More rare: Cycle or No-Hitter?

[BWebb]

I'm having trouble trying to find the number that each has been done in history, but which feat do you think is more rare? I know no-hitter seems like the obvious answer. But, this is what makes me think it might not be. In each game, there are 9 batters in each lineup and only 2 starting pitchers. So, each game has an opportunity for 18 cycles but only 2 no hitters. Therefore, wouldn't it be that if there are 9x more cycles than no hitters, they are happening at the same rate? So, if there are less than 9x more cycles than no-hitters, would that make the cycle more rare?

[Homer]

IIRC, there have actually been more no-hitters than cycles. Or at least the numbers are very close.

[Keats13]

I'm too lazy to count.

It looks like the cycle list is a little longer.
Cycles
No-hitters

[Punker]

I put two lists into excel and came up with :

Cycle: 136
No hitter: 297

I make no vouchers for the accuracy of those statistics.

[ThaSaltCracka]

since 2000, 22 players have hit for the cycle and 7 pitchers have thrown a no hitter. So at least for the past 5 years, cycles are more common.