Subgroups — C₁₁⋊(#1043⋊ΓL(1,𝔽₈)) #37528512

Order 23100.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₁⋊#1043	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₁×C₇⋊(C₅²⋊C₃)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₁₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
5	C₁₁⋊(#1043⋊C₃)	—	7	1	no	maximal, Hall
6	#1043⋊ΓL(1,𝔽₈)	—	11	1	no	maximal, Hall
7	C₅²×C₇×C₁₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₁₁×(C₅⋊D₅)⋊C₃	—	7	1	no	—
9	C₇×C₁₁⋊AGL(1,𝔽₅)	—	15	2	no	—
10	C₁₁⋊#1043	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
11	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	D₁₁	1	1	yes	characteristic
12	ΓL(1,𝔽₈)×Dic₁₁	—	25	1	no	maximal, Hall
13	C₁₁×C₅²⋊C₃	—	7	1	no	Hall
14	C₇×D₅×C₁₁	—	15	2	no	—
15	C₇×#1043	—	11	1	no	Hall
16	C₁₁×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₇⋊(C₅²⋊C₃)	Dic₁₁	1	1	yes	characteristic, complemented, Hall
18	C₂×C₁₁×ΓL(1,𝔽₈)	—	25	1	no	—
19	C₅×C₇×C₁₁	—	3	2	no	—
20	C₇×C₅⋊D₅	C₃×D₁₁	1	1	yes	characteristic
21	C₇×Dic₁₁	—	25	1	no	Hall
22	#1043⋊C₃	—	77	1	no	Hall
23	C₅²×C₁₁	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
24	C₁₁×ΓL(1,𝔽₈)	—	100	1	no	Hall
25	C₁₁⋊AGL(1,𝔽₅)	—	15	2	no	—
26	C₅²×C₇	C₃×Dic₁₁	1	1	yes	characteristic, complemented, Hall
27	C₂×C₇×C₁₁	—	25	1	no	—
28	(C₅⋊D₅)⋊C₃	—	7	1	no	—
29	C₇×AGL(1,𝔽₅)	—	165	2	no	—
30	C₃×Dic₁₁	—	175	1	no	Hall
31	D₅×C₁₁	—	15	2	no	—
32	#1043	—	11	1	no	Hall
33	C₄×ΓL(1,𝔽₈)	—	275	1	no	Hall
34	C₇×C₁₁	#1043⋊C₃	1	1	yes	characteristic, complemented, Hall
35	C₅²⋊C₃	—	7	1	no	Hall
36	C₇×D₅	—	15	2	no	—
37	C₂×C₃×C₁₁	—	175	1	no	—
38	C₅×C₁₁	—	3	2	no	—
39	C₅⋊D₅	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic
40	Dic₁₁	—	25	1	no	Hall
41	C₂×ΓL(1,𝔽₈)	—	25	1	no	—
42	C₅×C₇	—	3	2	no	—
43	C₃×C₁₁	—	700	1	no	Hall
44	C₄×C₇	—	275	1	no	Hall
45	C₅²	ΓL(1,𝔽₈)×Dic₁₁	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₁₁	—	25	1	no	—
47	ΓL(1,𝔽₈)	—	100	1	no	Hall
48	AGL(1,𝔽₅)	—	165	2	no	—
49	C₂×C₇	—	25	1	no	—
50	C₃×C₄	—	1925	1	no	Hall
51	C₁₁	#1043⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
52	D₅	—	15	2	no	—
53	C₇	C₁₁⋊(#1043⋊C₃)	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃	—	175	1	no	—
55	C₅	—	3	2	no	—
56	C₄	—	275	1	no	Sylow
57	C₃	—	700	1	no	Sylow
58	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₇×C₁₁	C₃×C₄
prime radical	C₁₁×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₂
cosocle	C₇×C₁₁×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₇×C₁₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₁₁⋊#1043	C₃	semidirect
3. C₁₁×C₇⋊(C₅²⋊C₃)	C₄	semidirect
7. C₅²×C₇×C₁₁	C₃×C₄	semidirect
10. C₁₁⋊#1043	ΓL(1,𝔽₈)	semidirect
17. C₇⋊(C₅²⋊C₃)	Dic₁₁	semidirect
23. C₅²×C₁₁	C₄×ΓL(1,𝔽₈)	semidirect
26. C₅²×C₇	C₃×Dic₁₁	semidirect
34. C₇×C₁₁	#1043⋊C₃	semidirect
45. C₅²	ΓL(1,𝔽₈)×Dic₁₁	semidirect
51. C₁₁	#1043⋊ΓL(1,𝔽₈)	semidirect
53. C₇	C₁₁⋊(#1043⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	2
C₇×C₁₁⋊#1043	3
C₁₁⋊(#1043⋊C₃)	7
#1043⋊ΓL(1,𝔽₈)	11
ΓL(1,𝔽₈)×Dic₁₁	25

Subgroups — C11⋊(#1043⋊ΓL(1,𝔽8)) #37528512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁⋊(#1043⋊ΓL(1,𝔽₈)) #37528512