Subgroups — C₁₇×D₁₁×AGL(1,𝔽₇) #30628107

Order 15708.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₁₁⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₁×C₁₇×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×ΓL(1,𝔽₈)×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×C₁₇×D₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₇×D₇₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₁₁×D₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₇×D₁₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×C₁₇×D₁₁	—	7	1	no	maximal, Hall
10	C₂×C₁₇×AGL(1,𝔽₇)	—	11	1	no	maximal, Hall
11	C₇×C₁₁×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₁₁×C₁₇	—	7	1	no	—
13	C₃×C₁₇×D₁₁	—	7	1	no	—
14	C₃×C₁₇×D₁₁	—	7	1	no	—
15	D₁₁×AGL(1,𝔽₇)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₁₇×D₁₁	—	7	1	no	Hall
17	C₂×C₁₇×ΓL(1,𝔽₈)	—	11	1	no	—
18	C₁₇×AGL(1,𝔽₇)	—	11	1	no	—
19	C₁₇×AGL(1,𝔽₇)	D₁₁	1	1	yes	characteristic, direct product factor
20	C₃×C₁₁×C₁₇	—	7	1	no	Hall
21	C₂×D₇×C₁₇	—	11	1	no	Hall
22	C₁₁⋊AGL(1,𝔽₇)	C₂×C₁₇	1	1	yes	characteristic, complemented
23	C₁₁×AGL(1,𝔽₇)	C₂×C₁₇	1	1	yes	characteristic, complemented
24	ΓL(1,𝔽₈)×D₁₁	C₂×C₁₇	1	1	yes	characteristic, complemented
25	C₂×C₁₁×C₁₇	—	7	1	no	—
26	C₁₇×D₁₁	—	7	1	no	—
27	C₁₇×D₁₁	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
28	C₁₇×ΓL(1,𝔽₈)	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
29	D₇×D₁₁	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₇×C₁₇	—	11	1	no	—
31	D₇×C₁₇	—	11	1	no	—
32	D₇×C₁₇	C₃×D₁₁	1	1	yes	characteristic, complemented
33	C₁₁×ΓL(1,𝔽₈)	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
34	C₂²×C₃×C₁₇	—	77	1	no	Hall
35	C₁₁×C₁₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
36	D₇₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
37	C₇×D₁₁	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
38	C₁₁×D₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
39	C₂×C₃×D₁₁	—	7	1	no	Hall
40	C₇×C₁₇	C₂×C₃×D₁₁	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₁₇	—	7	1	no	—
42	C₂×C₃×C₁₇	—	77	1	no	—
43	C₂×C₃×C₁₇	—	77	1	no	—
44	C₂×AGL(1,𝔽₇)	—	11	1	no	Hall
45	C₇×C₁₁	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
46	C₂²×C₁₇	—	77	1	no	Hall
47	C₂×C₃×C₁₁	—	7	1	no	—
48	C₃×D₁₁	—	7	1	no	—
49	C₃×D₁₁	—	7	1	no	—
50	C₃×C₁₇	—	7	1	no	Hall
51	C₂×D₁₁	—	7	1	no	Hall
52	C₂×ΓL(1,𝔽₈)	—	11	1	no	—
53	AGL(1,𝔽₇)	—	11	1	no	—
54	AGL(1,𝔽₇)	C₁₇×D₁₁	1	1	yes	characteristic, direct product factor
