Subgroups — C₁₁×SL(2,𝔽₅) #13050310

Order 1320.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
2	C₁₁×Dic₅	—	6	1	no	maximal
3	C₁₁×Dic₃	—	10	1	no	maximal
4	SL(2,𝔽₅)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₁₁	—	6	1	no	—
6	Q₈×C₁₁	—	5	1	no	Hall
7	C₂×C₃×C₁₁	—	10	1	no	—
8	C₅×C₁₁	—	6	1	no	Hall
9	C₄×C₁₁	—	15	1	no	—
10	C₃×C₁₁	—	10	1	no	Hall
11	SL(2,𝔽₃)	—	5	1	no	Hall
12	C₂×C₁₁	A₅	1	1	yes	maximal normal, characteristic
13	Dic₅	—	6	1	no	—
14	Dic₃	—	10	1	no	—
15	C₁₁	SL(2,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₅	—	6	1	no	—
17	Q₈	—	5	1	no	Sylow
18	C₂×C₃	—	10	1	no	—
19	C₅	—	6	1	no	Sylow
20	C₄	—	15	1	no	—
21	C₃	—	10	1	no	Sylow
22	C₂	C₁₁×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₁₁	A₅
commutator, hypocenter	SL(2,𝔽₅)	C₁₁
Frattini, cosocle	C₂	C₁₁×A₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₁₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. SL(2,𝔽₅)	C₁₁	direct (N × H)
15. C₁₁	SL(2,𝔽₅)	direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.