  We present relations between X-ray luminosity and velocity dispersion
(L-sigma), X-ray luminosity and gas mass (L-Mgas), and cluster radius and
velocity dispersion (r500-sigma) for 62 galaxy clusters in the HIFLUGCS, an
X-ray flux-limited sample minimizing bias toward any cluster morphology. Our
analysis in total is based on ~1.3Ms of clean X-ray XMM-Newton data and 13439
cluster member galaxies with redshifts. Cool cores are among the major
contributors to the scatter in the L-sigma relation. When the
cool-core-corrected X-ray luminosity is used the intrinsic scatter decreases to
0.27 dex. Even after the X-ray luminosity is corrected for the cool core, the
scatter caused by the presence of cool cores dominates for the low-mass
systems. The scatter caused by the non-cool-core clusters does not strongly
depend on the mass range, and becomes dominant in the high-mass regime. The
observed L-sigma relation agrees with the self-similar prediction, matches that
of a simulated sample with AGN feedback disregarding six clusters with <45
cluster members with spectroscopic redshifts, and shows a common trend of
increasing scatter toward the low-mass end, i.e., systems with sigma<500km/s. A
comparison of observations with simulations indicates an AGN-feedback-driven
impact in the low-mass regime. The best fits to the $L-M_{\rm gas}$ relations
for the disturbed clusters and undisturbed clusters in the observational sample
closely match those of the simulated samples with and without AGN feedback,
respectively. This suggests that one main cause of the scatter is AGN activity
providing feedback in different phases, e.g., during a feedback cycle. The
slope and scatter in the observed r500-sigma relation is similar to that of the
simulated sample with AGN feedback except for a small offset but still within
the scatter.
