TITLE: How is a group made up of simple groups?
QUESTION [94 upvotes]: I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers.
I've recently started self-studying subgroup series, which is supposed to explain the analogy, but I'm not completely sure I understand how "any group is made of simple groups".
Given a group $G$ with composition series $$ \{e\}=G_0 \triangleleft G_1\triangleleft \dots \triangleleft G_{r-1} \triangleleft G_r=G$$
then $G$ has associated the simple factor groups $H_{i+1}=G_{i+1}/G_i$. But how is it "built" from them?
Well, if we have those simple groups $H_i$ then we can say the subnormal subgroups in the composition series can be recovered by taking certain extensions of $H_i$: $$ 1 \to K_i \to G_i \to H_i \to 1$$
where $H_i = G_i/G_{i-1}$, $K_i\simeq G_{i-1}$.
Then $G$ is built from some uniquely determined (Jordan-Hölder) simple groups $H_i$ by taking extensions of these groups.
Is this description accurate?
The question now is: this description seems overly theoretical to me. I don't know how the extensions of $H_i$ look like, and I don't understand how $G$ puts these groups together. Can we describe more explicitly how a group $G$ is made of simple groups?
EDIT: I forgot a (not-so-tiny) detail. The previous explanation works for finite groups, or more in general for groups with a composition series. But what about groups which don't admit a composition series? Is it correct to say that they are built from simple groups?

REPLY [11 votes]: In Dummit & Foote's "Abstract Algebra" they briefly discuss the Hölder Program:


Classify all finite simple groups.
Find all ways of "putting simple groups together" to form other
  groups.


They write the following on part 2 of the program (the so-called extension problem for finite groups):

Part (2) of the Hölder Program, sometimes called the extension problem, was rather vaguely formulated. A more precise description of "putting two groups together" is: given groups $A$ and $B$, describe how to obtain all groups $G$ containing a normal subgroup $N$ such that $N \cong B$ and $G/N \cong A$. For instance, if $A=B=Z_2$, there are precisely two possibilities for $G$, namely $Z_4$ and $V_4$ [the Klein four grup] and the Hölder Program seeks to describe how the two groups of order 4 could have been built from two $Z_2$'s without a priori knowledge of the existence of the groups of order 4. This part of the Hölder Program is extremely difficult, even when the subgroups involved are of small order. For example, all composition factors of a group $G$ have order 2 if and only if $|G| = 2^n$, for some $n$ (...). It is known, however, that the number of nonisomorphic groups of order $2^n$ grows (exponentially) as a function of $2^n$, so the number of ways of putting groups of 2-power together is not bounded. Nonetheless, there are a wealth of interesting and powerful techniques in this subtle area which serve to unravel the structure of large classes of groups.