Gauge-fixing local H symmetry in supergravities

We discuss known maximal D-dimensional supergravities of two types: type I with G/H coset spaces and type II derived by compactification from higher dimensions without dualization, these have less manifest symmetries. In 4D and 6D in type I models we perform explicit gauge-fixing of local H symmetries in unitary gauges: symmetric, Iwasawa and partial Iwasawa. In 4D supergravity I in symmetric gauge global H-invariance and nonlinearly realized G-symmetry are valid on shell, classically. The global H-symmetry and G-symmetry in Iwasawa-type gauges in type I and in type II supergravities are not manifest, if at all present. This fact raises the issue of the gauge equivalence of the S-matrix of various gauge-fixed D-dimensional supergravities and its relation to the ones computable using superamplitude methods.