Subgroups — C₂×C₅×ΓL(1,𝔽₈)⋊D₇₁ #44447221

Order 29820.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×ΓL(1,𝔽₈)⋊D₇₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₄₉₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×ΓL(1,𝔽₈)×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×ΓL(1,𝔽₈)⋊D₇₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×D₄₉₇	C₂×C₃	1	2	yes	complemented
7	C₂×C₅×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅×D₇₁	—	7	1	no	maximal, Hall
9	ΓL(1,𝔽₈)⋊D₇₁	C₂×C₅	1	2	yes	direct product factor
10	C₂×ΓL(1,𝔽₈)×C₇₁	C₂×C₅	1	1	yes	characteristic, complemented
11	C₅×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅×D₇₁	—	7	2	no	—
13	C₂×C₃×C₅×C₇₁	—	7	1	no	—
14	C₂×D₄₉₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	ΓL(1,𝔽₈)×C₇₁	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂×C₅×D₇₁	—	7	1	no	Hall
17	C₃×C₅×C₇₁	—	7	1	no	Hall
18	C₂×C₇×C₇₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	D₄₉₇	C₂×C₃×C₅	1	2	yes	complemented
20	C₂×C₃×D₇₁	—	7	1	no	Hall
21	C₂×C₅×C₇₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₅×D₇₁	—	7	2	no	—
23	C₇×C₇₁	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₇₁	—	7	1	no	—
25	C₃×D₇₁	—	7	2	no	—
26	C₂×C₅×AGL(1,𝔽₇)	—	71	1	no	maximal, Hall
27	C₅×C₇₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
28	C₂×D₇₁	—	7	1	no	Hall
29	C₃×C₇₁	—	7	1	no	Hall
30	C₂×C₅×ΓL(1,𝔽₈)	D₇₁	1	1	yes	characteristic, complemented
31	C₅×AGL(1,𝔽₇)	—	71	2	no	—
32	C₂×C₇₁	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	D₇₁	—	7	2	no	—
34	C₂×C₅×D₇	—	71	1	no	Hall
35	C₅×ΓL(1,𝔽₈)	C₂×D₇₁	1	1	yes	characteristic, complemented, Hall
36	C₂×AGL(1,𝔽₇)	—	71	1	no	Hall
37	C₇₁	C₂×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₅×C₇	C₃×D₇₁	1	1	yes	characteristic, complemented
39	C₅×D₇	—	71	2	no	—
40	C₂²×C₃×C₅	—	497	1	no	Hall
41	C₂×ΓL(1,𝔽₈)	C₅×D₇₁	1	1	yes	characteristic, complemented
42	AGL(1,𝔽₇)	—	71	2	no	—
43	C₅×C₇	C₂×C₃×D₇₁	1	1	yes	characteristic, complemented, Hall
44	C₂×C₃×C₅	—	7	1	no	—
45	C₂×C₃×C₅	—	497	2	no	—
46	C₂×D₇	—	71	1	no	Hall
47	ΓL(1,𝔽₈)	C₂×C₅×D₇₁	1	1	yes	characteristic, complemented, Hall
48	C₂²×C₅	—	497	1	no	Hall
49	C₃×C₅	—	7	1	no	Hall
50	C₂×C₇	C₃×C₅×D₇₁	1	1	yes	characteristic, complemented
51	D₇	—	71	2	no	—
52	C₂²×C₃	—	497	1	no	Hall
53	C₂×C₅	—	497	2	no	—
54	C₂×C₅	ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor
55	C₇	C₂×C₃×C₅×D₇₁	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	7	1	no	—
57	C₂×C₃	—	497	2	no	—
58	C₅	C₂×ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	497	1	no	Sylow
60	C₃	—	7	1	no	Sylow
61	C₂	—	497	2	no	—
62	C₂	C₅×ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	ΓL(1,𝔽₈)⋊D₇₁
commutator, cosocle, hypocenter	C₇×C₇₁	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇×C₇₁	C₂×C₃
LCS	G ▷ C₇×C₇₁
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×ΓL(1,𝔽₈)⋊D₇₁	C₂	direct (N × H)
2. C₂×C₅×ΓL(1,𝔽₈)×C₇₁	C₂	semidirect
3. C₂×C₅×D₄₉₇	C₃	semidirect
4. C₅×ΓL(1,𝔽₈)×C₇₁	C₂²	semidirect
5. C₂×ΓL(1,𝔽₈)⋊D₇₁	C₅	direct (N × H)
6. C₅×D₄₉₇	C₂×C₃	semidirect
7. C₂×C₅×C₇×C₇₁	C₂×C₃	semidirect
9. ΓL(1,𝔽₈)⋊D₇₁	C₂×C₅	direct (N × H)
10. C₂×ΓL(1,𝔽₈)×C₇₁	C₂×C₅	semidirect
11. C₅×C₇×C₇₁	C₂²×C₃	semidirect
14. C₂×D₄₉₇	C₃×C₅	semidirect
15. ΓL(1,𝔽₈)×C₇₁	C₂²×C₅	semidirect
18. C₂×C₇×C₇₁	C₂×C₃×C₅	semidirect
19. D₄₉₇	C₂×C₃×C₅	semidirect
21. C₂×C₅×C₇₁	AGL(1,𝔽₇)	semidirect
23. C₇×C₇₁	C₂²×C₃×C₅	semidirect
27. C₅×C₇₁	C₂×AGL(1,𝔽₇)	semidirect
30. C₂×C₅×ΓL(1,𝔽₈)	D₇₁	semidirect
32. C₂×C₇₁	C₅×AGL(1,𝔽₇)	semidirect
35. C₅×ΓL(1,𝔽₈)	C₂×D₇₁	semidirect
37. C₇₁	C₂×C₅×AGL(1,𝔽₇)	semidirect
38. C₂×C₅×C₇	C₃×D₇₁	semidirect
41. C₂×ΓL(1,𝔽₈)	C₅×D₇₁	semidirect
43. C₅×C₇	C₂×C₃×D₇₁	semidirect
47. ΓL(1,𝔽₈)	C₂×C₅×D₇₁	semidirect
50. C₂×C₇	C₃×C₅×D₇₁	semidirect
54. C₂×C₅	ΓL(1,𝔽₈)⋊D₇₁	direct (N × H)
55. C₇	C₂×C₃×C₅×D₇₁	semidirect
58. C₅	C₂×ΓL(1,𝔽₈)⋊D₇₁	direct (N × H)
62. C₂	C₅×ΓL(1,𝔽₈)⋊D₇₁	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×ΓL(1,𝔽₈)⋊D₇₁	2
C₂×C₅×ΓL(1,𝔽₈)×C₇₁	2
C₂×C₅×D₄₉₇	3
C₂×ΓL(1,𝔽₈)⋊D₇₁	5
C₂×C₃×C₅×D₇₁	7
C₂×C₅×AGL(1,𝔽₇)	71

Subgroups — C2×C5×ΓL(1,𝔽8)⋊D71 #44447221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×ΓL(1,𝔽₈)⋊D₇₁ #44447221