Hybrid III-V-on-silicon semiconductor optical amplifiers with high-gain and high-output-power are important in many applications such as transceivers, integrated microwave photonics and photonic beamforming. In this work we present the design, fabrication and characterization of high-gain, high-output-power III-V-on-silicon semiconductor optical amplifiers. The amplifiers support a hybrid III-V/Si optical mode to reduce confinement in the active region and increase the saturation power. A small-signal gain of 27 dB, a saturation power of 17.24 dBm and an on-chip output power of 17.5 dBm is measured for a current density of 4.9 kA/cm2 (power consumption of 540 mW) at room temperature for an amplifier with a total length of 1.45 mm. The amplifiers were realized using a 6 quantum well InGaAsP active region, which was previously used to fabricate high-speed directly modulated DFB lasers, enabling their co-integration.