Group #20276884 — subgroups

Subgroups — #20276884

Order 6784.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811020	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810740	C₂	1	1	yes	maximal, characteristic
3	#16810446	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810445	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810443	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810433	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#360	C₂	1	1	yes	maximal, characteristic
8	#13133224	—	2	1	no	—
9	#13133216	—	2	1	no	—
10	#13133157	C₂²	1	1	yes	characteristic
11	#13133153	C₂²	1	1	yes	characteristic
12	#13133149	C₂²	1	1	yes	characteristic
13	#13133148	—	2	1	no	—
14	#13133144	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	#135×C₅₃	C₂²	1	1	yes	characteristic
17	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	#106×C₅₃	C₂²	1	1	yes	characteristic
19	#41×C₅₃	—	2	1	no	—
20	C₂×Dic₁₀₆	—	4	1	no	—
21	Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
22	SD₁₆×C₅₃	—	2	1	no	—
23	D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄⋊Dic₅₃	—	2	1	no	—
27	#11719919	—	4	1	no	—
28	C₄×Dic₅₃	D₄	1	1	yes	characteristic
29	#11719917	—	2	1	no	—
30	#11719917	—	4	1	no	—
31	C₂×#88322	—	4	1	no	—
32	C₂×#88322	D₄	1	1	yes	characteristic
33	C₁₆×C₅₃	—	2	1	no	—
34	#11719908	—	2	1	no	—
35	Q₈×C₅₃	—	2	1	no	—
36	D₄×C₅₃	—	2	1	no	—
37	D₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	1	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	Dic₁₀₆	—	8	2	no	—
44	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₃	#41	1	1	yes	characteristic
46	#88322	—	4	1	no	—
47	#88322	SD₁₆	1	1	yes	characteristic
48	#88322	SD₁₆	1	1	yes	characteristic
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₅₃	—	4	1	no	—
52	C₄×C₅₃	—	8	1	no	—
53	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₅₃	—	2	2	no	—
56	Dic₅₃	—	8	1	no	—
57	#2230	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	8	1	no	—
60	C₂×C₅₃	#476	1	1	yes	characteristic
61	#498	—	53	1	no	—
62	#443	—	53	1	no	—
63	#365	—	53	1	no	—
64	#365	—	53	1	no	—
65	#364	—	53	1	no	—
66	#360	—	53	1	no	—
67	#360	D₅₃	1	1	yes	characteristic
68	C₅₃	#2230	1	1	yes	characteristic, complemented, Sylow
69	#135	C₂×D₅₃	1	1	yes	characteristic
70	#131	—	106	1	no	—
71	C₄⋊Q₈	—	106	1	no	—
72	M₃₂	—	53	1	no	—
73	M₃₂	C₂×D₅₃	1	1	yes	characteristic
74	#108	—	53	1	no	—
75	#106	—	53	1	no	—
76	#106	—	106	1	no	—
77	#106	C₂×D₅₃	1	1	yes	characteristic
78	#104	—	106	1	no	—
79	C₄×C₈	—	53	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	212	1	no	—
82	Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
85	M₁₆	—	106	1	no	—
86	M₁₆	—	212	1	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	—	212	1	no	—
89	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	106	1	no	—
92	C₄⋊C₄	—	212	1	no	—
93	C₄²	—	53	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	106	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	424	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	106	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
105	C₈	—	53	1	no	—
106	C₈	—	53	1	no	—
107	C₈	—	212	1	no	—
108	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
109	C₈	#11719947	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#13133241	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	106	2	no	—
115	C₄	—	424	1	no	—
116	C₄	#13133194	1	1	yes	characteristic
117	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#16810911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811020

C₂

yes

maximal, characteristic, complemented

#16810740

C₂

yes

maximal, characteristic

#16810446

C₂

yes

maximal, characteristic, complemented

#16810445

C₂

yes

maximal, characteristic, complemented

#16810443

C₂

yes

maximal, characteristic, complemented

#16810433

C₂

yes

maximal, characteristic

C₅₃×#360

C₂

yes

maximal, characteristic

#13133224

—

#13133216

—

#13133157

C₂²

yes

characteristic

#13133153

C₂²

yes

characteristic

#13133149

C₂²

yes

characteristic

#13133148

—

#13133144

—

C₈×Dic₅₃

C₂²

yes

characteristic

#135×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

Q₈

yes

characteristic, complemented

SD₁₆×C₅₃

—

D₈×C₅₃

Q₈

yes

characteristic, complemented

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₁₆×C₅₃

—

#11719908

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

C₈×C₅₃

yes

—

SD₁₆

yes

characteristic

#88322

SD₁₆

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2230

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2230

yes

characteristic, complemented, Sylow

#135

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₄⋊Q₈

—

106

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

—

—

—

106

—

#106

C₂×D₅₃

yes

characteristic

#104

—

106

—

C₄×C₈

—

#41

—

C₂×Q₈

—

212

—

Q₁₆

Dic₁₀₆

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₀₆

yes

characteristic, complemented

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

Q₈

—

Q₈

—

424

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

212

—

104

C₂×C₄

D₄×D₅₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

212

—

108

C₈

C₂×Dic₁₀₆

yes

characteristic

109

C₈

#11719947

yes

characteristic

110

C₂²

—

111

C₂²

#13133241

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

106

—

115

C₄

—

424

—

116

C₄

#13133194

yes

characteristic

117

C₄

SD₁₆×D₅₃

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#16810911

yes

characteristic

	subgroup	G/H
center	C₂	#16810911
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#360	C₂
socle	C₂×C₅₃	#476
hypercenter	#360	D₅₃
hypocenter	C₅₃	#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#16810911

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#360

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #16811020	C₂	semidirect
3. #16810446	C₂	semidirect
4. #16810445	C₂	semidirect
5. #16810443	C₂	semidirect
21. Q₁₆×C₅₃	Q₈	semidirect
23. D₈×C₅₃	Q₈	semidirect
68. C₅₃	#2230	semidirect
82. Q₁₆	Dic₁₀₆	semidirect
84. D₈	Dic₁₀₆	semidirect

H ≅ G/N

type

1. #16811020

C₂

semidirect

3. #16810446

C₂

semidirect

4. #16810445

C₂

semidirect

5. #16810443

C₂

semidirect

21. Q₁₆×C₅₃

Q₈

semidirect

23. D₈×C₅₃

Q₈

semidirect

68. C₅₃

#2230

semidirect

82. Q₁₆

Dic₁₀₆

semidirect

84. D₈

Dic₁₀₆

semidirect

Subgroups — #20276884

Conjugacy classes of subgroups

Complements