The paper combines two major contemporary systems and control methodologies to obtain a unique \(\epsilon\)-Nash equilibrium for optimal execution problems within the stock market, namely Mean Field Game (MFG) theory and Hybrid Optimal Control (HOC) theory. Following standard financial models, the stock market is studied in this paper as a large population non-cooperative game where each trader has stochastic linear dynamics with quadratic costs. We consider the case where there exists one major trader with significant influence on market movements together with a large number of minor traders (within two subpopulations), each with individually asymptotically negligible effect on the market. The traders are coupled in their dynamics and cost functions by the market's average trading rate (a component of the system mean field) and the hybrid feature enters via the indexing of the cessation of trading by one or both subpopulations of minor traders by discrete states. Optimal stopping time strategies together with best response trading policies for all traders are established with respect to their individual cost criteria by an application of LQG HOC theory.