Subgroups — D₅×AGL(1,𝔽₇)×C₇₁ #44447207

Order 29820.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×C₅⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×AGL(1,𝔽₇)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₇×C₇₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅×ΓL(1,𝔽₈)×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×AGL(1,𝔽₇)×C₇₁	—	5	1	no	maximal, Hall
7	D₃₅×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₅×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×D₇×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₅×C₇₁	—	7	1	no	maximal, Hall
11	AGL(1,𝔽₇)×C₇₁	—	5	1	no	—
12	AGL(1,𝔽₇)×C₇₁	D₅	1	1	yes	characteristic, direct product factor
13	C₂×ΓL(1,𝔽₈)×C₇₁	—	5	1	no	—
14	C₅×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	C₃×D₅×C₇₁	—	7	1	no	—
16	C₃×D₅×C₇₁	—	7	1	no	—
17	C₂×C₃×C₅×C₇₁	—	7	1	no	—
18	C₂×D₇×C₇₁	—	5	1	no	Hall
19	ΓL(1,𝔽₈)×C₇₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
20	C₂×D₅×C₇₁	—	7	1	no	Hall
21	C₃×C₅×C₇₁	—	7	1	no	Hall
22	C₂×C₇×C₇₁	—	5	1	no	—
23	D₇×C₇₁	—	5	1	no	—
24	D₇×C₇₁	C₃×D₅	1	1	yes	characteristic, complemented
25	C₂²×C₃×C₇₁	—	35	1	no	Hall
26	C₂×C₅×C₇₁	—	7	1	no	—
27	D₅×C₇₁	—	7	1	no	—
28	D₅×C₇₁	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	C₇×C₇₁	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₇₁	—	7	1	no	—
31	C₂×C₃×C₇₁	—	35	1	no	—
32	C₂×C₃×C₇₁	—	35	1	no	—
33	D₅×AGL(1,𝔽₇)	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
34	C₅×C₇₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₇₁	—	35	1	no	Hall
36	C₃×C₇₁	—	7	1	no	Hall
37	C₅⋊AGL(1,𝔽₇)	C₂×C₇₁	1	1	yes	characteristic, complemented
38	D₅×ΓL(1,𝔽₈)	C₂×C₇₁	1	1	yes	characteristic, complemented
39	C₅×AGL(1,𝔽₇)	C₂×C₇₁	1	1	yes	characteristic, complemented
40	C₂×C₇₁	—	5	1	no	—
41	C₂×C₇₁	—	7	1	no	—
42	C₂×C₇₁	—	35	1	no	—
43	D₅×D₇	C₃×C₇₁	1	1	yes	characteristic, complemented, Hall
44	C₅×ΓL(1,𝔽₈)	C₂²×C₇₁	1	1	yes	characteristic, complemented, Hall
45	C₂×AGL(1,𝔽₇)	—	5	1	no	Hall
46	C₇₁	D₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
47	D₃₅	C₂×C₃×C₇₁	1	1	yes	characteristic, complemented
48	C₅×D₇	C₂×C₃×C₇₁	1	1	yes	characteristic, complemented
49	C₇×D₅	C₂×C₃×C₇₁	1	1	yes	characteristic, complemented
50	C₂×C₃×D₅	—	7	1	no	Hall
51	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
52	AGL(1,𝔽₇)	—	5	1	no	—
53	AGL(1,𝔽₇)	D₅×C₇₁	1	1	yes	characteristic, direct product factor
54	C₅×C₇	C₂²×C₃×C₇₁	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	—	7	1	no	—
56	C₃×D₅	—	7	1	no	—
57	C₃×D₅	—	7	1	no	—
58	C₂×D₇	—	5	1	no	Hall
59	ΓL(1,𝔽₈)	C₂×D₅×C₇₁	1	1	yes	characteristic, complemented, Hall
60	C₂×D₅	—	7	1	no	Hall
61	C₃×C₅	—	7	1	no	Hall
62	C₂×C₇	—	5	1	no	—
63	D₇	—	5	1	no	—
64	D₇	C₃×D₅×C₇₁	1	1	yes	characteristic, complemented
65	C₂²×C₃	—	35	1	no	Hall
66	C₂×C₅	—	7	1	no	—
67	D₅	—	7	1	no	—
68	D₅	AGL(1,𝔽₇)×C₇₁	1	1	yes	characteristic, direct product factor
69	C₇	C₂×C₃×D₅×C₇₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	1	no	—
71	C₂×C₃	—	35	1	no	—
72	C₂×C₃	—	35	1	no	—
73	C₅	C₂×AGL(1,𝔽₇)×C₇₁	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	35	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	5	1	no	—
77	C₂	—	7	1	no	—
78	C₂	—	35	1	no	—

