In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer
n
n
and a subgroup
G
⊆
G
L
2
(
Z
/
n
Z
)
G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z})
with surjective determinant, we provide a definition for
G
G
to represent an
(
a
,
b
)
(a,b)
-entanglement and give additional criteria for
G
G
to represent an explained or unexplained
(
a
,
b
)
(a,b)
-entanglement.
Using these new definitions, we determine the tuples
(
(
p
,
q
)
,
T
)
((p,q),T)
, with
p
>
q
∈
Z
p>q\in \mathbb {Z}
distinct primes and
T
T
a finite group, such that there are infinitely many non-
Q
¯
\overline {\mathbb {Q}}
-isomorphic elliptic curves over
Q
\mathbb {Q}
with an unexplained
(
p
,
q
)
(p,q)
-entanglement of type
T
T
. Furthermore, for each possible combination of entanglement level
(
p
,
q
)
(p,q)
and type
T
T
, we completely classify the elliptic curves defined over
Q
\mathbb {Q}
with that combination by constructing the corresponding modular curve and
j
j
-map.