Haskell, 56 bytes

_!1=[[]]
i!n=[j:f|j<-[i..n],mod n j<1,f<-j!div n j]
(2!)

(2!)(1338557220::Int) prints in five minutes on my laptop, when compiled with ghc -O3.

Haskell, 62 bytes, but much faster

i!n|i*i>n=[[n]]|0<1=[i:f|mod n i<1,f<-i!div n i]++(i+1)!n
(2!)

(2!)(1338557220::Int) prints in a quarter of a second on my laptop, when compiled with ghc -O3.

Pyth, 29 bytes

Msam+Ldgd/Hdf!%HT>S@H2tG]]Hg2

M                                def g(G, H):
                   @H2             square root of H
                  S                1-indexed range up to floor
                 >    tG           all but first G − 1 elements
            f                      filter for elements T such that:
              %HT                    H mod T
             !                       is false (0)
   m                               map for elements d:
       gd/Hd                         g(d, H/d)
    +Ld                              prepend d to each element
  a                     ]]H        append [[H]]
 s                                 concatenate
                           g2Q   print g(2, input)

Try it online

Runs in twenty seconds for 1338557220 on my laptop.

JavaScript (ES6), 83 bytes

f=(n,a=[],m=2,i=m)=>{for(;i*i<=n;i++)n%i<1&&f(n/i,[...a,i],i);console.log(...a,n)}

Only borrowed @AndersKaseorg's square root trick because it ended up saving me bytes overall. Prints 1 for an input of 1, otherwise doesn't print 1s.

Python 2, 252 313 312 311 145 141 137 135 103 bytes

This is largely based on Anders Kaseorg's Pyth answer. Any golfing suggestions welcome.

import math
g=lambda n,m=2:[[n]]+[[d]+j for d in range(m,int(math.sqrt(n))+1)if n%d<1for j in g(n/d,d)]

Jelly, 15 bytes

ÆfŒ!ŒṖ€;/×/€Ṣ€Q

Try it online!

ÆfŒ!ŒṖ€;/×/€Ṣ€Q   Main monadic chain; argument: z
                  temp ← z
Æf                temp ← prime factors of temp (the array of primes whose product is temp)
  Œ!              temp ← all permutations of temp
    ŒṖ€           temp ← unique partitions for each array in temp
       ;/         temp ← eh, kind of flatten
         ×/€      temp ← for each partition in temp, product of each group
            Ṣ€    temp ← sorted each in temp
              Q   temp ← unique items in temp