A classification of curious Galois groups as direct products

Abstract: Let $N$ be a positive integer. Let $\operatorname{H}$ be a group of level $N$ and let $E$ be an elliptic curve defined over the rationals with $\textit{j}{E} \neq 0, 1728$. Then the image $\overline{\rho}{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$, of the mod-$N$ Galois representation attached to $E$, is conjugate to a subgroup of $\operatorname{H}$ if and only if $E$ corresponds to a non-cuspidal rational point on the modular curve $\operatorname{X}{\operatorname{H}}$ generated by $\operatorname{H}$. In this article, we are interested when $\overline{\rho}{E,N}(G_{\mathbb{Q}})$ is precisely $\operatorname{H}$. More precisely, we classify all groups $\operatorname{H}$ that are direct products of subgroups $\operatorname{H}$ for which $\operatorname{X}{\operatorname{H}}$ contains infinitely many non-cuspidal rational points but there is no elliptic curve $E/\mathbb{Q}$ such that $\overline{\rho}{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$ is conjugate to $\operatorname{H}$.