International maritime organization (IMO) is regulating the $CO_2$ emissions from ships by energy efficient design index (EEDI). Although the conventional LNG-fuelled propulsion has significantly reduced the green-house gas emissions, it cannot fully satisfy the emission regulations that will be strengthened gradually with time. This study proposed an $LNG-LH_2$ (liquid hydrogen) hybrid propulsion system for an LNG carrier to keep the ship in compliance with the $CO_2$ emissions regulations. The hybrid system was designed for a 267K LNG carrier and evaluated in terms of the EEDI, hybrid fuel ratio, installation space, safety, and economics. System design parameters were determined to satisfy the EEDI of Phases 4 and 5, which should be reduced by 14 and 28 % from the current level, respectively. In Phase 4, the mass ratio of $LH_2$ to LNG was estimated 3% with the power of 6 MW and the LH2 tank of 700 $m^3$. In Phase 5, those values were increased to 6%, 13 MW and 1,500 $m^3$, respectively. However, the volume growth induced by the fuel cells and $LH_2$ tank was insignificant because they should have been installed in the open space on the deck as per the rules. The economic feasibility presented the price of $LH_2$ fuel, which secured economic operation of the hybrid system. The life cycle cost (LCC) of the hybrid system was estimated higher by 8% than the LNG system, and the breakeven price of $LH_2$ should be reduced to the level of 2.0 $/kg for Phase 4. In Phase 5, the LCC increased to 19% with the breakeven price decreased to 1.6 $/kg. In order to apply the feasible hybrid systems, $LH_2$ infrastructure should be built and LH2 fuel prices should be lowered.