I know that the general process for these types of problems is to first apply the Fundamental Theorem of Finite Abelian Groups (FTFAG), then find some semi-direct products.
First, if we assume $G$ is abelian, then it is isomorphic to $\mathbb{Z}_5 \times \mathbb{Z}_7 \times \mathbb{Z}_{19}$ by FTFAG. There is no other abelian groups of the proper order.
I am stuck, now, on finding the appropriate semi-direct products to find the non-abelian groups of the proper order. By the Sylow counting theorems, I know that the Sylow-$5$, Sylow-$7$, and Sylow-$19$ subgroups are all normal in $G$, but I can't seem to come up with a proper homomorphism to construct a semidirect product.
In addition to this specific problem, I would appreciate any help when it comes to constructing groups of a certain order as a semidirect product in general.
If the Sylow subgroups are all normal, it follows that $G\cong C_5×C_7×C_{19}$.
The Sylow subgroups in this case are all cyclic, hence abelian. Thus $G$ is abelian. (More can actually be said: by the Chinese remainder theorem, $G$ is cyclic.)