Subgroups — ΓL(1,𝔽₈)×#704 #13132402

Order 1680.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×#170	C₂	1	2	yes	maximal, complemented
3	C₇×#704	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	Dic₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	Dic₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	M₁₆×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
8	C₂×C₇×Dic₅	C₂×C₃	1	1	yes	characteristic
9	C₇×#170	C₂×C₃	1	2	yes	complemented
10	C₃×#704	—	7	1	no	maximal, Hall
11	C₂×C₅×ΓL(1,𝔽₈)	—	2	1	no	—
12	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×ΓL(1,𝔽₈)	—	5	1	no	—
14	C₈×ΓL(1,𝔽₈)	—	5	2	no	—
15	C₂²×C₅×C₇	C₃×C₄	1	1	yes	characteristic
16	C₇×Dic₅	C₃×C₄	1	1	yes	characteristic
17	C₇×Dic₅	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₃×Dic₅	—	7	1	no	—
19	C₃×#170	—	7	2	no	—
20	C₇×M₁₆	—	5	1	no	Hall
21	C₅×ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂²×ΓL(1,𝔽₈)	AGL(1,𝔽₅)	1	1	yes	characteristic
23	C₄×ΓL(1,𝔽₈)	—	5	1	no	—
24	C₄×ΓL(1,𝔽₈)	—	5	1	no	—
25	#704	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₅×C₇	—	2	1	no	—
27	C₂×C₅×C₇	C₂×C₃×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₅	—	7	1	no	—
29	C₃×Dic₅	—	7	1	no	—
30	C₃×Dic₅	—	7	1	no	—
31	C₂×C₄×C₇	—	5	1	no	—
32	C₇×C₈	—	5	2	no	—
33	C₃×M₁₆	—	35	1	no	Hall
34	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₂×ΓL(1,𝔽₈)	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂×Dic₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	#170	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
38	C₅×C₇	C₃×M₁₆	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	—	7	1	no	—
40	C₂×C₃×C₅	—	14	1	no	—
41	C₂²×C₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₄×C₇	—	5	1	no	—
43	C₄×C₇	—	5	1	no	—
44	C₂×C₃×C₄	—	35	1	no	—
45	C₃×C₈	—	35	2	no	—
46	ΓL(1,𝔽₈)	#704	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₅	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	Dic₅	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
49	Dic₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	M₁₆	—	5	1	no	Sylow
51	C₃×C₅	—	7	1	no	Hall
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂²×C₃	—	7	1	no	—
55	C₃×C₄	—	35	1	no	—
56	C₃×C₄	—	35	1	no	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₂×C₄	—	5	1	no	—
60	C₈	—	5	2	no	—
61	C₇	C₃×#704	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	7	1	no	—
63	C₂×C₃	—	14	1	no	—
64	C₅	M₁₆×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
65	C₂²	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₄	—	5	1	no	—
67	C₄	—	5	1	no	—
68	C₃	—	7	1	no	Sylow
69	C₂	—	2	1	no	—
70	C₂	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)
commutator, socle	C₂×C₅×C₇	C₂×C₃×C₄
Fitting	C₂²×C₅×C₇	C₃×C₄
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₄
cosocle	C₇×Dic₅	C₂²×C₃
hypercenter	C₂²	AGL(1,𝔽₅)×ΓL(1,𝔽₈)
hypocenter	C₅×C₇	C₃×M₁₆
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×#170	C₂	semidirect
3. C₇×#704	C₃	semidirect
9. C₇×#170	C₂×C₃	semidirect
21. C₅×ΓL(1,𝔽₈)	M₁₆	semidirect
25. #704	ΓL(1,𝔽₈)	direct (N × H)
37. #170	C₂×ΓL(1,𝔽₈)	semidirect
38. C₅×C₇	C₃×M₁₆	semidirect
46. ΓL(1,𝔽₈)	#704	direct (N × H)
61. C₇	C₃×#704	semidirect
64. C₅	M₁₆×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₅×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#170	2
C₇×#704	3
M₁₆×ΓL(1,𝔽₈)	5
C₃×#704	7

Subgroups — ΓL(1,𝔽8)×#704 #13132402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#704 #13132402