Electron transport was studied in an open square quantum dot with a dimension typical for current experiments. A numerical analysis of the probability density distribution inside the dot was performed which enabled us to unambiguously map the resonant states which dominate the conductance of the structure. It was shown that, despite of the presence of dot openings, transport through the dot is effectively mediated by just a few (or even a single) eigenstates of the corresponding closed structure. In a single-mode regime in the leads, the broadening of the resonant levels is typically smaller than the mean energy level spacing, Δ. On the contrary, in the many-mode regime this broadening typically exceeds Δ and has an irregular, essentially non-Lorentzian, character. It was demonstrated that in the latter case eigenlevel spacing statistics of the corresponding closed system are not relevant to the averaged transport properties of the dot. This conclusion seems to have a number of experimental as well as numerical verifications.