  The Lorentz Invariance Violation (LIV), a proposed consequence of certain
quantum gravity (QG) scenarios, could instigate an energy-dependent group
velocity for ultra-relativistic particles. This energy dependence, although
suppressed by the massive QG energy scale $E_\mathrm{QG}$, expected to be on
the level of the Planck energy $1.22 \times 10^{19}$ GeV, is potentially
detectable in astrophysical observations. In this scenario, the cosmological
distances traversed by photons act as an amplifier for this effect. By
leveraging the observation of a remarkable flare from the blazar Mrk\,421,
recorded at energies above 100 GeV by the MAGIC telescopes on the night of
April 25 to 26, 2014, we look for time delays scaling linearly and
quadratically with the photon energies. Using for the first time in LIV studies
a binned-likelihood approach we set constraints on the QG energy scale. For the
linear scenario, we set $95\%$ lower limits $E_\mathrm{QG}>2.7\times10^{17}$
GeV for the subluminal case and $E_\mathrm{QG}> 3.6 \times10^{17}$ GeV for the
superluminal case. For the quadratic scenario, the $95\%$ lower limits for the
subluminal and superluminal cases are $E_\mathrm{QG}>2.6 \times10^{10}$ GeV and
$E_\mathrm{QG}>2.5\times10^{10}$ GeV, respectively.
