The effect of suprathermal particles is examined on arbitrary amplitude nonlinear low-frequency electrostatic solitary waves propagating obliquely to the ambient magnetic field of the terrestrial magnetosphere using the Sagdeev pseudopotential formalism. The four-component plasma model consists of cold singly charged oxygen ions, hot protons, and two distinct groups of electrons with the latter three having kappa velocity distributions. Our numerical investigations based on comparatively low values for the spectral indices of the two electron species and the hot proton distributions as inferred from time history of events and macroscale interactions during substorms mission data reveal that the coexistence of compressive and rarefactive solitons is suppressed in comparison to the behavior in the presence of Maxwellian species (infinitely large values of kappa). Also, the existence of positive potential solitons is suppressed in an intermediate range of cold oxygen-ion densities. The differences are attributed to the occurrence of singularities in the Sagdeev potential for the chosen set of spectral indices.