The investigation of plasma motions in the solar chromosphere is crucial for understanding the transport of mechanical energy from the interior of the Sun to the outer atmosphere and into interplanetary space. We report the finding of large-amplitude oscillatory transverse motions prevailing in the non-spicular Halpha chromosphere of a small quiet region near the solar disk center. The observation was carried out on 2018 August 25 with the Microlensed Hyperspectral Imager (MiHI) installed as an extension to the spectrograph at the Swedish Solar Telescope (SST). MiHi produced high-resolution Stokes spectra of the Halpha line over a two-dimensional array of points (sampled every 0.066 on the image plane) every 1.33 s for about 17 min. We extracted the Dopple-shift-insensitive intensity data of the line core by applying a bisector fit to Stoke I line profiles. From our time--distance analysis of the intensity data, we find a variety of transverse motions with velocity amplitudes of up to 40 in fan fibrils and tiny filaments. In particular, in the fan fibrils, large-amplitude transverse MHD waves were seen to occur with a mean velocity amplitude of 25 and a mean period of 5.8 min, propagating at a speed of 40 These waves are nonlinear and display group behavior. We estimate the wave energy flux in the upper chromosphere at $3 10^6$ erg $ s$^ $. Our results contribute to the advancement of our understanding of the properties of transverse MHD waves in the solar chromosphere.