TITLE: Groups without factorization
QUESTION [8 upvotes]: A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \  | \  a \in A, b \in B \}$.    
The paper Factorisations of sporadic simple groups (by Michael Giudici) provides the classification of all the factorizations of the sporadic simple groups. We observe there that only $11$ sporadic simple groups (among $26$) admit a factorization. So the remaining $15$ (which are $J_1$, $M_{22}$, $J_3$, $McL$, $O'N$, $Co_3$, $Co_2$, $F_5$, $Ly$, $F_3$, $Fi_{23}$, $J_4$, $F_{3+}$, $F_2$, $F_1$) admit no factorization.  
Obviously, a cyclic group admits no factorization if and only if it is of prime power order.
Question: What are all the (other) finite groups without factorization?
(Is there an official name for such a group?)  
Proposition: Let $G$ be a group without factorization. If the intersection of all the maximal subgroups of $G$ is the trivial subgroup then $G$ is simple.
proof: Assume $G$ non-simple and let $N$ be a non-trivial proper normal subgroup of $G$. If every maximal subgroup of $G$ contains $N$ then their intersection also, which contradicts the assumption. So there is a maximal subgroup $M$ of $G$  not containing $N$. It follows that $NM=G$, contradiction. $\square$
Sub-question: must a non-simple finite group without factorization be cyclic (of prime power order)?
Lemma: A finite group $G$ admits a unique maximal subgroup $M$ iff it is cyclic of prime power order.
proof:  Let $g \in G \setminus M$ and $H = \langle g \rangle$. If $H \neq G$ then there must exist a maximal subgroup $M'$ of $G$ containing $H$, but $M=M'$ by assumption, so $g \in M$, contradiction. So $G = H$ is cyclic. It is moreover of prime power order by Chinese Remainder Theorem. $\square$
Bonus question: What do we know about the infinite groups without factorization?

REPLY [3 votes]: The smallest non-abelian group without factorization is simple of order $1092$: it is $A_1(13)$.
Using the answer of Geoff and by browsing the book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl), the finite simple groups without factorization are the following:  

Every cyclic group of prime order. 
Alternating: none
Chevalley groups: $A_n(q)$ with $n=1$, $q \equiv 1 (\textrm{mod}\ 4)$ and $q\neq 5,9,29$, or with $n \ge 2$ even and $(n+1) | (q-1)$; $E_6(q)$; $E_7(q)$; $E_8(q)$; $F_4(q)$ with $q \neq 2^n$; $G_2(q) $ with  $q>4 $ and $q \neq 3^n$.
Twisted Chevalley groups: $^2A_{2n}(q^2)$ except $(n,q) = (1,3),(1,5),(4,2)$; $^2B_2(2^{2n+1})$; $^2D_{2n}(q^2)$; $^3D_4(q^3)$; $^2E_6(q^2)$; $^2F_4(2^{2n+1})$; $^2G_2(3^{2n+1})$.    
Sporadic: $J_1$, $M_{22}$, $J_3$, $McL$, $O'N$, $Co_3$, $Co_2$, $F_5$, $Ly$, $F_3$, $Fi_{23}$, $J_4$, $F_{3+}$, $F_2$, $F_1$.

Before any use: this list requires a double checking.