If you look at the eigenmode in the fiber, there is non mode group due to each mode has its own propagation constant even though they are very close.
Let us talk about the linear polarization mode in weakly guide fiber. Some modes will linearly combine since their propagation constants can be seen as identical and then they form LP mode, e.g. TE01, TM01 and HE21 generate LP11 mode. Actually, such LP11 mode is indeed a mode group because it contains degenerated modes, polarization degenerated and spatial degenerated, i.e. LP11x, LP11y, LP11o, LP11e. Hence LP11 mode group has four modes: LP11ox, LP11oy, LP11ex and LP11ey.
Being in your mind, there are only eigenmodes exist in the fiber. Other types of mode are just another way to express them.