I stumbled upon these very nice looking notes by Brian Lawrence on the period of the Fibonacci numbers over finite fields. In them, he shows that the period of the Fibonacci sequence over $\mathbb{F}_p$ divides $p$ or $p-1$ or $p+1$.

I am wondering if there are explicit lower bounds on this period. Is it true, for instance, that as $p \to \infty$, so does the order?

A quick calculation on my computer shows that there are some "large" primes with period under 100.

9901 66
19489 58
28657 92