55	C₂×C₁₇	—	7	1	no	—
56	C₂×C₁₇	—	11	1	no	—
57	C₂×C₁₇	—	77	1	no	—
58	C₃×C₁₁	—	7	1	no	Hall
59	C₂×D₇	—	11	1	no	Hall
60	C₂×C₁₁	—	7	1	no	—
61	D₁₁	—	7	1	no	—
62	D₁₁	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
63	ΓL(1,𝔽₈)	C₂×C₁₇×D₁₁	1	1	yes	characteristic, complemented, Hall
64	C₁₇	D₁₁×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₇	—	11	1	no	—
66	D₇	—	11	1	no	—
67	D₇	C₃×C₁₇×D₁₁	1	1	yes	characteristic, complemented
68	C₂²×C₃	—	77	1	no	Hall
69	C₁₁	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
70	C₇	C₂×C₃×C₁₇×D₁₁	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	1	no	—
72	C₂×C₃	—	77	1	no	—
73	C₂×C₃	—	77	1	no	—
74	C₂²	—	77	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	7	1	no	—
77	C₂	—	11	1	no	—
78	C₂	—	77	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	D₁₁×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×C₁₁	C₂²×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₁×C₁₇	C₂²×C₃
LCS	G ▷ C₇×C₁₁
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₁₁⋊AGL(1,𝔽₇)	C₂	semidirect
2. C₁₁×C₁₇×AGL(1,𝔽₇)	C₂	semidirect
3. C₁₇×ΓL(1,𝔽₈)×D₁₁	C₂	semidirect
4. D₇×C₁₇×D₁₁	C₃	semidirect
5. C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
6. C₁₇×D₇₇	C₂×C₃	semidirect
7. C₁₁×D₇×C₁₇	C₂×C₃	semidirect
8. C₇×C₁₇×D₁₁	C₂×C₃	semidirect
11. C₇×C₁₁×C₁₇	C₂²×C₃	semidirect
15. D₁₁×AGL(1,𝔽₇)	C₁₇	direct (N × H)
19. C₁₇×AGL(1,𝔽₇)	D₁₁	direct (N × H)
22. C₁₁⋊AGL(1,𝔽₇)	C₂×C₁₇	semidirect
23. C₁₁×AGL(1,𝔽₇)	C₂×C₁₇	semidirect
24. ΓL(1,𝔽₈)×D₁₁	C₂×C₁₇	semidirect
27. C₁₇×D₁₁	AGL(1,𝔽₇)	direct (N × H)
28. C₁₇×ΓL(1,𝔽₈)	C₂×D₁₁	semidirect
29. D₇×D₁₁	C₃×C₁₇	semidirect
32. D₇×C₁₇	C₃×D₁₁	semidirect
33. C₁₁×ΓL(1,𝔽₈)	C₂²×C₁₇	semidirect
35. C₁₁×C₁₇	C₂×AGL(1,𝔽₇)	semidirect
36. D₇₇	C₂×C₃×C₁₇	semidirect
37. C₇×D₁₁	C₂×C₃×C₁₇	semidirect
38. C₁₁×D₇	C₂×C₃×C₁₇	semidirect
40. C₇×C₁₇	C₂×C₃×D₁₁	semidirect
45. C₇×C₁₁	C₂²×C₃×C₁₇	semidirect
54. AGL(1,𝔽₇)	C₁₇×D₁₁	direct (N × H)
62. D₁₁	C₁₇×AGL(1,𝔽₇)	direct (N × H)
63. ΓL(1,𝔽₈)	C₂×C₁₇×D₁₁	semidirect
64. C₁₇	D₁₁×AGL(1,𝔽₇)	direct (N × H)
67. D₇	C₃×C₁₇×D₁₁	semidirect
69. C₁₁	C₂×C₁₇×AGL(1,𝔽₇)	semidirect
70. C₇	C₂×C₃×C₁₇×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₁₁⋊AGL(1,𝔽₇)	2
C₁₁×C₁₇×AGL(1,𝔽₇)	2
C₁₇×ΓL(1,𝔽₈)×D₁₁	2
D₇×C₁₇×D₁₁	3
C₂×C₃×C₁₇×D₁₁	7
C₂×C₁₇×AGL(1,𝔽₇)	11
D₁₁×AGL(1,𝔽₇)	17

Subgroups — C17×D11×AGL(1,𝔽7) #30628107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×D₁₁×AGL(1,𝔽₇) #30628107