Subgroups — C₅×C₄⋊Dic₂₁ #13132330

Order 1680.

104 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	C₅×C₄⋊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	—	2	2	no	—
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	C₅×C₄⋊Dic₃	—	7	1	no	maximal, Hall
11	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
13	C₂×C₃×C₅×C₇	Q₈	1	1	yes	characteristic
14	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic
15	C₂×Dic₂₁	C₂×C₅	1	2	yes	—
16	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
17	C₄×C₅×C₇	Dic₃	1	2	yes	complemented
18	C₅×Dic₇	—	6	2	no	—
19	C₂×C₃×C₄×C₅	D₇	1	1	yes	characteristic
20	C₂×C₅×Dic₃	—	7	2	no	—
21	C₄⋊Dic₇	—	3	1	no	Hall
22	C₃×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
23	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
24	C₃×C₄×C₇	C₄×C₅	1	2	yes	complemented
25	Dic₂₁	—	2	2	no	—
26	C₅×C₄⋊C₄	—	21	1	no	Hall
27	C₂×C₅×C₇	Dic₆	1	1	yes	characteristic
28	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
29	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
30	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
31	C₃×C₄×C₅	Dic₇	1	2	yes	complemented
32	C₅×Dic₃	—	14	2	no	—
33	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic
34	C₂×Dic₇	—	3	2	no	—
35	C₄⋊Dic₃	—	7	1	no	Hall
36	C₂×C₃×C₇	C₂×C₄×C₅	1	1	yes	characteristic
37	C₂×C₃×C₇	C₅×D₄	1	1	yes	characteristic
38	C₂×C₃×C₇	C₅×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅	—	21	2	no	—
40	C₂×C₄×C₅	D₂₁	1	1	yes	characteristic
41	C₅×C₇	C₄⋊Dic₃	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×C₅	Dic₁₄	1	1	yes	characteristic
43	C₂×C₃×C₅	D₂₈	1	1	yes	characteristic
44	C₂×C₃×C₅	C₂×Dic₇	1	1	yes	characteristic
45	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
46	C₄×C₇	C₅×Dic₃	1	2	yes	complemented
47	Dic₇	—	6	2	no	—
48	C₂×C₃×C₄	C₅×D₇	1	1	yes	characteristic
49	C₂×Dic₃	—	7	2	no	—
50	C₃×C₇	C₅×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
51	C₂²×C₅	C₂×D₂₁	1	1	yes	characteristic
52	C₄×C₅	—	42	2	no	—
53	C₄×C₅	Dic₂₁	1	2	yes	complemented
54	C₄⋊C₄	—	21	1	no	Sylow
55	C₃×C₅	C₄⋊Dic₇	1	1	yes	characteristic, complemented, Hall
56	C₂×C₇	C₅×Dic₆	1	1	yes	characteristic
57	C₂×C₇	C₅×D₁₂	1	1	yes	characteristic
58	C₂×C₇	C₂×C₅×Dic₃	1	1	yes	characteristic
59	C₂²×C₃	C₂×C₅×D₇	1	1	yes	characteristic
60	C₃×C₄	C₅×Dic₇	1	2	yes	complemented
61	Dic₃	—	14	2	no	—
62	C₂×C₅	Dic₄₂	1	1	yes	characteristic
63	C₂×C₅	D₈₄	1	1	yes	characteristic
64	C₂×C₅	C₂×Dic₂₁	1	1	yes	characteristic
65	C₂×C₄	—	21	2	no	—
66	C₂×C₄	C₅×D₂₁	1	1	yes	characteristic
67	C₇	C₅×C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃	C₅×Dic₁₄	1	1	yes	characteristic
69	C₂×C₃	C₅×D₂₈	1	1	yes	characteristic
70	C₂×C₃	C₂×C₅×Dic₇	1	1	yes	characteristic
71	C₅	C₄⋊Dic₂₁	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	C₂×C₅×D₂₁	1	1	yes	characteristic
73	C₄	—	42	2	no	—
74	C₄	C₅×Dic₂₁	1	2	yes	complemented
75	C₃	C₅×C₄⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
76	C₂	C₅×Dic₄₂	1	1	yes	characteristic
77	C₂	C₅×D₈₄	1	1	yes	characteristic
78	C₂	C₂×C₅×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂×C₄×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
hypercenter	C₂×C₄×C₅	D₂₁
hypocenter	C₃×C₇	C₅×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₅ ◁ 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
7. C₄⋊Dic₂₁	C₅	direct (N × H)
17. C₄×C₅×C₇	Dic₃	semidirect
22. C₃×C₅×C₇	C₄⋊C₄	semidirect
24. C₃×C₄×C₇	C₄×C₅	semidirect
31. C₃×C₄×C₅	Dic₇	semidirect
41. C₅×C₇	C₄⋊Dic₃	semidirect
46. C₄×C₇	C₅×Dic₃	semidirect
50. C₃×C₇	C₅×C₄⋊C₄	semidirect
53. C₄×C₅	Dic₂₁	semidirect
55. C₃×C₅	C₄⋊Dic₇	semidirect
60. C₃×C₄	C₅×Dic₇	semidirect
67. C₇	C₅×C₄⋊Dic₃	semidirect
71. C₅	C₄⋊Dic₂₁	direct (N × H)
74. C₄	C₅×Dic₂₁	semidirect
75. C₃	C₅×C₄⋊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
C₅×C₄⋊Dic₇	3
C₄⋊Dic₂₁	5
C₅×C₄⋊Dic₃	7

Subgroups — C5×C4⋊Dic21 #13132330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₄⋊Dic₂₁ #13132330