Subgroups — C₅×C₇×#208 #13132311

Order 1680.

80 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	C₅×C₇×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₅×C₇×M₁₆	—	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	C₇×#208	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
9	C₅×C₇×C₈	—	3	2	no	—
10	C₅×#208	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×C₇	—	2	1	no	—
12	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic
14	C₇×C₃⋊C₈	C₂×C₅	1	2	yes	complemented
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	C₅×C₃⋊C₈	C₂×C₇	1	2	yes	complemented
20	C₇×M₁₆	—	3	1	no	Hall
21	C₃×C₅×C₇	M₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₇	C₄×C₅	1	1	yes	characteristic
23	C₃×C₄×C₇	C₄×C₅	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
25	C₅×M₁₆	—	3	1	no	Hall
26	C₂×C₅×C₇	—	2	1	no	—
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	C₅×D₃	1	1	yes	characteristic
32	C₇×C₈	—	3	2	no	—
33	#208	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₇	—	2	1	no	—
35	C₂×C₃×C₇	C₂×C₄×C₅	1	1	yes	characteristic
36	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic
37	C₅×C₈	—	3	2	no	—
38	C₅×C₇	#208	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	C₂×C₄×C₇	1	1	yes	characteristic
41	C₂²×C₇	C₅×Dic₃	1	1	yes	characteristic
42	C₄×C₇	C₅×Dic₃	1	1	yes	characteristic
43	C₄×C₇	C₂×C₅×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×C₅×C₇	1	1	yes	characteristic
45	C₃⋊C₈	C₂×C₅×C₇	1	2	yes	complemented
46	C₃×C₇	C₅×M₁₆	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	M₁₆	—	3	1	no	Sylow
51	C₃×C₅	C₇×M₁₆	1	1	yes	characteristic, complemented, Hall
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	C₂×C₅×Dic₃	1	1	yes	characteristic
54	C₂²×C₃	C₄×C₅×C₇	1	1	yes	characteristic
55	C₃×C₄	C₄×C₅×C₇	1	1	yes	characteristic
56	C₃×C₄	C₂²×C₅×C₇	1	1	yes	characteristic
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂×C₇×Dic₃	1	1	yes	characteristic
59	C₂×C₄	C₅×D₃×C₇	1	1	yes	characteristic
60	C₈	—	3	2	no	—
61	C₇	C₅×#208	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	C₂×C₄×C₅×C₇	1	1	yes	characteristic
64	C₅	C₇×#208	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₅×C₇×Dic₃	1	1	yes	characteristic
66	C₄	C₅×C₇×Dic₃	1	1	yes	characteristic
67	C₄	C₂×C₅×D₃×C₇	1	1	yes	characteristic
68	C₃	C₅×C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	1	no	—
70	C₂	C₂×C₅×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅×C₇	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₇
Frattini	C₄	C₂×C₅×D₃×C₇
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical	C₂²×C₃×C₅×C₇	C₄
cosocle	C₃×C₄	C₂²×C₅×C₇
hypercenter	C₂×C₄×C₅×C₇	D₃
hypocenter	C₃	C₅×C₇×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅×C₇ ◁ C₂×C₄×C₅×C₇

Complements

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

N	H ≅ G/N	type
2. C₅×C₇×C₃⋊C₈	C₂	semidirect
7. C₇×#208	C₅	direct (N × H)
10. C₅×#208	C₇	direct (N × H)
14. C₇×C₃⋊C₈	C₂×C₅	semidirect
19. C₅×C₃⋊C₈	C₂×C₇	semidirect
21. C₃×C₅×C₇	M₁₆	semidirect
33. #208	C₅×C₇	direct (N × H)
38. C₅×C₇	#208	direct (N × H)
45. C₃⋊C₈	C₂×C₅×C₇	semidirect
46. C₃×C₇	C₅×M₁₆	semidirect
51. C₃×C₅	C₇×M₁₆	semidirect
61. C₇	C₅×#208	direct (N × H)
64. C₅	C₇×#208	direct (N × H)
68. C₃	C₅×C₇×M₁₆	semidirect

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
C₅×C₇×C₃⋊C₈	2
C₅×C₇×M₁₆	3
C₇×#208	5
C₅×#208	7

Subgroups — C5×C7×#208 #13132311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×#208 #13132311