I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.
For example, in ncatlab's article,
The Monster group was predicted to exist by Bernd Fischer and Robert Griess in 1973, as a simple group containing the Fischer groups and some other sporadic simple groups as subquotients. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of $M$ and discovered other properties and subgroups, assuming that it existed.
The Monster was predicted by Bernd Fischer (unpublished) and Robert Griess (1976) in about 1973 as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months the order of $M$ was found by Griess using the Thompson order formula [...]
Its existence is a non-trivial fact: when the original moonshine conjectures were made, mathematicians suspected its existence, and had been able to work out its character table, but could not prove it actually existed. They did know that the dimensions of the smallest irreducible representations would be 1, 196883; and 21296876.
It surprises me that this object could have been predicted before being rigorously discovered, due to it being often described as very complicated and highly nonobvious (or at least its construction).
Take for instance the description in this AMS review of Moonshine Beyond the Monster:
The proof of the moonshine conjectures depends on several coincidences. Even the existence of the monster seems to be a ﬂuke in any of the known constructions: these all depend on long, strange calculations that just happen to work for no obvious reason, and would not have been done if the monster had not already been suspected to exist.
It'd be cool to be acquainted with this part of the story in some more detail at an accessible level, if possible, though I realize it may necessarily involve heavy machinery or convoluted calculations.