There is no such continuum. See 

Z. Waraszkiewicz, *Sur un problème de M.H. Hahn*, Fund. Math. 22 (1934) 180–205.

Apparently, Waraszkiewicz constructs a continuum $X$ so that no continuum can be mapped continuously onto every subcontinuum of $X$. This is extended to a stronger non-existence result in

S.B. Nadler, *The nonexistence of almost continuous surjections between certain continua*, Topology and its Applications, 154 (2007) 1008-1014.