Subgroups — C₄×C₅×C₇×Dic₃ #13132312

Order 1680.

120 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	C₂×C₅×C₇×Dic₃	C₂	1	2	yes	maximal
3	C₄²×C₅×C₇	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₃×C₄×C₅×C₇	C₄	1	2	yes	complemented
6	C₅×C₇×Dic₃	C₄	1	4	yes	direct product factor
7	C₄×C₇×Dic₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₇	—	3	2	no	—
9	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
10	C₄×C₅×Dic₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×C₇	C₂×C₄	1	2	yes	—
12	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic
14	C₂×C₇×Dic₃	C₂×C₅	1	2	yes	—
15	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
16	C₄×C₅×C₇	—	3	4	no	—
17	C₄×C₅×C₇	Dic₃	1	2	yes	direct product factor
18	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic
19	C₂×C₅×Dic₃	C₂×C₇	1	2	yes	—
20	C₄²×C₇	—	3	1	no	Hall
21	C₃×C₅×C₇	C₄²	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
23	C₃×C₄×C₇	C₄×C₅	1	2	yes	complemented
24	C₇×Dic₃	C₄×C₅	1	4	yes	direct product factor
25	C₄²×C₅	—	3	1	no	Hall
26	C₂×C₅×C₇	C₄×D₃	1	2	yes	—
27	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
29	C₃×C₄×C₅	C₄×C₇	1	2	yes	complemented
30	C₅×Dic₃	C₄×C₇	1	4	yes	direct product factor
31	C₂×C₄×C₇	—	3	2	no	—
32	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic
33	C₄×Dic₃	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₇	C₂×C₄×C₅	1	2	yes	—
35	C₂×C₃×C₇	C₂×C₄×C₅	1	1	yes	characteristic
36	C₂×C₄×C₅	—	3	2	no	—
37	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic
38	C₅×C₇	C₄×Dic₃	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₃×C₅	C₂×C₄×C₇	1	2	yes	—
40	C₂×C₃×C₅	C₂×C₄×C₇	1	1	yes	characteristic
41	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
42	C₄×C₇	—	3	4	no	—
43	C₄×C₇	C₅×Dic₃	1	2	yes	direct product factor
44	C₂×C₃×C₄	C₂×C₅×C₇	1	1	yes	characteristic
45	C₂×Dic₃	C₂×C₅×C₇	1	2	yes	—
46	C₃×C₇	C₄²×C₅	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₅	C₂×D₃×C₇	1	1	yes	characteristic
48	C₄×C₅	—	3	4	no	—
49	C₄×C₅	C₇×Dic₃	1	2	yes	direct product factor
50	C₄²	—	3	1	no	Sylow
51	C₃×C₅	C₄²×C₇	1	1	yes	characteristic, complemented, Hall
52	C₂×C₇	C₄×C₅×D₃	1	2	yes	—
53	C₂×C₇	C₂×C₅×Dic₃	1	1	yes	characteristic
54	C₂²×C₃	C₂²×C₅×C₇	1	1	yes	characteristic
55	C₃×C₄	C₄×C₅×C₇	1	2	yes	complemented
56	Dic₃	C₄×C₅×C₇	1	4	yes	direct product factor
57	C₂×C₅	C₄×D₃×C₇	1	2	yes	—
58	C₂×C₅	C₂×C₇×Dic₃	1	1	yes	characteristic
59	C₂×C₄	—	3	2	no	—
60	C₂×C₄	C₅×D₃×C₇	1	1	yes	characteristic
61	C₇	C₄×C₅×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃	C₂×C₄×C₅×C₇	1	2	yes	—
63	C₂×C₃	C₂×C₄×C₅×C₇	1	1	yes	characteristic
64	C₅	C₄×C₇×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₂×C₅×D₃×C₇	1	1	yes	characteristic
66	C₄	—	3	4	no	—
67	C₄	C₅×C₇×Dic₃	1	2	yes	direct product factor
68	C₃	C₄²×C₅×C₇	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₄×C₅×D₃×C₇	1	2	yes	—
70	C₂	C₂×C₅×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅×C₇	D₃
commutator, hypocenter	C₃	C₄²×C₅×C₇
Frattini	C₂²	C₂×C₅×D₃×C₇
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
cosocle	C₂²×C₃	C₂²×C₅×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅×C₇

Complements

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

N	H ≅ G/N	type
5. C₃×C₄×C₅×C₇	C₄	semidirect
6. C₅×C₇×Dic₃	C₄	direct (N × H)
7. C₄×C₇×Dic₃	C₅	direct (N × H)
10. C₄×C₅×Dic₃	C₇	direct (N × H)
17. C₄×C₅×C₇	Dic₃	direct (N × H)
21. C₃×C₅×C₇	C₄²	semidirect
23. C₃×C₄×C₇	C₄×C₅	semidirect
24. C₇×Dic₃	C₄×C₅	direct (N × H)
29. C₃×C₄×C₅	C₄×C₇	semidirect
30. C₅×Dic₃	C₄×C₇	direct (N × H)
33. C₄×Dic₃	C₅×C₇	direct (N × H)
38. C₅×C₇	C₄×Dic₃	direct (N × H)
43. C₄×C₇	C₅×Dic₃	direct (N × H)
46. C₃×C₇	C₄²×C₅	semidirect
49. C₄×C₅	C₇×Dic₃	direct (N × H)
51. C₃×C₅	C₄²×C₇	semidirect
55. C₃×C₄	C₄×C₅×C₇	semidirect
56. Dic₃	C₄×C₅×C₇	direct (N × H)
61. C₇	C₄×C₅×Dic₃	direct (N × H)
64. C₅	C₄×C₇×Dic₃	direct (N × H)
67. C₄	C₅×C₇×Dic₃	direct (N × H)
68. C₃	C₄²×C₅×C₇	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
C₂×C₅×C₇×Dic₃	2
C₄²×C₅×C₇	3
C₄×C₇×Dic₃	5
C₄×C₅×Dic₃	7

Subgroups — C4×C5×C7×Dic3 #13132312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₇×Dic₃ #13132312