Subgroups — C₂×C₃⋊(C₇⋊#170) #13132628

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₇⋊#170)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇⋊#170	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	C₄	1	1	yes	characteristic
5	C₃×C₇×Dic₅	C₄	1	1	yes	characteristic
6	C₃×C₇×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₇⋊C₈)	—	5	1	no	maximal, Hall
8	C₂×C₇×Dic₅	D₃	1	1	yes	characteristic
9	C₇⋊#170	—	3	2	no	—
10	C₂×C₃⋊#170	—	7	1	no	maximal, Hall
11	C₂×C₃×C₅×C₇	C₈	1	2	yes	complemented
12	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₇	—	5	1	no	—
14	C₃⋊(C₇⋊C₈)	—	5	2	no	—
15	C₂²×C₅×C₇	Dic₃	1	1	yes	characteristic
16	C₇×Dic₅	Dic₃	1	1	yes	characteristic
17	C₇×Dic₅	C₂×D₃	1	1	yes	characteristic
18	C₂×C₃×Dic₅	D₇	1	1	yes	characteristic
19	C₃⋊#170	—	7	2	no	—
20	C₂×C₇⋊C₈	—	15	1	no	Hall
21	C₃×C₅×C₇	C₂×C₈	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic
23	C₃×C₄×C₇	—	5	1	no	—
24	C₃×C₄×C₇	—	5	1	no	—
25	C₂×#170	—	21	1	no	Hall
26	C₂×C₅×C₇	C₃⋊C₈	1	2	yes	complemented
27	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	Dic₇	1	1	yes	characteristic
29	C₃×Dic₅	Dic₇	1	1	yes	characteristic
30	C₃×Dic₅	C₂×D₇	1	1	yes	characteristic
31	C₂×C₄×C₇	—	5	1	no	—
32	C₇⋊C₈	—	15	2	no	—
33	C₂×C₃⋊C₈	—	35	1	no	Hall
34	C₂×C₃×C₇	#170	1	2	yes	complemented
35	C₂×C₃×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂×Dic₅	D₂₁	1	1	yes	characteristic
37	#170	—	21	2	no	—
38	C₅×C₇	C₂×C₃⋊C₈	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	C₇⋊C₈	1	2	yes	complemented
40	C₂×C₃×C₅	C₂×Dic₇	1	1	yes	characteristic
41	C₂²×C₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
42	C₄×C₇	—	5	1	no	—
43	C₄×C₇	—	5	1	no	—
44	C₂×C₃×C₄	—	5	1	no	—
45	C₃⋊C₈	—	35	2	no	—
46	C₃×C₇	C₂×#170	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₅	Dic₂₁	1	1	yes	characteristic
48	Dic₅	Dic₂₁	1	1	yes	characteristic
49	Dic₅	C₂×D₂₁	1	1	yes	characteristic
50	C₂×C₈	—	105	1	no	Sylow
51	C₃×C₅	C₂×C₇⋊C₈	1	1	yes	characteristic, complemented, Hall
52	C₂×C₇	C₃⋊#170	1	2	yes	complemented
53	C₂×C₇	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
54	C₂²×C₃	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₃×C₄	—	5	1	no	—
56	C₃×C₄	—	5	1	no	—
57	C₂×C₅	C₃⋊(C₇⋊C₈)	1	2	yes	complemented
58	C₂×C₅	C₂×Dic₂₁	1	1	yes	characteristic
59	C₂×C₄	—	5	1	no	—
60	C₈	—	105	2	no	—
61	C₇	C₂×C₃⋊#170	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	C₇⋊#170	1	2	yes	complemented
63	C₂×C₃	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₅	C₂×C₃⋊(C₇⋊C₈)	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₃⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic
66	C₄	—	5	1	no	—
67	C₄	—	5	1	no	—
68	C₃	C₂×C₇⋊#170	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₃⋊(C₇⋊#170)	1	2	yes	direct product factor
70	C₂	C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₇⋊#170)	C₂	direct (N × H)
11. C₂×C₃×C₅×C₇	C₈	semidirect
21. C₃×C₅×C₇	C₂×C₈	semidirect
26. C₂×C₅×C₇	C₃⋊C₈	semidirect
34. C₂×C₃×C₇	#170	semidirect
38. C₅×C₇	C₂×C₃⋊C₈	semidirect
39. C₂×C₃×C₅	C₇⋊C₈	semidirect
46. C₃×C₇	C₂×#170	semidirect
51. C₃×C₅	C₂×C₇⋊C₈	semidirect
52. C₂×C₇	C₃⋊#170	semidirect
57. C₂×C₅	C₃⋊(C₇⋊C₈)	semidirect
61. C₇	C₂×C₃⋊#170	semidirect
62. C₂×C₃	C₇⋊#170	semidirect
64. C₅	C₂×C₃⋊(C₇⋊C₈)	semidirect
68. C₃	C₂×C₇⋊#170	semidirect
69. C₂	C₃⋊(C₇⋊#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×C₃×C₇×Dic₅	2
C₃⋊(C₇⋊#170)	2
C₂×C₇⋊#170	3
C₂×C₃⋊(C₇⋊C₈)	5
C₂×C₃⋊#170	7

Subgroups — C2×C3⋊(C7⋊#170) #13132628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₇⋊#170) #13132628