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?