Subgroups — C₂×Dic₇₀⋊C₃ #13132575

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₈)⋊Dic₅	C₂	1	2	yes	maximal
3	Dic₇₀⋊C₃	C₂	1	4	yes	maximal, direct product factor
4	C₂×Dic₇₀	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
7	ΓL(1,𝔽₈)⋊Dic₅	C₂²	1	4	yes	—
8	C₂×ΓL(1,𝔽₈)⋊Q₈	—	5	1	no	maximal, Hall
9	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×Dic₃₅	C₂×C₃	1	2	yes	—
11	Dic₇₀	C₂×C₃	1	4	yes	complemented
12	C₂×C₃×Dic₁₀	—	7	1	no	maximal, Hall
13	C₂×C₅×ΓL(1,𝔽₈)	Q₈	1	2	yes	complemented
14	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
15	C₂×C₄×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic
16	C₂×ΓL(1,𝔽₈)⋊C₄	—	5	2	no	—
17	ΓL(1,𝔽₈)⋊Q₈	—	5	4	no	—
18	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂²×C₃	1	2	yes	—
20	Dic₃₅	C₂²×C₃	1	4	yes	—
21	C₂×C₃×C₄×C₅	—	7	1	no	—
22	C₂×C₃×Dic₅	—	7	2	no	—
23	C₃×Dic₁₀	—	7	4	no	—
24	C₂×Dic₁₄	—	5	1	no	Hall
25	C₅×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
26	C₂²×ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic
27	C₄×ΓL(1,𝔽₈)	C₂×D₅	1	2	yes	—
28	ΓL(1,𝔽₈)⋊C₄	—	5	4	no	—
29	C₂×Dic₁₀	—	7	1	no	Hall
30	C₂×C₅×C₇	C₃×Q₈	1	2	yes	complemented
31	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₃×C₅	—	7	1	no	—
33	C₃×C₄×C₅	—	7	2	no	—
34	C₃×Dic₅	—	7	4	no	—
35	C₂×C₄×C₇	C₃×D₅	1	1	yes	characteristic
36	C₂×Dic₇	—	5	2	no	—
37	Dic₁₄	—	5	4	no	—
38	C₂×C₃×Q₈	—	35	1	no	Hall
39	C₂×ΓL(1,𝔽₈)	Dic₁₀	1	2	yes	complemented
40	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	1	yes	characteristic
41	C₂×C₄×C₅	AGL(1,𝔽₇)	1	1	yes	characteristic
42	C₂×Dic₅	—	7	2	no	—
43	Dic₁₀	—	7	4	no	—
44	C₅×C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	2	no	—
46	C₂×C₃×C₅	—	7	1	no	—
47	C₂²×C₇	C₂×C₃×D₅	1	1	yes	characteristic
48	C₄×C₇	C₂×C₃×D₅	1	2	yes	—
49	Dic₇	—	5	4	no	—
50	C₃×Q₈	—	35	4	no	—
51	C₂×C₃×C₄	—	7	1	no	—
52	C₂×C₃×C₄	—	35	2	no	—
53	ΓL(1,𝔽₈)	C₂×Dic₁₀	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₅	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
55	C₄×C₅	C₂×AGL(1,𝔽₇)	1	2	yes	—
56	Dic₅	—	7	4	no	—
57	C₂×Q₈	—	35	1	no	Sylow
58	C₃×C₅	—	7	1	no	Hall
59	C₂×C₇	C₃×Dic₁₀	1	2	yes	complemented
60	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
61	C₂²×C₃	—	7	1	no	—
62	C₃×C₄	—	7	2	no	—
63	C₃×C₄	—	35	4	no	—
64	C₂×C₅	ΓL(1,𝔽₈)⋊Q₈	1	2	yes	complemented
65	C₂×C₅	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
66	Q₈	—	35	4	no	—
67	C₂×C₄	—	35	2	no	—
68	C₂×C₄	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic
69	C₇	C₂×C₃×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	2	no	—
71	C₂×C₃	—	7	1	no	—
72	C₅	C₂×ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²	C₂×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic
74	C₄	—	35	4	no	—
75	C₄	C₂×C₅⋊AGL(1,𝔽₇)	1	2	yes	—
76	C₃	—	7	1	no	Sylow
77	C₂	Dic₇₀⋊C₃	1	2	yes	direct product factor
78	C₂	C₂²×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₅⋊AGL(1,𝔽₇)
commutator, cosocle	C₂×C₅×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₅⋊AGL(1,𝔽₇)
Fitting	C₂×C₄×C₅×C₇	C₂×C₃
socle	C₂²×C₅×C₇	C₂²×C₃
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₂²
hypercenter	C₂×C₄	C₅⋊AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₂×C₃×Q₈
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. Dic₇₀⋊C₃	C₂	direct (N × H)
4. C₂×Dic₇₀	C₃	semidirect
11. Dic₇₀	C₂×C₃	semidirect
13. C₂×C₅×ΓL(1,𝔽₈)	Q₈	semidirect
25. C₅×ΓL(1,𝔽₈)	C₂×Q₈	semidirect
30. C₂×C₅×C₇	C₃×Q₈	semidirect
39. C₂×ΓL(1,𝔽₈)	Dic₁₀	semidirect
44. C₅×C₇	C₂×C₃×Q₈	semidirect
53. ΓL(1,𝔽₈)	C₂×Dic₁₀	semidirect
59. C₂×C₇	C₃×Dic₁₀	semidirect
64. C₂×C₅	ΓL(1,𝔽₈)⋊Q₈	semidirect
69. C₇	C₂×C₃×Dic₁₀	semidirect
72. C₅	C₂×ΓL(1,𝔽₈)⋊Q₈	semidirect
77. C₂	Dic₇₀⋊C₃	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×C₄×C₅×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊Dic₅	2
Dic₇₀⋊C₃	2
C₂×Dic₇₀	3
C₂×ΓL(1,𝔽₈)⋊Q₈	5
C₂×C₃×Dic₁₀	7

Subgroups — C2×Dic70⋊C3 #13132575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₇₀⋊C₃ #13132575