Subgroups — C₄×C₅×Dic₂₁ #13132328

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×Dic₂₁	C₂	1	2	yes	maximal
3	C₄×C₅×Dic₇	—	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×Dic₂₁	C₄	1	4	yes	direct product factor
7	C₄×Dic₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
9	C₂×C₅×Dic₇	—	3	2	no	—
10	C₄×C₅×Dic₃	—	7	1	no	maximal, 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×Dic₂₁	C₂×C₅	1	2	yes	—
15	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
16	C₄×C₅×C₇	Dic₃	1	2	yes	complemented
17	C₅×Dic₇	—	3	4	no	—
18	C₂×C₃×C₄×C₅	D₇	1	1	yes	characteristic
19	C₂×C₅×Dic₃	—	7	2	no	—
20	C₄×Dic₇	—	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	Dic₂₁	C₄×C₅	1	4	yes	direct product factor
25	C₄²×C₅	—	21	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	C₂×D₇	1	1	yes	characteristic
29	C₃×C₄×C₅	Dic₇	1	2	yes	complemented
30	C₅×Dic₃	—	7	4	no	—
31	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic
32	C₂×Dic₇	—	3	2	no	—
33	C₄×Dic₃	—	7	1	no	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	—	21	2	no	—
37	C₂×C₄×C₅	D₂₁	1	1	yes	characteristic
38	C₅×C₇	C₄×Dic₃	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	C₄×D₇	1	2	yes	—
40	C₂×C₃×C₅	C₂×Dic₇	1	1	yes	characteristic
41	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
42	C₄×C₇	C₅×Dic₃	1	2	yes	complemented
43	Dic₇	—	3	4	no	—
44	C₂×C₃×C₄	C₅×D₇	1	1	yes	characteristic
45	C₂×Dic₃	—	7	2	no	—
46	C₃×C₇	C₄²×C₅	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₅	C₂×D₂₁	1	1	yes	characteristic
48	C₄×C₅	—	21	4	no	—
49	C₄×C₅	Dic₂₁	1	2	yes	direct product factor
50	C₄²	—	21	1	no	Sylow
51	C₃×C₅	C₄×Dic₇	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	C₂×C₅×D₇	1	1	yes	characteristic
55	C₃×C₄	C₅×Dic₇	1	2	yes	complemented
56	Dic₃	—	7	4	no	—
57	C₂×C₅	C₄×D₂₁	1	2	yes	—
58	C₂×C₅	C₂×Dic₂₁	1	1	yes	characteristic
59	C₂×C₄	—	21	2	no	—
60	C₂×C₄	C₅×D₂₁	1	1	yes	characteristic
61	C₇	C₄×C₅×Dic₃	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	C₄×C₅×D₇	1	2	yes	—
63	C₂×C₃	C₂×C₅×Dic₇	1	1	yes	characteristic
64	C₅	C₄×Dic₂₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₂×C₅×D₂₁	1	1	yes	characteristic
66	C₄	—	21	4	no	—
67	C₄	C₅×Dic₂₁	1	2	yes	direct product factor
68	C₃	C₄×C₅×Dic₇	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₄×C₅×D₂₁	1	2	yes	—
70	C₂	C₂×C₅×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₂₁
commutator, hypocenter	C₃×C₇	C₄²×C₅
Frattini	C₂²	C₂×C₅×D₂₁
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
cosocle	C₂²×C₃×C₇	C₂²×C₅
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄×C₅

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×Dic₂₁	C₄	direct (N × H)
7. C₄×Dic₂₁	C₅	direct (N × H)
16. C₄×C₅×C₇	Dic₃	semidirect
21. C₃×C₅×C₇	C₄²	semidirect
23. C₃×C₄×C₇	C₄×C₅	semidirect
24. Dic₂₁	C₄×C₅	direct (N × H)
29. C₃×C₄×C₅	Dic₇	semidirect
38. C₅×C₇	C₄×Dic₃	semidirect
42. C₄×C₇	C₅×Dic₃	semidirect
46. C₃×C₇	C₄²×C₅	semidirect
49. C₄×C₅	Dic₂₁	direct (N × H)
51. C₃×C₅	C₄×Dic₇	semidirect
55. C₃×C₄	C₅×Dic₇	semidirect
61. C₇	C₄×C₅×Dic₃	semidirect
64. C₅	C₄×Dic₂₁	direct (N × H)
67. C₄	C₅×Dic₂₁	direct (N × H)
68. C₃	C₄×C₅×Dic₇	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×Dic₂₁	2
C₄×C₅×Dic₇	3
C₄×Dic₂₁	5
C₄×C₅×Dic₃	7

Subgroups — C4×C5×Dic21 #13132328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×Dic₂₁ #13132328