Subgroups — C₂×C₇×D₆₀ #13132882

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₁₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₆₀	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×D₂₀	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×D₁₅	—	2	4	no	—
7	C₂×C₇×D₁₅	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅×C₇	C₂²	1	2	yes	complemented
9	C₂×C₇×D₁₂	—	5	1	no	maximal, Hall
10	C₂²×C₇×D₅	—	3	2	no	—
11	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic, complemented
12	C₇×D₂₀	—	3	4	no	—
13	C₂×D₆₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₇	D₄	1	2	yes	complemented
15	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₇×D₁₅	—	2	4	no	—
17	C₂²×D₃×C₇	—	5	2	no	—
18	C₂×C₃×C₄×C₇	D₅	1	1	yes	characteristic, complemented
19	C₇×D₁₂	—	5	4	no	—
20	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	C₂×C₇×D₅	—	3	4	no	—
22	C₂×C₇×D₅	—	6	4	no	—
23	C₄×C₅×C₇	C₂×D₃	1	2	yes	complemented
24	C₂²×D₁₅	C₂×C₇	1	2	yes	complemented
25	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic, complemented
26	D₆₀	C₂×C₇	1	4	yes	direct product factor
27	C₂×C₇×D₄	—	15	1	no	Hall
28	C₃×C₅×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
30	C₂×D₃×C₇	—	5	4	no	—
31	C₂×D₃×C₇	—	10	4	no	—
32	C₃×C₄×C₇	C₂×D₅	1	2	yes	complemented
33	C₂×D₂₀	—	3	1	no	Hall
34	C₂×C₅×C₇	D₁₂	1	2	yes	complemented
35	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₇×D₅	—	6	4	no	—
37	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
38	C₂×D₁₅	—	2	4	no	—
39	C₂×D₁₅	C₂²×C₇	1	4	yes	complemented
40	C₃×C₄×C₅	C₂²×C₇	1	2	yes	complemented
41	C₂³×C₇	—	15	2	no	—
42	C₇×D₄	—	15	4	no	—
43	C₂×C₄×C₇	D₁₅	1	1	yes	characteristic, complemented
44	C₂×D₁₂	—	5	1	no	Hall
45	C₂×C₃×C₇	D₂₀	1	2	yes	complemented
46	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
47	D₃×C₇	—	10	4	no	—
48	C₂²×D₅	—	3	2	no	—
49	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic, complemented
50	D₂₀	—	3	4	no	—
51	C₅×C₇	C₂×D₁₂	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	C₇×D₄	1	2	yes	complemented
53	C₂×C₃×C₅	C₂³×C₇	1	1	yes	characteristic
54	D₁₅	—	2	4	no	—
55	C₂²×C₇	—	15	4	no	—
56	C₂²×C₇	—	30	4	no	—
57	C₂²×C₇	C₂×D₁₅	1	1	yes	characteristic
58	C₄×C₇	C₂×D₁₅	1	2	yes	complemented
59	C₂²×D₃	—	5	2	no	—
60	C₂×C₃×C₄	C₇×D₅	1	1	yes	characteristic, complemented
61	D₁₂	—	5	4	no	—
62	C₃×C₇	C₂×D₂₀	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	C₂×D₃×C₇	1	1	yes	characteristic
64	C₂×D₅	—	3	4	no	—
65	C₂×D₅	—	6	4	no	—
66	C₄×C₅	C₂×D₃×C₇	1	2	yes	complemented
67	C₂×D₄	—	15	1	no	Sylow
