Subgroups — C₅×C₇×C₁₇⋊AGL(1,𝔽₇) #38735940

Order 24990.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₇×D₁₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₇⋊AGL(1,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₅×C₁₇⋊AGL(1,𝔽₇)	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₅×C₇×D₁₇	—	7	1	no	maximal
7	C₇×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅×C₇×C₁₇	—	7	1	no	—
9	C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₇	1	1	yes	characteristic, complemented
10	C₇×D₁₁₉	C₃×C₅	1	1	yes	characteristic, complemented, Hall
11	C₅×C₇×AGL(1,𝔽₇)	—	17	1	no	maximal, Hall
12	C₅×D₁₁₉	C₃×C₇	1	1	yes	characteristic, complemented
13	C₅×C₇×D₁₇	—	7	1	no	—
14	C₇²×C₁₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	C₅×C₇×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented, Hall
16	C₃×C₇×D₁₇	—	7	1	no	—
17	C₁₇⋊AGL(1,𝔽₇)	C₅×C₇	1	1	yes	characteristic, direct product factor
18	C₅×C₇×C₁₇	—	6	1	no	—
19	C₅×C₇×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
20	C₅×C₇×C₁₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
21	C₃×C₅×D₁₇	—	7	1	no	Hall
22	C₅×C₇×D₇	—	17	1	no	Hall
23	C₃×C₇×C₁₇	—	7	1	no	—
24	C₁₇×ΓL(1,𝔽₈)	C₂×C₅×C₇	1	1	yes	characteristic, complemented
25	C₇×AGL(1,𝔽₇)	—	17	1	no	Hall
26	C₃×C₅×C₁₇	—	7	1	no	Hall
27	C₅×C₇²	C₃×D₁₇	1	1	yes	characteristic, complemented, Hall
28	D₁₁₉	C₃×C₅×C₇	1	1	yes	characteristic, complemented
29	C₇×D₁₇	—	7	1	no	—
30	C₂×C₃×C₅×C₇	—	119	1	no	—
31	C₅×AGL(1,𝔽₇)	—	17	1	no	—
32	C₅×D₁₇	—	7	1	no	Hall
33	C₇×ΓL(1,𝔽₈)	C₅×D₁₇	1	1	yes	characteristic, complemented, Hall
34	C₇×C₁₇	—	6	1	no	—
35	C₇×C₁₇	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
36	C₇×C₁₇	C₂×C₃×C₅×C₇	1	1	yes	characteristic, complemented
37	C₃×C₅×C₇	—	7	1	no	—
38	C₅×ΓL(1,𝔽₈)	C₇×D₁₇	1	1	yes	characteristic, complemented
39	C₃×D₁₇	—	7	1	no	Hall
40	C₇×D₇	—	17	1	no	Hall
41	C₅×C₁₇	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
42	C₂×C₅×C₇	—	119	1	no	—
43	C₅×D₇	—	17	1	no	—
44	C₃×C₁₇	—	7	1	no	Hall
45	C₇²	C₃×C₅×D₁₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×C₇	—	119	1	no	—
47	AGL(1,𝔽₇)	—	17	1	no	—
48	C₅×C₇	—	6	1	no	—
49	C₅×C₇	C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
50	C₅×C₇	C₃×C₇×D₁₇	1	1	yes	characteristic, complemented
51	D₁₇	—	7	1	no	Hall
52	C₂×C₃×C₅	—	119	1	no	Hall
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	C₅×C₇×D₁₇	1	1	yes	characteristic, complemented
55	C₁₇	C₅×C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₅	—	7	1	no	Hall
57	C₂×C₇	—	119	1	no	—
58	D₇	—	17	1	no	—
59	C₂×C₅	—	119	1	no	Hall
60	C₇	—	6	1	no	—
61	C₇	C₃×C₅×C₇×D₁₇	1	1	yes	characteristic, complemented
62	C₇	C₅×C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
63	C₂×C₃	—	119	1	no	Hall
64	C₅	C₇×C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
65	C₃	—	7	1	no	Sylow
66	C₂	—	119	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₇	C₁₇⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×C₁₇	C₂×C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₇	C₂×C₃
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
2. C₅×C₇×D₁₁₉	C₃	semidirect
3. C₇×C₁₇⋊AGL(1,𝔽₇)	C₅	direct (N × H)
4. C₅×C₇²×C₁₇	C₂×C₃	semidirect
5. C₅×C₁₇⋊AGL(1,𝔽₇)	C₇	direct (N × H)
7. C₇×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	semidirect
9. C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₇	semidirect
10. C₇×D₁₁₉	C₃×C₅	semidirect
12. C₅×D₁₁₉	C₃×C₇	semidirect
14. C₇²×C₁₇	C₂×C₃×C₅	semidirect
15. C₅×C₇×ΓL(1,𝔽₈)	D₁₇	semidirect
17. C₁₇⋊AGL(1,𝔽₇)	C₅×C₇	direct (N × H)
19. C₅×C₇×C₁₇	AGL(1,𝔽₇)	semidirect
20. C₅×C₇×C₁₇	C₂×C₃×C₇	semidirect
24. C₁₇×ΓL(1,𝔽₈)	C₂×C₅×C₇	semidirect
27. C₅×C₇²	C₃×D₁₇	semidirect
28. D₁₁₉	C₃×C₅×C₇	semidirect
33. C₇×ΓL(1,𝔽₈)	C₅×D₁₇	semidirect
35. C₇×C₁₇	C₅×AGL(1,𝔽₇)	semidirect
36. C₇×C₁₇	C₂×C₃×C₅×C₇	semidirect
38. C₅×ΓL(1,𝔽₈)	C₇×D₁₇	semidirect
41. C₅×C₁₇	C₇×AGL(1,𝔽₇)	semidirect
45. C₇²	C₃×C₅×D₁₇	semidirect
49. C₅×C₇	C₁₇⋊AGL(1,𝔽₇)	direct (N × H)
50. C₅×C₇	C₃×C₇×D₁₇	semidirect
54. ΓL(1,𝔽₈)	C₅×C₇×D₁₇	semidirect
55. C₁₇	C₅×C₇×AGL(1,𝔽₇)	semidirect
61. C₇	C₃×C₅×C₇×D₁₇	semidirect
62. C₇	C₅×C₁₇⋊AGL(1,𝔽₇)	direct (N × H)
64. C₅	C₇×C₁₇⋊AGL(1,𝔽₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₇×C₁₇×ΓL(1,𝔽₈)	2
C₅×C₇×D₁₁₉	3
C₇×C₁₇⋊AGL(1,𝔽₇)	5
C₅×C₁₇⋊AGL(1,𝔽₇)	7
C₃×C₅×C₇×D₁₇	7
C₅×C₇×AGL(1,𝔽₇)	17

Subgroups — C5×C7×C17⋊AGL(1,𝔽7) #38735940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×C₁₇⋊AGL(1,𝔽₇) #38735940