Subgroups — C₂×ΓL(1,𝔽₈)×#170 #13132401

Order 1680.

88 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, direct product factor
3	C₂×C₇×#170	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	C₂×C₈×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	C₂×C₃×#170	—	7	1	no	maximal, Hall
11	C₂×C₅×ΓL(1,𝔽₈)	C₈	1	2	yes	complemented
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	C₂×C₇×C₈	—	5	1	no	Hall
21	C₅×ΓL(1,𝔽₈)	C₂×C₈	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	C₂×#170	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₅×C₇	C₃×C₈	1	2	yes	complemented
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	C₂×C₃×C₈	—	35	1	no	Hall
34	C₂×ΓL(1,𝔽₈)	#170	1	2	yes	direct product factor
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	direct product factor
38	C₅×C₇	C₂×C₃×C₈	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	—	7	2	no	—
40	C₂×C₃×C₅	—	7	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)	C₂×#170	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	C₂×C₈	—	5	1	no	Sylow
51	C₃×C₅	—	7	1	no	Hall
52	C₂×C₇	C₃×#170	1	2	yes	complemented
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	C₈×ΓL(1,𝔽₈)	1	2	yes	complemented
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	C₂×C₃×#170	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	7	2	no	—
63	C₂×C₃	—	7	1	no	—
64	C₅	C₂×C₈×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	ΓL(1,𝔽₈)×#170	1	2	yes	direct product factor
70	C₂	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	AGL(1,𝔽₅)×ΓL(1,𝔽₈)
commutator, hypocenter	C₅×C₇	C₂×C₃×C₈
Frattini	C₂	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)
Fitting, socle	C₂²×C₅×C₇	C₃×C₄
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₄
cosocle	C₇×Dic₅	C₂²×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×#170	C₂	direct (N × H)
3. C₂×C₇×#170	C₃	semidirect
9. C₇×#170	C₂×C₃	semidirect
11. C₂×C₅×ΓL(1,𝔽₈)	C₈	semidirect
21. C₅×ΓL(1,𝔽₈)	C₂×C₈	semidirect
25. C₂×#170	ΓL(1,𝔽₈)	direct (N × H)
26. C₂×C₅×C₇	C₃×C₈	semidirect
34. C₂×ΓL(1,𝔽₈)	#170	direct (N × H)
37. #170	C₂×ΓL(1,𝔽₈)	direct (N × H)
38. C₅×C₇	C₂×C₃×C₈	semidirect
46. ΓL(1,𝔽₈)	C₂×#170	direct (N × H)
52. C₂×C₇	C₃×#170	semidirect
57. C₂×C₅	C₈×ΓL(1,𝔽₈)	semidirect
61. C₇	C₂×C₃×#170	semidirect
64. C₅	C₂×C₈×ΓL(1,𝔽₈)	semidirect
69. C₂	ΓL(1,𝔽₈)×#170	direct (N × H)

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
C₂×C₇×#170	3
C₂×C₈×ΓL(1,𝔽₈)	5
C₂×C₃×#170	7

Subgroups — C2×ΓL(1,𝔽8)×#170 #13132401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×#170 #13132401