68	C₃×C₅	C₂×C₇×D₄	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	—	30	4	no	—
70	C₂×C₇	D₆₀	1	2	yes	direct product factor
71	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
72	C₂²×C₃	C₂×C₇×D₅	1	1	yes	characteristic
73	C₂×D₃	—	5	4	no	—
74	C₂×D₃	—	10	4	no	—
75	C₃×C₄	C₂×C₇×D₅	1	2	yes	complemented
76	C₂×C₅	C₇×D₁₂	1	2	yes	complemented
77	C₂×C₅	C₂²×D₃×C₇	1	1	yes	characteristic
78	D₅	—	6	4	no	—
79	C₂³	—	15	2	no	—
80	D₄	—	15	4	no	—
81	C₂×C₄	C₇×D₁₅	1	1	yes	characteristic, complemented
82	C₇	C₂×D₆₀	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	C₇×D₂₀	1	2	yes	complemented
84	C₂×C₃	C₂²×C₇×D₅	1	1	yes	characteristic
85	D₃	—	10	4	no	—
86	C₅	C₂×C₇×D₁₂	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	15	4	no	—
88	C₂²	—	30	4	no	—
89	C₂²	C₂×C₇×D₁₅	1	1	yes	characteristic
90	C₄	C₂×C₇×D₁₅	1	2	yes	complemented
91	C₃	C₂×C₇×D₂₀	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	30	4	no	—
93	C₂	C₇×D₆₀	1	2	yes	direct product factor
94	C₂	C₂²×C₇×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂³×C₇
Frattini	C₂	C₂²×C₇×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₄×C₇	D₁₅
hypocenter	C₃×C₅	C₂×C₇×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×D₁₅	C₂	semidirect
2. C₂×C₃×C₄×C₅×C₇	C₂	semidirect
3. C₇×D₆₀	C₂	direct (N × H)
7. C₂×C₇×D₁₅	C₂²	semidirect
8. C₃×C₄×C₅×C₇	C₂²	semidirect
11. C₂×C₄×C₅×C₇	D₃	semidirect
13. C₂×D₆₀	C₇	direct (N × H)
14. C₂×C₃×C₅×C₇	D₄	semidirect
18. C₂×C₃×C₄×C₇	D₅	semidirect
23. C₄×C₅×C₇	C₂×D₃	semidirect
24. C₂²×D₁₅	C₂×C₇	semidirect
25. C₂×C₃×C₄×C₅	C₂×C₇	semidirect
26. D₆₀	C₂×C₇	direct (N × H)
28. C₃×C₅×C₇	C₂×D₄	semidirect
32. C₃×C₄×C₇	C₂×D₅	semidirect
34. C₂×C₅×C₇	D₁₂	semidirect
39. C₂×D₁₅	C₂²×C₇	semidirect
40. C₃×C₄×C₅	C₂²×C₇	semidirect
43. C₂×C₄×C₇	D₁₅	semidirect
45. C₂×C₃×C₇	D₂₀	semidirect
49. C₂×C₄×C₅	D₃×C₇	semidirect
51. C₅×C₇	C₂×D₁₂	semidirect
52. C₂×C₃×C₅	C₇×D₄	semidirect
58. C₄×C₇	C₂×D₁₅	semidirect
60. C₂×C₃×C₄	C₇×D₅	semidirect
62. C₃×C₇	C₂×D₂₀	semidirect
66. C₄×C₅	C₂×D₃×C₇	semidirect
68. C₃×C₅	C₂×C₇×D₄	semidirect
70. C₂×C₇	D₆₀	direct (N × H)
75. C₃×C₄	C₂×C₇×D₅	semidirect
76. C₂×C₅	C₇×D₁₂	semidirect
81. C₂×C₄	C₇×D₁₅	semidirect
82. C₇	C₂×D₆₀	direct (N × H)
83. C₂×C₃	C₇×D₂₀	semidirect
86. C₅	C₂×C₇×D₁₂	semidirect
90. C₄	C₂×C₇×D₁₅	semidirect
91. C₃	C₂×C₇×D₂₀	semidirect
93. C₂	C₇×D₆₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₁₅	2
C₂×C₃×C₄×C₅×C₇	2
C₇×D₆₀	2
C₂×C₇×D₂₀	3
C₂×C₇×D₁₂	5
C₂×D₆₀	7

Subgroups — C2×C7×D60 #13132882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×D₆₀ #13132882