  In this paper, a new graph structure called the \textit{coprime order graph}
of a finite group $G$ denoted by $\Theta(G)$ has been introduced. The
\textit{coprime graph} of a finite group introduced by Ma, Wei, and Yang
[\textit{The coprime graph of a group. International Journal of Group Theory,
3(3), pp.13-23.}] is a subgraph of the \textit{coprime order graph} introduced
in this paper. The vertex set of $\Theta(G)$ is $G$, and any two vertices $x,y$
in $\Theta(G)$ are adjacent if and only if $\gcd(o(x),o(y))$ is equal to $1$ or
a prime number. We study how the graph properties of $\Theta(G)$ and group
properties of $G$ are related among themselves. We provide a necessary and
sufficient condition for $\Theta(G)$ to be Eulerian for any finite group $G$.
We also study $\Theta(G)$ for certain finite groups like $\mathbb Z_n$ and
$\mbox D_n$ and derive conditions when it is connected, complete, planar, and
Hamiltonian for various $n\in \mathbb N$. We also study the vertex connectivity
of $\Theta(\mathbb Z_n)$ for various $n\in \mathbb N.$ Finally, we have
computed the signless Laplacian spectrum of $\Theta(G)$ when $G=\mathbb Z_n$
and $G=\mbox D_n$ for $n\in \{pq,p^m\}$ where $p,q$ are distinct primes and
$m\in \mathbb{N}$.
