Overtone crystals are designed to run like that.
Although all crystals do have some activity at the overtone frequencies, crystals designed for fundamental mode will not work at all well at overtone.
The reasons are:-
Adjustment. The overtone frequencies are not exact multiples of the fundamental frequency. During manufacture, the final frequency is adjusted with the crystal running at the overtone frequency.
Activity. Overtone crystals, especially 5th and higher overtone crystals, are ground with finer abrasive or polished before the electrodes are applied. This improves the activity and makes it much easier for the crystal to oscillate at the overtone frequency. Polishing makes little difference for fundamental crystals so it is very rarely done as it is a slow process.
Temperature stability. The angle that the quartz disk is cut from the crystal has a huge effect on the temperature stability of the final crystal. The optimum angle is different for fundamental crystals and for overtones, by about 0.1 degrees of angle, or about 20 - 30 ppm over 60 degrees C.
All these differences are most pronounced between fundamental and 3rd overtone crystals, getting less and less significant between higher overtones, so running a 5th overtone at 7th will work quite well.
However, high overtones are difficult to design oscillators for. Each overtone has less activity than the lower ones. What that means is that a crystal bought as a 5th overtone 100 MHz will be easier to make oscillate at 60 MHz on its 3rd overtone than it will be at its design frequency. It will be easier still to make it oscillate at 20 MHz at its fundamental frequency. The oscillator has to be tuned to prevent it running at the wrong overtone. For a fundamental oscillator, that is easy as it is the lowest frequency and the oscillators tend to have less gain at higher frequencies. At higher overtones, the ratio between the frequencies is less, so the tuning of the oscillator becomes more difficult.