Subgroups — C₅×C₇×#204 #13132307

Order 1680.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₇×SD₁₆	—	3	1	no	maximal, Hall
5	C₂×C₅×D₃×C₇	—	2	1	no	—
6	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₅×C₇×Dic₃	—	2	1	no	—
8	C₇×#204	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×Q₈	—	3	1	no	—
10	C₅×C₇×D₄	—	3	1	no	—
11	C₅×C₇×C₈	D₃	1	1	yes	characteristic, complemented
12	C₅×#204	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
14	C₅×D₃×C₇	—	4	1	no	—
15	C₇×D₁₂	C₂×C₅	1	1	yes	characteristic
16	C₇×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
17	C₃×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂²×C₅×C₇	—	6	1	no	—
19	C₄×C₅×C₇	—	6	1	no	—
20	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	C₅×D₁₂	C₂×C₇	1	1	yes	characteristic
22	C₅×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
23	C₃×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
24	C₇×SD₁₆	—	3	1	no	Hall
25	C₃×C₅×C₇	SD₁₆	1	1	yes	characteristic, complemented, Hall
26	C₂×D₃×C₇	—	2	1	no	—
27	C₃×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
28	C₇×Dic₃	—	2	1	no	—
29	C₅×SD₁₆	—	3	1	no	Hall
30	C₂×C₅×C₇	—	12	1	no	—
31	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
32	C₂×C₅×D₃	—	2	1	no	—
33	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
34	C₅×Dic₃	—	2	1	no	—
35	C₇×Q₈	—	3	1	no	—
36	C₇×D₄	—	3	1	no	—
37	C₇×C₈	C₅×D₃	1	1	yes	characteristic, complemented
38	#204	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₃×C₇	C₅×D₄	1	1	yes	characteristic
40	D₃×C₇	—	4	1	no	—
41	C₅×Q₈	—	3	1	no	—
42	C₅×D₄	—	3	1	no	—
43	C₅×C₈	D₃×C₇	1	1	yes	characteristic, complemented
44	C₅×C₇	#204	1	1	yes	characteristic, direct product factor, Hall
45	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
46	C₅×D₃	—	4	1	no	—
47	C₂²×C₇	—	6	1	no	—
48	C₄×C₇	—	6	1	no	—
49	C₄×C₇	C₂×C₅×D₃	1	1	yes	characteristic
50	D₁₂	C₂×C₅×C₇	1	1	yes	characteristic
51	Dic₆	C₂×C₅×C₇	1	1	yes	characteristic, complemented
52	C₃×C₈	C₂×C₅×C₇	1	1	yes	characteristic, complemented
53	C₃×C₇	C₅×SD₁₆	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₅	—	6	1	no	—
55	C₄×C₅	—	6	1	no	—
56	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
57	SD₁₆	—	3	1	no	Sylow
58	C₃×C₅	C₇×SD₁₆	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	—	12	1	no	—
60	C₂×C₇	C₅×D₁₂	1	1	yes	characteristic
61	C₂×D₃	—	2	1	no	—
62	C₃×C₄	C₂²×C₅×C₇	1	1	yes	characteristic
63	Dic₃	—	2	1	no	—
64	C₂×C₅	—	12	1	no	—
65	C₂×C₅	C₇×D₁₂	1	1	yes	characteristic
66	Q₈	—	3	1	no	—
67	D₄	—	3	1	no	—
68	C₈	C₅×D₃×C₇	1	1	yes	characteristic, complemented
69	C₇	C₅×#204	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	C₅×C₇×D₄	1	1	yes	characteristic
71	D₃	—	4	1	no	—
72	C₅	C₇×#204	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	—	6	1	no	—
74	C₄	—	6	1	no	—
75	C₄	C₂×C₅×D₃×C₇	1	1	yes	characteristic
76	C₃	C₅×C₇×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	12	1	no	—
78	C₂	C₅×C₇×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₇	D₁₂
commutator, cosocle	C₃×C₄	C₂²×C₅×C₇
Frattini	C₄	C₂×C₅×D₃×C₇
Fitting	C₃×C₅×C₇×C₈	C₂
socle	C₂×C₃×C₅×C₇	D₄
prime radical	C₅×C₇×D₁₂	C₂
hypercenter	C₅×C₇×C₈	D₃
hypocenter	C₃	C₅×C₇×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₇ ◁ C₄×C₅×C₇ ◁ C₅×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₇×Dic₆	C₂	semidirect
3. C₃×C₅×C₇×C₈	C₂	semidirect
8. C₇×#204	C₅	direct (N × H)
11. C₅×C₇×C₈	D₃	semidirect
12. C₅×#204	C₇	direct (N × H)
16. C₇×Dic₆	C₂×C₅	semidirect
17. C₃×C₇×C₈	C₂×C₅	semidirect
22. C₅×Dic₆	C₂×C₇	semidirect
23. C₃×C₅×C₈	C₂×C₇	semidirect
25. C₃×C₅×C₇	SD₁₆	semidirect
37. C₇×C₈	C₅×D₃	semidirect
38. #204	C₅×C₇	direct (N × H)
43. C₅×C₈	D₃×C₇	semidirect
44. C₅×C₇	#204	direct (N × H)
51. Dic₆	C₂×C₅×C₇	semidirect
52. C₃×C₈	C₂×C₅×C₇	semidirect
53. C₃×C₇	C₅×SD₁₆	semidirect
58. C₃×C₅	C₇×SD₁₆	semidirect
68. C₈	C₅×D₃×C₇	semidirect
69. C₇	C₅×#204	direct (N × H)
72. C₅	C₇×#204	direct (N × H)
76. C₃	C₅×C₇×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×D₁₂	2
C₅×C₇×Dic₆	2
C₃×C₅×C₇×C₈	2
C₅×C₇×SD₁₆	3
C₇×#204	5
C₅×#204	7

Subgroups — C5×C7×#204 #13132307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×#204 #13132307