The history of Dc's cosmology goes back to way at the beginning of dc's start when it had a universe filled with characters such as the golden age Superman and Jay garrick flash.The dc universe had one universe until the issue #123 of flash sept 1961 called "the flash of two worlds",where barry allen vibrates to another world confirming the existence of a multiverse.
Dc then explained the origin of the concept of the multiverse in further comics,it went something like this.A scientist peered back to the beginning of the of the universe and the act of peering back to the beginning caused a the universe to separate into an infinite multiverse.
This act created a being that would then destroy the multiverse.
At the end of this event the multiverse was gone and only a universe was left one main universe.
This would last for another few years in the meantime dc had introduced a concept called hypertime .This was in the 90's.It was an attempt to bring back the multiverse.But since the multiverse was destroyed by the crisis event dc had to find a way to make the crisis on infinite earths have meaning without just bringing back something called the multiverse.So they decided why not we have alternate timelines from the main all equally real.What dc didn't realise is that this is a multiverse in essence.But the y didn't call it that it was called hypertime.Because they went on the premise that these are infinite different parallel timelines branching from the main timeline .While not really the same thing as the multiverse theory it still kinda is a multiverse theory with a different name.Because not only could you travel but it operate as a multiverse.The only distinction that could be made different from the multiverse theory is that hypertime some timelines could merge into the main continuity if they fit.Hypertime was a tool to explain different books that were not canon but readers wanted to be canon in a sense.So books like kingdom were canon via the use of hypertime.Because of the concept of infinite parallel timelines.
The concept of hypertime was eventually abandoned in 2005 and they went back to one universe,when Dc decided to celebrate crisis on infinite earths with there title "Infinite Crisis" which was a continuation of the crisis on infinite earths where the remaining survivors of the crisis went to a paradise dimension to live out there lives.Eventually the story was told in infinite crisis that they were watching the main universe from their dimension they didn't like how the heroes were going about problems so they left their dimension to "fix" and bring about a perfect universe.Long story short superboy punch reality and changed it.
Hypertime was then used again to explain the arrival of a multiverse,and dc changed again from a universe to 52 universes and to this day it has been still 52 universes.
What has changed though is an existence of a second multiverse called the dark multiverse which has been something that is being told in dc stories as of the date of this article.
Hypertime has made a recent comeback.And with it an explanation.With the fact that dc is sticking to its 52 universes as the main multiverse then how does hypertime fit in.Well remember that hypertime is the existence of an infinite branch of parallel timelines branching from the main timeline.So the 52 universes have branches of infinite different timelines.Take the 52 universes as a multiverse and then imagine that as the main timeline and then everything is explained and fits together well.Dc seems to be making a distinction between the mutliverse,i.e the 52 universes and hypertime which in principle they are very similar the only distinction is that hypertime some timelines can merge with the main timeline so in essence every story that has been told is canon.That is what dc is trying to get at with the start.Not that they are infinite earths in the multiverse but infinite different ways of reality in hypertime.
But in conclusion infinite universes.
In essence Dc has infinite universes.
Hypertime explained by Grant morrison.