Subgroups — ΓL(1,𝔽₈)×#713 #13132572

Order 1680.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₂₀	C₂	1	3	yes	maximal, complemented
3	C₄×D₅×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
4	C₇×#713	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×D₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
6	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
7	Dic₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	#41×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₇×D₂₀	C₂×C₃	1	3	yes	complemented
11	C₄×C₇×D₅	C₂×C₃	1	3	yes	complemented
12	C₃×#713	—	7	1	no	maximal, Hall
13	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
14	D₅×ΓL(1,𝔽₈)	—	2	3	no	—
15	Q₈×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
16	D₄×ΓL(1,𝔽₈)	—	5	3	no	—
17	C₂×C₄×ΓL(1,𝔽₈)	—	5	3	no	—
18	C₂×C₇×D₅	C₂²×C₃	1	3	yes	—
19	C₄×C₅×C₇	C₂²×C₃	1	3	yes	—
20	C₇×Dic₅	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₅×Q₈	—	7	1	no	—
22	C₃×D₂₀	—	7	3	no	—
23	C₃×C₄×D₅	—	7	3	no	—
24	C₇×#41	—	5	1	no	Hall
25	C₅×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic, complemented, Hall
26	C₂²×ΓL(1,𝔽₈)	—	5	3	no	—
27	C₄×ΓL(1,𝔽₈)	—	5	1	no	—
28	C₄×ΓL(1,𝔽₈)	C₂×D₅	1	3	yes	—
29	#713	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
31	C₇×D₅	—	2	3	no	—
32	C₂×C₃×D₅	—	7	3	no	—
33	C₃×C₄×C₅	—	7	3	no	—
34	C₃×Dic₅	—	7	1	no	—
35	C₇×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
36	C₇×D₄	—	5	3	no	—
37	C₂×C₄×C₇	—	5	3	no	—
38	C₃×#41	—	35	1	no	Hall
39	C₂×ΓL(1,𝔽₈)	—	10	3	no	—
40	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	1	yes	characteristic
41	C₅×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	D₂₀	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
43	C₄×D₅	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
44	C₅×C₇	C₃×#41	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	1	no	—
46	C₃×D₅	—	14	3	no	—
47	C₂²×C₇	—	5	3	no	—
48	C₄×C₇	—	5	1	no	—
49	C₄×C₇	C₂×C₃×D₅	1	3	yes	—
50	C₃×Q₈	—	7	1	no	—
51	C₃×D₄	—	35	3	no	—
52	C₂×C₃×C₄	—	35	3	no	—
53	ΓL(1,𝔽₈)	#713	1	1	yes	characteristic, direct product factor, Hall
54	C₂×D₅	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
55	C₄×C₅	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
56	Dic₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
57	#41	—	5	1	no	Sylow
58	C₃×C₅	—	7	1	no	Hall
59	C₂×C₇	—	10	3	no	—
60	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
61	C₂²×C₃	—	35	3	no	—
62	C₃×C₄	—	7	3	no	—
63	C₃×C₄	—	35	1	no	—
64	C₂×C₅	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	D₅	—	2	3	no	—
66	Q₈	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
67	D₄	—	5	3	no	—
68	C₂×C₄	—	5	3	no	—
69	C₇	C₃×#713	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	1	no	—
71	C₂×C₃	—	70	3	no	—
72	C₅	#41×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	3	no	—
74	C₄	—	5	1	no	—
75	C₄	C₂×D₅×ΓL(1,𝔽₈)	1	3	yes	—
76	C₃	—	7	1	no	Sylow
77	C₂	—	10	3	no	—
78	C₂	C₂²×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₅×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂×C₅×C₇	C₂³×C₃
Fitting	C₅×C₇×Q₈	C₂×C₃
hypercenter	Q₈	D₅×ΓL(1,𝔽₈)
hypocenter	C₅×C₇	C₃×#41
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)×D₂₀	C₂	semidirect
3. C₄×D₅×ΓL(1,𝔽₈)	C₂	semidirect
4. C₇×#713	C₃	semidirect
9. C₅×C₇×Q₈	C₂×C₃	semidirect
10. C₇×D₂₀	C₂×C₃	semidirect
11. C₄×C₇×D₅	C₂×C₃	semidirect
15. Q₈×ΓL(1,𝔽₈)	D₅	semidirect
25. C₅×ΓL(1,𝔽₈)	#41	semidirect
29. #713	ΓL(1,𝔽₈)	direct (N × H)
35. C₇×Q₈	C₃×D₅	semidirect
41. C₅×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
42. D₂₀	C₂×ΓL(1,𝔽₈)	semidirect
43. C₄×D₅	C₂×ΓL(1,𝔽₈)	semidirect
44. C₅×C₇	C₃×#41	semidirect
53. ΓL(1,𝔽₈)	#713	direct (N × H)
66. Q₈	D₅×ΓL(1,𝔽₈)	semidirect
69. C₇	C₃×#713	semidirect
72. C₅	#41×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
C₅×Q₈×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×D₂₀	2
C₄×D₅×ΓL(1,𝔽₈)	2
C₇×#713	3
#41×ΓL(1,𝔽₈)	5
C₃×#713	7

Subgroups — ΓL(1,𝔽8)×#713 #13132572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#713 #13132572