Quotients of a composition series of a group G

Question

Let $G$ be a group (finite or otherwise) and $(H_i)_{0\leq i\leq n}$ a composition series of $G$. Is it possible to have $$H_i/H_{i+1}\simeq H_j/H_{j+1}$$ for $i\ne j$? In general, do the quotients of a composition series have any noteworthy relation to each other?

Answer 1

Of course you can have distinct factors that are isomorphic.

Just consider the cyclic group of four elements $G=\{1,c,c^2,c^3\}$ and

$\{1\}\subseteq \{1, c^2\}\subseteq G$.

The factors do not have any deep connection to each other.

For any finite indexed collection of simple groups $G_i$, you can find all of them as different composition factors of $\prod_{i=1}^n G_i$