  The coprime hypergraph of integers on $n$ vertices $CHI_k(n)$ is defined via
vertex set $\{1,2,\dots,n\}$ and hyperedge set
$\{\{v_1,v_2,\dots,v_{k+1}\}\subseteq\{1,2,\dots,n\}:\gcd(v_1,v_2,\dots,v_{k+1})=1\}$.
In this article we present ideas on how to construct maximal subgraphs in
$CHI_k(n)$. This continues the author's earlier work, which dealt with bounds
on the size and structural properties of these subgraphs. We succeed in the
cases $k\in\{1,2,3\}$ and give promising ideas for $k\geq 4$.
