Subgroups — C₅×D₈×ΓL(1,𝔽₈) #13132252

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
2	C₅×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇×D₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
5	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	D₈×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×C₇×D₄	C₂×C₃	1	2	yes	complemented
8	C₅×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₅×D₈	—	7	1	no	maximal, Hall
10	C₂×C₅×ΓL(1,𝔽₈)	—	4	2	no	—
11	C₂×C₅×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
12	D₄×ΓL(1,𝔽₈)	C₂×C₅	1	2	yes	complemented
13	C₈×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂²×C₅×C₇	—	2	2	no	—
15	C₄×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
16	C₃×C₅×D₄	—	7	2	no	—
17	C₃×C₅×C₈	—	7	1	no	—
18	C₇×D₈	C₃×C₅	1	1	yes	characteristic, complemented, Hall
19	C₅×ΓL(1,𝔽₈)	D₈	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
21	C₄×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
22	C₅×D₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₅×C₇	—	4	2	no	—
24	C₂×C₅×C₇	C₃×D₄	1	1	yes	characteristic
25	C₂²×C₃×C₅	—	14	2	no	—
26	C₃×C₄×C₅	—	7	1	no	—
27	C₇×D₄	C₂×C₃×C₅	1	2	yes	complemented
28	C₇×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
29	C₃×D₈	—	7	1	no	Hall
30	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
31	C₂×ΓL(1,𝔽₈)	C₅×D₄	1	1	yes	characteristic
32	C₅×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
33	C₅×C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₅×C₇	C₃×D₈	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	—	7	1	no	—
36	C₂×C₃×C₅	—	28	2	no	—
37	C₂²×C₇	—	2	2	no	—
38	C₄×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
39	C₃×D₄	—	7	2	no	—
40	C₃×C₈	—	7	1	no	—
41	ΓL(1,𝔽₈)	C₅×D₈	1	1	yes	characteristic, direct product factor, Hall
42	C₂²×C₅	—	2	2	no	—
43	C₄×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	D₈	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
45	C₃×C₅	—	7	1	no	Hall
46	C₂×C₇	—	4	2	no	—
47	C₂×C₇	C₃×C₅×D₄	1	1	yes	characteristic
48	C₂²×C₃	—	14	2	no	—
49	C₃×C₄	—	7	1	no	—
50	C₂×C₅	—	4	2	no	—
51	C₂×C₅	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	D₄	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
53	C₈	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₇	C₃×C₅×D₈	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	7	1	no	—
56	C₂×C₃	—	28	2	no	—
57	C₅	D₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	2	2	no	—
59	C₄	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
60	C₃	—	7	1	no	Sylow
61	C₂	—	4	2	no	—
62	C₂	C₅×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂²×C₃×C₅
Frattini	C₄	C₂²×C₅×ΓL(1,𝔽₈)
Fitting	C₅×C₇×D₈	C₃
socle	C₂×C₅×C₇	C₃×D₄
hypercenter	C₅×D₈	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₅×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₈

Complements

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

N	H ≅ G/N	type
1. C₅×D₄×ΓL(1,𝔽₈)	C₂	semidirect
2. C₅×C₈×ΓL(1,𝔽₈)	C₂	semidirect
3. C₅×C₇×D₈	C₃	semidirect
6. D₈×ΓL(1,𝔽₈)	C₅	direct (N × H)
7. C₅×C₇×D₄	C₂×C₃	semidirect
8. C₅×C₇×C₈	C₂×C₃	semidirect
12. D₄×ΓL(1,𝔽₈)	C₂×C₅	semidirect
13. C₈×ΓL(1,𝔽₈)	C₂×C₅	semidirect
18. C₇×D₈	C₃×C₅	semidirect
19. C₅×ΓL(1,𝔽₈)	D₈	direct (N × H)
22. C₅×D₈	ΓL(1,𝔽₈)	direct (N × H)
27. C₇×D₄	C₂×C₃×C₅	semidirect
28. C₇×C₈	C₂×C₃×C₅	semidirect
32. C₅×D₄	C₂×ΓL(1,𝔽₈)	semidirect
33. C₅×C₈	C₂×ΓL(1,𝔽₈)	semidirect
34. C₅×C₇	C₃×D₈	semidirect
41. ΓL(1,𝔽₈)	C₅×D₈	direct (N × H)
44. D₈	C₅×ΓL(1,𝔽₈)	direct (N × H)
52. D₄	C₂×C₅×ΓL(1,𝔽₈)	semidirect
53. C₈	C₂×C₅×ΓL(1,𝔽₈)	semidirect
54. C₇	C₃×C₅×D₈	semidirect
57. C₅	D₈×ΓL(1,𝔽₈)	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×D₄×ΓL(1,𝔽₈)	2
C₅×C₈×ΓL(1,𝔽₈)	2
C₅×C₇×D₈	3
D₈×ΓL(1,𝔽₈)	5
C₃×C₅×D₈	7

Subgroups — C5×D8×ΓL(1,𝔽8) #13132252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₈×ΓL(1,𝔽₈) #13132252