Subgroups — C₁₁×C₁₇×C₃⋊AGL(1,𝔽₇) #37555639

Order 23562.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×C₁₇×AGL(1,𝔽₇)	—	3	1	no	maximal
3	C₁₁×C₁₇×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₁×C₁₇×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₁₁×C₁₇×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇×C₁₁×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×D₃×C₁₁×C₁₇	—	7	1	no	maximal, Hall
8	C₁₁×D₇×C₁₇	—	3	1	no	Hall
9	C₁₇×C₃⋊AGL(1,𝔽₇)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃²×C₁₁×C₁₇	—	7	1	no	Hall
11	C₁₁×C₃⋊AGL(1,𝔽₇)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₇×C₁₁×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₁₁×C₁₇	—	21	1	no	—
14	D₃×C₁₁×C₁₇	—	7	1	no	—
15	C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₁	1	1	yes	characteristic, complemented, Hall
16	C₁₇×D₂₁	C₃×C₁₁	1	1	yes	characteristic, complemented
17	C₁₇×AGL(1,𝔽₇)	—	3	1	no	—
18	C₃×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
19	C₃×C₁₁×C₁₇	—	7	1	no	—
20	C₃×C₁₁×C₁₇	—	14	1	no	—
21	C₃×C₁₁×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₁₁×D₂₁	C₃×C₁₇	1	1	yes	characteristic, complemented
23	C₁₁×AGL(1,𝔽₇)	—	3	1	no	—
24	C₂×C₁₁×C₁₇	—	21	1	no	Hall
25	C₃×C₇×C₁₇	C₂×C₃×C₁₁	1	1	yes	characteristic, complemented
26	C₁₇×ΓL(1,𝔽₈)	—	2	1	no	—
27	C₁₇×ΓL(1,𝔽₈)	D₃×C₁₁	1	1	yes	characteristic, complemented
28	C₃×D₃×C₁₇	—	7	1	no	Hall
29	D₇×C₁₇	—	3	1	no	Hall
30	C₃×C₇×C₁₁	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
31	C₁₁×ΓL(1,𝔽₈)	—	2	1	no	—
32	C₁₁×ΓL(1,𝔽₈)	D₃×C₁₇	1	1	yes	characteristic, complemented
33	C₃×D₃×C₁₁	—	7	1	no	Hall
34	C₁₁×C₁₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Hall
35	C₁₁×D₇	—	3	1	no	Hall
36	C₃²×C₁₇	—	7	1	no	Hall
37	C₃⋊AGL(1,𝔽₇)	C₁₁×C₁₇	1	1	yes	characteristic, direct product factor, Hall
38	C₇×C₁₇	C₃×D₃×C₁₁	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₁₇	—	21	1	no	—
40	D₃×C₁₇	—	7	1	no	—
41	C₃²×C₁₁	—	7	1	no	Hall
42	C₇×C₁₁	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Hall
43	C₂×C₃×C₁₁	—	21	1	no	—
44	D₃×C₁₁	—	7	1	no	—
45	C₃×ΓL(1,𝔽₈)	C₂×C₁₁×C₁₇	1	1	yes	characteristic, complemented, Hall
46	C₃×C₁₇	—	7	1	no	—
47	C₃×C₁₇	—	14	1	no	—
48	C₃×C₁₇	C₁₁×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
49	D₂₁	C₃×C₁₁×C₁₇	1	1	yes	characteristic, complemented
50	AGL(1,𝔽₇)	—	3	1	no	—
51	C₂×C₁₇	—	21	1	no	Hall
52	C₃×C₁₁	—	7	1	no	—
53	C₃×C₁₁	—	14	1	no	—
54	C₃×C₁₁	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
55	C₂×C₁₁	—	21	1	no	Hall
56	C₃×C₇	C₂×C₃×C₁₁×C₁₇	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	2	1	no	—
58	ΓL(1,𝔽₈)	D₃×C₁₁×C₁₇	1	1	yes	characteristic, complemented
59	C₃×D₃	—	7	1	no	Hall
60	C₁₇	C₁₁×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
61	D₇	—	3	1	no	Hall
62	C₁₁	C₁₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
63	C₃²	—	7	1	no	Sylow
64	C₇	C₃×D₃×C₁₁×C₁₇	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	21	1	no	—
66	D₃	—	7	1	no	—
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₁₁×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
70	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁×C₁₇	C₃⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₁₁×C₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₁×C₁₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₁₁×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
3. C₁₁×C₁₇×D₂₁	C₃	semidirect
5. C₁₁×C₁₇×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇×C₁₁×C₁₇	C₂×C₃	semidirect
9. C₁₇×C₃⋊AGL(1,𝔽₇)	C₁₁	direct (N × H)
11. C₁₁×C₃⋊AGL(1,𝔽₇)	C₁₇	direct (N × H)
12. C₇×C₁₁×C₁₇	C₃×D₃	semidirect
15. C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₁	semidirect
16. C₁₇×D₂₁	C₃×C₁₁	semidirect
18. C₃×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₇	semidirect
21. C₃×C₁₁×C₁₇	AGL(1,𝔽₇)	semidirect
22. C₁₁×D₂₁	C₃×C₁₇	semidirect
25. C₃×C₇×C₁₇	C₂×C₃×C₁₁	semidirect
27. C₁₇×ΓL(1,𝔽₈)	D₃×C₁₁	semidirect
30. C₃×C₇×C₁₁	C₂×C₃×C₁₇	semidirect
32. C₁₁×ΓL(1,𝔽₈)	D₃×C₁₇	semidirect
34. C₁₁×C₁₇	C₃⋊AGL(1,𝔽₇)	direct (N × H)
37. C₃⋊AGL(1,𝔽₇)	C₁₁×C₁₇	direct (N × H)
38. C₇×C₁₇	C₃×D₃×C₁₁	semidirect
42. C₇×C₁₁	C₃×D₃×C₁₇	semidirect
45. C₃×ΓL(1,𝔽₈)	C₂×C₁₁×C₁₇	semidirect
48. C₃×C₁₇	C₁₁×AGL(1,𝔽₇)	semidirect
49. D₂₁	C₃×C₁₁×C₁₇	semidirect
54. C₃×C₁₁	C₁₇×AGL(1,𝔽₇)	semidirect
56. C₃×C₇	C₂×C₃×C₁₁×C₁₇	semidirect
58. ΓL(1,𝔽₈)	D₃×C₁₁×C₁₇	semidirect
60. C₁₇	C₁₁×C₃⋊AGL(1,𝔽₇)	direct (N × H)
62. C₁₁	C₁₇×C₃⋊AGL(1,𝔽₇)	direct (N × H)
64. C₇	C₃×D₃×C₁₁×C₁₇	semidirect
69. C₃	C₁₁×C₁₇×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₁×C₁₇×ΓL(1,𝔽₈)	2
C₁₁×C₁₇×AGL(1,𝔽₇)	3
C₁₁×C₁₇×D₂₁	3
C₃×D₃×C₁₁×C₁₇	7
C₁₇×C₃⋊AGL(1,𝔽₇)	11
C₁₁×C₃⋊AGL(1,𝔽₇)	17

Subgroups — C11×C17×C3⋊AGL(1,𝔽7) #37555639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×C₁₇×C₃⋊AGL(1,𝔽₇) #37555639