This dissertation is broadly concerned with the subsystem problem for subshifts on countable amenable groups: given a subshift $X$, what are its subsystems and can they be classified or characterized by a simple criterion? In particular, we focus on the restricted case where $X$ is a subshift of finite type (SFT). We pursue a variety of approaches to the problem. Firstly, we utilize the theory of topological entropy to demonstrate that an SFT with positive entropy exhibits a ubiquity of subsystems. Specifically, we prove that for any countable amenable group $G$, if $X$ is a $G$-SFT with positive topological entropy $h(X) > 0$, then the entropies of the SFT subsystems of $X$ are dense in the interval $[0, h(X)]$. In fact, we prove a ``relative" version of the same result: if $X$ is a $G$-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs $Z$ for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic subshifts. These results generalize the results of Desai for $G = \mathbb{Z}^d$. Secondly, we present an embedding theorem which provides conditions under which a given subshift may be realized as a subsystem of a given SFT. Namely, the result we obtain is as follows. Let $G$ be a countable amenable group with the comparison property. Let $X$ be a strongly aperiodic subshift over $G$. Let $Y$ be a strongly irreducible shift of finite type over $G$ which has no global period, meaning that the shift action is faithful on $Y$. If $h(X) < h(Y)$ and $Y$ contains at least one factor of $X$, then $X$ embeds into $Y$. This result partially extends the classical result of Krieger for $G = \mathbb{Z}$ and the results of Lightwood for $G = \mathbb{Z}^d$ for $d \geq 2$. Our proofs rely on recent developments in the theory of tilings and quasi-tilings of amenable groups due to Downarowicz, Huczek, and Zhang.