Subgroups — C₅×D₁₆₈ #13132324

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₈₄	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₅₆	—	3	1	no	maximal, Hall
4	C₂×C₅×D₂₁	—	2	2	no	—
5	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
6	D₁₆₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₂₈	—	3	2	no	—
8	C₅×C₇×C₈	D₃	1	1	yes	characteristic, complemented
9	C₅×D₂₄	—	7	1	no	maximal, Hall
10	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
11	C₅×D₂₁	—	4	2	no	—
12	D₈₄	C₂×C₅	1	2	yes	complemented
13	C₃×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂×C₅×D₇	—	6	2	no	—
15	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
16	C₅×D₁₂	—	7	2	no	—
17	C₃×C₅×C₈	D₇	1	1	yes	characteristic, complemented
18	D₅₆	—	3	1	no	Hall
19	C₃×C₅×C₇	D₈	1	1	yes	characteristic, complemented, Hall
20	C₂×D₂₁	—	2	2	no	—
21	C₃×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
22	C₅×D₈	—	21	1	no	Hall
23	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
24	C₅×D₇	—	12	2	no	—
25	C₂×C₅×D₃	—	14	2	no	—
26	C₃×C₄×C₅	C₂×D₇	1	1	yes	characteristic
27	D₂₈	—	3	2	no	—
28	C₇×C₈	C₅×D₃	1	1	yes	characteristic, complemented
29	D₂₄	—	7	1	no	Hall
30	C₂×C₃×C₇	C₅×D₄	1	1	yes	characteristic
31	D₂₁	—	4	2	no	—
32	C₅×D₄	—	21	2	no	—
33	C₅×C₈	D₂₁	1	1	yes	characteristic, complemented
34	C₅×C₇	D₂₄	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	D₂₈	1	1	yes	characteristic
36	C₅×D₃	—	28	2	no	—
37	C₂×D₇	—	6	2	no	—
38	C₄×C₇	C₂×C₅×D₃	1	1	yes	characteristic
39	D₁₂	—	7	2	no	—
40	C₃×C₈	C₅×D₇	1	1	yes	characteristic, complemented
41	C₃×C₇	C₅×D₈	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₅	—	42	2	no	—
43	C₄×C₅	C₂×D₂₁	1	1	yes	characteristic
44	D₈	—	21	1	no	Sylow
45	C₃×C₅	D₅₆	1	1	yes	characteristic, complemented, Hall
46	C₂×C₇	C₅×D₁₂	1	1	yes	characteristic
47	D₇	—	12	2	no	—
48	C₂×D₃	—	14	2	no	—
49	C₃×C₄	C₂×C₅×D₇	1	1	yes	characteristic
50	C₂×C₅	—	84	2	no	—
51	C₂×C₅	D₈₄	1	1	yes	characteristic
52	D₄	—	21	2	no	—
53	C₈	C₅×D₂₁	1	1	yes	characteristic, complemented
54	C₇	C₅×D₂₄	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	C₅×D₂₈	1	1	yes	characteristic
56	D₃	—	28	2	no	—
57	C₅	D₁₆₈	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	42	2	no	—
59	C₄	C₂×C₅×D₂₁	1	1	yes	characteristic
60	C₃	C₅×D₅₆	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	84	2	no	—
62	C₂	C₅×D₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	D₈₄
commutator, cosocle	C₃×C₄×C₇	C₂²×C₅
Frattini	C₄	C₂×C₅×D₂₁
Fitting	C₃×C₅×C₇×C₈	C₂
socle	C₂×C₃×C₅×C₇	D₄
hypercenter	C₅×C₈	D₂₁
hypocenter	C₃×C₇	C₅×D₈
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈

Complements

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

N	H ≅ G/N	type
1. C₅×D₈₄	C₂	semidirect
2. C₃×C₅×C₇×C₈	C₂	semidirect
6. D₁₆₈	C₅	direct (N × H)
8. C₅×C₇×C₈	D₃	semidirect
12. D₈₄	C₂×C₅	semidirect
13. C₃×C₇×C₈	C₂×C₅	semidirect
17. C₃×C₅×C₈	D₇	semidirect
19. C₃×C₅×C₇	D₈	semidirect
28. C₇×C₈	C₅×D₃	semidirect
33. C₅×C₈	D₂₁	semidirect
34. C₅×C₇	D₂₄	semidirect
40. C₃×C₈	C₅×D₇	semidirect
41. C₃×C₇	C₅×D₈	semidirect
45. C₃×C₅	D₅₆	semidirect
53. C₈	C₅×D₂₁	semidirect
54. C₇	C₅×D₂₄	semidirect
57. C₅	D₁₆₈	direct (N × H)
60. C₃	C₅×D₅₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×D₈₄	2
C₃×C₅×C₇×C₈	2
C₅×D₅₆	3
D₁₆₈	5
C₅×D₂₄	7

Subgroups — C5×D168 #13132324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₁₆₈ #13132324