Subgroups — C₁₁×#15668765 #44568659

Order 32340.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₁₁×#3861	C₂	1	1	yes	maximal, characteristic
2	C₁₁×C₇⋊(C₇⋊AGL(1,𝔽₅))	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₁×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₁×#10596874	—	5	1	no	maximal, Hall
5	C₇²×D₅×C₁₁	C₂×C₃	1	1	yes	characteristic
6	C₁₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	7	8	no	maximal
7	C₂×C₁₁×#3861	—	5	1	no	—
8	#15668765	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇²×C₁₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	D₅×C₁₁×ΓL(1,𝔽₈)	—	7	8	no	—
11	C₁₁×C₇⋊Dic₇	—	5	1	no	Hall
12	C₁₁×#3861	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₁₁×C₇⋊AGL(1,𝔽₅)	—	7	8	no	—
14	D₅×#3861	C₂×C₁₁	1	1	yes	characteristic
15	C₅×C₁₁×ΓL(1,𝔽₈)	—	7	8	no	—
16	C₂×C₇²×C₁₁	—	5	1	no	—
17	C₇⋊(C₇⋊AGL(1,𝔽₅))	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
18	C₁₁×ΓL(1,𝔽₈)⋊C₄	—	35	8	no	—
19	C₇×D₅×C₁₁	AGL(1,𝔽₇)	1	8	yes	—
20	C₅×#3861	C₄×C₁₁	1	1	yes	characteristic, complemented, Hall
21	C₃×C₁₁×AGL(1,𝔽₅)	—	49	1	no	Hall
22	#10596874	—	5	1	no	Hall
23	C₇²×C₁₁	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
24	C₇²×D₅	C₂×C₃×C₁₁	1	1	yes	characteristic
25	C₂×C₁₁×ΓL(1,𝔽₈)	—	35	8	no	—
26	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	7	8	no	—
27	C₅×C₇×C₁₁	ΓL(1,𝔽₈)⋊C₄	1	8	yes	complemented
28	C₃×D₅×C₁₁	—	49	1	no	—
29	C₁₁×Dic₇	—	35	8	no	—
30	C₂×#3861	—	5	1	no	—
31	C₅×C₇²	C₃×C₄×C₁₁	1	1	yes	characteristic, complemented, Hall
32	C₁₁×ΓL(1,𝔽₈)	—	7	8	no	—
33	C₁₁×AGL(1,𝔽₅)	—	49	1	no	Hall
34	D₅×ΓL(1,𝔽₈)	—	7	8	no	—
35	C₇⋊Dic₇	—	5	1	no	Hall
36	C₃×C₅×C₁₁	—	49	1	no	Hall
37	C₂×C₇×C₁₁	—	5	8	no	—
38	#3861	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
39	C₇⋊AGL(1,𝔽₅)	—	7	8	no	—
40	C₃×C₄×C₁₁	—	245	1	no	Hall
41	D₅×C₁₁	#64424	1	1	yes	characteristic
42	C₅×ΓL(1,𝔽₈)	—	7	8	no	—
43	C₂×C₇²	—	5	1	no	—
44	ΓL(1,𝔽₈)⋊C₄	—	35	8	no	—
45	C₇×C₁₁	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	8	yes	complemented
46	C₇×D₅	C₁₁×AGL(1,𝔽₇)	1	8	yes	—
47	C₂×C₃×C₁₁	—	245	1	no	—
48	C₃×AGL(1,𝔽₅)	—	49	1	no	Hall
49	C₅×C₁₁	#10596874	1	1	yes	characteristic, complemented, Hall
50	C₇²	C₃×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
51	C₄×C₁₁	—	245	1	no	Hall
52	C₂×ΓL(1,𝔽₈)	—	35	8	no	—
53	C₅×C₇	C₁₁×ΓL(1,𝔽₈)⋊C₄	1	8	yes	complemented
54	C₃×C₁₁	—	49	1	no	Hall
55	C₃×D₅	—	49	1	no	—
56	Dic₇	—	35	8	no	—
57	C₂×C₁₁	—	5	1	no	—
58	ΓL(1,𝔽₈)	—	7	8	no	—
59	AGL(1,𝔽₅)	—	49	1	no	Hall
60	C₃×C₅	—	49	1	no	Hall
61	C₂×C₇	—	5	8	no	—
62	C₃×C₄	—	245	1	no	Hall
63	C₁₁	#15668765	1	1	yes	characteristic, direct product factor, Sylow
64	D₅	C₁₁×#64424	1	1	yes	characteristic
65	C₇	C₁₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	8	yes	complemented
66	C₂×C₃	—	245	1	no	—
67	C₅	C₁₁×#10596874	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	245	1	no	Sylow
69	C₃	—	49	1	no	Sylow
70	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁	#15668765
commutator, hypocenter	C₅×C₇²	C₃×C₄×C₁₁
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₁	C₃×C₄
prime radical	D₅×C₁₁×#3861	C₂
cosocle	C₇²×D₅	C₂×C₃×C₁₁
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₁₁

Complements

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

N	H ≅ G/N	type
2. C₁₁×C₇⋊(C₇⋊AGL(1,𝔽₅))	C₃	semidirect
3. C₅×C₁₁×#3861	C₄	semidirect
8. #15668765	C₁₁	direct (N × H)
9. C₅×C₇²×C₁₁	C₃×C₄	semidirect
12. C₁₁×#3861	AGL(1,𝔽₅)	semidirect
17. C₇⋊(C₇⋊AGL(1,𝔽₅))	C₃×C₁₁	semidirect
20. C₅×#3861	C₄×C₁₁	semidirect
23. C₇²×C₁₁	C₃×AGL(1,𝔽₅)	semidirect
27. C₅×C₇×C₁₁	ΓL(1,𝔽₈)⋊C₄	semidirect
31. C₅×C₇²	C₃×C₄×C₁₁	semidirect
38. #3861	C₁₁×AGL(1,𝔽₅)	semidirect
45. C₇×C₁₁	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	semidirect
49. C₅×C₁₁	#10596874	semidirect
50. C₇²	C₃×C₁₁×AGL(1,𝔽₅)	semidirect
53. C₅×C₇	C₁₁×ΓL(1,𝔽₈)⋊C₄	semidirect
63. C₁₁	#15668765	direct (N × H)
65. C₇	C₁₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	semidirect
67. C₅	C₁₁×#10596874	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₁₁×#3861	2
C₁₁×C₇⋊(C₇⋊AGL(1,𝔽₅))	3
C₁₁×#10596874	5
C₁₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	7
#15668765	11

Subgroups — C11×#15668765 #44568659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#15668765 #44568659