching the popular content close to end users and using coded transmissions. One of the chief issues of most coded caching schemes in literature is the issue of large $\textit{subpacketization}$, i.e., they require each file to be divided into a large number of subfiles. In this work, we present a coded caching scheme using line graphs of bipartite graphs in conjunction with projective geometries over finite fields. The presented scheme achieves a rate $Θ(\frac{K}{\log_q{K}})$ ($K$ being the number of users, $q$ is some prime power) with $\textit{subexponential}$ subpacketization $q^{O((\log_q{K})^2)}$ when cached fraction is upper bounded by a constant ($\frac{M}{N}\leq \frac{1}{q^α}$) for some positive integer $α$). Compared to earlier schemes, the presented scheme has a lower subpacketization (albeit possessing a higher rate). We also present a new subpacketization dependent lower bound on the rate for caching schemes in which each subfile is cached in the same number of users. Compared to the previously known bounds, this bound seems to perform better for a range of parameters of the caching system.