I constructed the groups of order $15,400=2^3\cdot 5^2\cdot 7\cdot 11$ with GAP and noticed that every such group has a normal subgroup of order $275=5^2\cdot 11$.
Can this be proven by hand ?
Motivation :
With this information, we can conclude via the Hall-theorem, that every group of order $15,400$ is a semidirect product of a group of order $275=5^2\cdot11$ and a group of order $56=2^3\cdot 7$.
Furthermore, using the Burnside theorem, it follows that every group of order $15,400$ is solvable.