Subgroups — C₅×D₃×C₇×C₈ #13132305

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₅×C₇×C₈	D₃
commutator, hypocenter	C₃	C₂×C₅×C₇×C₈
Frattini	C₄	C₂×C₅×D₃×C₇
Fitting	C₃×C₅×C₇×C₈	C₂
socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical	C₂×C₅×D₃×C₇	C₄
cosocle	C₃×C₄	C₂²×C₅×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₅×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₇×C₈	C₂	semidirect
3. C₅×C₇×C₃⋊C₈	C₂	semidirect
8. D₃×C₇×C₈	C₅	direct (N × H)
11. C₅×C₇×C₈	D₃	direct (N × H)
12. C₅×D₃×C₈	C₇	direct (N × H)
14. C₅×D₃×C₇	C₈	direct (N × H)
16. C₃×C₇×C₈	C₂×C₅	semidirect
17. C₇×C₃⋊C₈	C₂×C₅	semidirect
22. C₃×C₅×C₈	C₂×C₇	semidirect
23. C₅×C₃⋊C₈	C₂×C₇	semidirect
25. C₃×C₅×C₇	C₂×C₈	semidirect
37. C₇×C₈	C₅×D₃	direct (N × H)
38. D₃×C₈	C₅×C₇	direct (N × H)
40. D₃×C₇	C₅×C₈	direct (N × H)
43. C₅×C₈	D₃×C₇	direct (N × H)
44. C₅×C₇	D₃×C₈	direct (N × H)
46. C₅×D₃	C₇×C₈	direct (N × H)
51. C₃×C₈	C₂×C₅×C₇	semidirect
52. C₃⋊C₈	C₂×C₅×C₇	semidirect
53. C₃×C₇	C₂×C₅×C₈	semidirect
58. C₃×C₅	C₂×C₇×C₈	semidirect
68. C₈	C₅×D₃×C₇	direct (N × H)
69. C₇	C₅×D₃×C₈	direct (N × H)
71. D₃	C₅×C₇×C₈	direct (N × H)
72. C₅	D₃×C₇×C₈	direct (N × H)
76. C₃	C₂×C₅×C₇×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₃×C₇	2
C₃×C₅×C₇×C₈	2
C₅×C₇×C₃⋊C₈	2
C₂×C₅×C₇×C₈	3
D₃×C₇×C₈	5
C₅×D₃×C₈	7

Subgroups — C5×D3×C7×C8 #13132305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×C₇×C₈ #13132305