The following question is "ideologically related" to the [one I recently asked][1]. 

For a prime $p$, let $M_p$ denote the least common multiple of the orders modulo $p$ of all odd prime divisors of $p-1$:
  $$ M_p := {\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q\ \text{is an odd prime}\}. $$
I am interested in the primes $p\equiv5\pmod 8$, and I want to show that, normally, $M_p>\sqrt p$ holds for such primes. In the range $5\le p<100,000,000$, there are only three exceptions (primes $p\equiv 5\pmod 8$ with $M_p<\sqrt p$): namely, $5$, $13$, and $148,997$. Are there any more such exceptional primes and if so, is the set of all these primes finite?

Notice that allowing $p\equiv 1\pmod 8$ would make every Fermat prime a bold exception.


[1]: https://mathoverflow.net/questions/185683/why-gcd-rm-ord-pq-colon-q-mid-p-1-likes-to-be-large