Subgroups — C₂×C₇×C₃⋊#168 #13132342

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊#168	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×#168	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅×C₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₃⋊C₈	—	5	1	no	maximal, Hall
8	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
9	C₇×#168	—	3	2	no	—
10	C₂×C₃⋊#168	C₇	1	1	yes	maximal, characteristic, direct product factor, 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	D₅	1	1	yes	characteristic
14	C₇×C₃⋊C₈	—	5	2	no	—
15	C₂²×C₅×C₇	Dic₃	1	1	yes	characteristic
16	C₄×C₅×C₇	Dic₃	1	1	yes	characteristic
17	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
18	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic
19	C₃⋊#168	C₂×C₇	1	2	yes	direct product factor
20	C₂×C₇×C₈	—	15	1	no	Hall
21	C₃×C₅×C₇	C₂×C₈	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₇	Dic₅	1	1	yes	characteristic
23	C₃×C₄×C₇	Dic₅	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
25	C₂×#168	—	3	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	C₄×C₇	1	1	yes	characteristic
29	C₃×C₄×C₅	C₄×C₇	1	1	yes	characteristic
30	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
31	C₂×C₄×C₇	D₁₅	1	1	yes	characteristic
32	C₇×C₈	—	15	2	no	—
33	C₂×C₃⋊C₈	—	5	1	no	Hall
34	C₂×C₃×C₇	#168	1	2	yes	complemented
35	C₂×C₃×C₇	C₂×Dic₅	1	1	yes	characteristic
36	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic
37	#168	—	3	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×C₄×C₇	1	1	yes	characteristic
41	C₂²×C₇	Dic₁₅	1	1	yes	characteristic
42	C₄×C₇	Dic₁₅	1	1	yes	characteristic
43	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
44	C₂×C₃×C₄	C₇×D₅	1	1	yes	characteristic
45	C₃⋊C₈	—	5	2	no	—
46	C₃×C₇	C₂×#168	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₅	C₇×Dic₃	1	1	yes	characteristic
48	C₄×C₅	C₇×Dic₃	1	1	yes	characteristic
49	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
50	C₂×C₈	—	15	1	no	Sylow
51	C₃×C₅	C₂×C₇×C₈	1	1	yes	characteristic, complemented, Hall
52	C₂×C₇	C₃⋊#168	1	2	yes	direct product factor
53	C₂×C₇	C₂×Dic₁₅	1	1	yes	characteristic
54	C₂²×C₃	C₇×Dic₅	1	1	yes	characteristic
55	C₃×C₄	C₇×Dic₅	1	1	yes	characteristic
56	C₃×C₄	C₂×C₇×D₅	1	1	yes	characteristic
57	C₂×C₅	C₇×C₃⋊C₈	1	2	yes	complemented
58	C₂×C₅	C₂×C₇×Dic₃	1	1	yes	characteristic
59	C₂×C₄	C₇×D₁₅	1	1	yes	characteristic
60	C₈	—	15	2	no	—
61	C₇	C₂×C₃⋊#168	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃	C₇×#168	1	2	yes	complemented
63	C₂×C₃	C₂×C₇×Dic₅	1	1	yes	characteristic
64	C₅	C₂×C₇×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₇×Dic₁₅	1	1	yes	characteristic
66	C₄	C₇×Dic₁₅	1	1	yes	characteristic
67	C₄	C₂×C₇×D₁₅	1	1	yes	characteristic
68	C₃	C₂×C₇×#168	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₇×C₃⋊#168	1	2	yes	direct product factor
70	C₂	C₂×C₇×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₇	D₁₅
commutator, hypocenter	C₃×C₅	C₂×C₇×C₈
Frattini	C₄	C₂×C₇×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle, prime radical	C₂²×C₃×C₅×C₇	C₄
cosocle	C₃×C₄×C₅	C₂²×C₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₇×C₃⋊#168	C₂	direct (N × H)
10. C₂×C₃⋊#168	C₇	direct (N × H)
11. C₂×C₃×C₅×C₇	C₈	semidirect
19. C₃⋊#168	C₂×C₇	direct (N × H)
21. C₃×C₅×C₇	C₂×C₈	semidirect
26. C₂×C₅×C₇	C₃⋊C₈	semidirect
34. C₂×C₃×C₇	#168	semidirect
38. C₅×C₇	C₂×C₃⋊C₈	semidirect
39. C₂×C₃×C₅	C₇×C₈	semidirect
46. C₃×C₇	C₂×#168	semidirect
51. C₃×C₅	C₂×C₇×C₈	semidirect
52. C₂×C₇	C₃⋊#168	direct (N × H)
57. C₂×C₅	C₇×C₃⋊C₈	semidirect
61. C₇	C₂×C₃⋊#168	direct (N × H)
62. C₂×C₃	C₇×#168	semidirect
64. C₅	C₂×C₇×C₃⋊C₈	semidirect
68. C₃	C₂×C₇×#168	semidirect
69. C₂	C₇×C₃⋊#168	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×C₄×C₅×C₇	2
C₇×C₃⋊#168	2
C₂×C₇×#168	3
C₂×C₇×C₃⋊C₈	5
C₂×C₃⋊#168	7

Subgroups — C2×C7×C3⋊#168 #13132342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₃⋊#168 #13132342