We study support recovery for a $k \times k$ principal submatrix with elevated mean $λ/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $λ>0$ is a universal constant, and we assume $k = N ρ$ for some constant $ρ\in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $λ{\geq C} \sqrt{\frac{1}ρ \log \frac{1}ρ}$, {while such recovery is information theoretically impossible if $λ= o( \sqrt{\frac{1}ρ \log \frac{1}ρ} )$}. The MLE is computationally intractable in general, and in fact, for $ρ>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}ρ \log \frac{1}ρ } \ll λ\ll \frac{1}{ρ^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $λ> 1/ρ$, a simple spectral method recovers a constant proportion of the hidden submatrix.