  In this paper, we initiate a systematic study of entanglements of division
fields from a group theoretic perspective. For a positive integer $n$ and a
subgroup $G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$ with surjective
determinant, we provide a definition for $G$ to represent an
$(a,b)$-entanglement and give additional criteria for $G$ to represent an
explained or unexplained $(a,b)$-entanglement.
  Using these new definitions, we determine the tuples $((p,q),T)$, with
$p<q\in\mathbb{Z}$ distinct primes and $T$ a finite group, such that there are
infinitely many non-$\bar{\mathbb{Q}}$-isomorphic elliptic curves over
$\mathbb{Q}$ with an unexplained $(p,q)$-entanglement of type $T$. Furthermore,
for each possible combination of entanglement level $(p,q)$ and type $T$, we
completely classify the elliptic curves defined over $\mathbb{Q}$ with that
combination by constructing the corresponding modular curve and $j$-map.
