/usr/share/qt4/bin/designer is in qt4-designer 4:4.8.7+dfsg-5ubuntu2.
This file is owned by root:root, with mode 0o777.
It is a symlink to /usr/share/qt4/bin/../../../lib/x86_64-linux-gnu/qt4/bin/designer
/usr/share/qt4/bin/../../../lib/x86_64-linux-gnu/qt4/bin/designer