	subgroup	G/H
center, hypercenter	C₇₁	D₅×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₅×C₇	C₂²×C₃×C₇₁
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₇₁	C₂²×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₇₁

Complements

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

N	H ≅ G/N	type
1. C₇₁×C₅⋊AGL(1,𝔽₇)	C₂	semidirect
2. D₅×ΓL(1,𝔽₈)×C₇₁	C₂	semidirect
3. C₅×AGL(1,𝔽₇)×C₇₁	C₂	semidirect
4. D₅×D₇×C₇₁	C₃	semidirect
5. C₅×ΓL(1,𝔽₈)×C₇₁	C₂²	semidirect
7. D₃₅×C₇₁	C₂×C₃	semidirect
8. C₇×D₅×C₇₁	C₂×C₃	semidirect
9. C₅×D₇×C₇₁	C₂×C₃	semidirect
12. AGL(1,𝔽₇)×C₇₁	D₅	direct (N × H)
14. C₅×C₇×C₇₁	C₂²×C₃	semidirect
19. ΓL(1,𝔽₈)×C₇₁	C₂×D₅	semidirect
24. D₇×C₇₁	C₃×D₅	semidirect
28. D₅×C₇₁	AGL(1,𝔽₇)	direct (N × H)
29. C₇×C₇₁	C₂×C₃×D₅	semidirect
33. D₅×AGL(1,𝔽₇)	C₇₁	direct (N × H)
34. C₅×C₇₁	C₂×AGL(1,𝔽₇)	semidirect
37. C₅⋊AGL(1,𝔽₇)	C₂×C₇₁	semidirect
38. D₅×ΓL(1,𝔽₈)	C₂×C₇₁	semidirect
39. C₅×AGL(1,𝔽₇)	C₂×C₇₁	semidirect
43. D₅×D₇	C₃×C₇₁	semidirect
44. C₅×ΓL(1,𝔽₈)	C₂²×C₇₁	semidirect
46. C₇₁	D₅×AGL(1,𝔽₇)	direct (N × H)
47. D₃₅	C₂×C₃×C₇₁	semidirect
48. C₅×D₇	C₂×C₃×C₇₁	semidirect
49. C₇×D₅	C₂×C₃×C₇₁	semidirect
53. AGL(1,𝔽₇)	D₅×C₇₁	direct (N × H)
54. C₅×C₇	C₂²×C₃×C₇₁	semidirect
59. ΓL(1,𝔽₈)	C₂×D₅×C₇₁	semidirect
64. D₇	C₃×D₅×C₇₁	semidirect
68. D₅	AGL(1,𝔽₇)×C₇₁	direct (N × H)
69. C₇	C₂×C₃×D₅×C₇₁	semidirect
73. C₅	C₂×AGL(1,𝔽₇)×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×C₅⋊AGL(1,𝔽₇)	2
D₅×ΓL(1,𝔽₈)×C₇₁	2
C₅×AGL(1,𝔽₇)×C₇₁	2
D₅×D₇×C₇₁	3
C₂×AGL(1,𝔽₇)×C₇₁	5
C₂×C₃×D₅×C₇₁	7
D₅×AGL(1,𝔽₇)	71

Subgroups — D5×AGL(1,𝔽7)×C71 #44447207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×AGL(1,𝔽₇)×C₇₁ #44447207