  Let $K \subset \mathbb{C}$ be a number field, and let $\mathcal{O}_{K,N} =
\mathcal{O}_{K}[N^{-1}]$ be its ring of $N$-integers. Recently, Lawrence and
Venkatesh proposed a general strategy for proving the Shafarevich conjecture
for the fibres of a smooth projective family $f : X \to S$ defined over
$\mathcal{O}_{K,N}$. To carry out their strategy, one needs to be able to
decide whether the algebraic monodromy group $\mathbf{H}_{Z}$ of any
positive-dimensional geometrically irreducible subvariety $Z \subset
S_{\mathbb{C}}$ is "large enough", in the sense that a certain orbit of
$\mathbf{H}_{Z}$ in a variety of Hodge flags has dimension bounded from below
by a certain quantity. In this article we give an effective method for deciding
this question. Combined with the effective methods of Lawrence-Venkatesh for
understanding semisimplifications of global Galois representations using
$p$-adic Hodge theory, this gives a fully effective strategy for solving
Shafarevich-type problems for arbitrary families $f$.
