Group #20276931 — subgroups

Subgroups — #20276931

Order 6784.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16810877	C₂	1	1	yes	maximal, characteristic
2	#16810670	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810445	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810437	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810435	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810404	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
9	C₂×#11719912	C₂²	1	1	yes	characteristic
10	M₁₆×D₅₃	—	2	1	no	—
11	#13133192	—	2	1	no	—
12	#13133157	C₂²	1	1	yes	characteristic
13	#13133153	C₂²	1	1	yes	characteristic
14	#13133152	—	2	1	no	—
15	#13133137	—	2	1	no	—
16	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	#108×C₅₃	C₂²	1	1	yes	characteristic
18	#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₂₁₂	—	2	1	no	—
20	C₂×D₂₁₂	—	4	1	no	—
21	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	#11719921	—	4	1	no	—
26	#11719917	—	2	1	no	—
27	C₂×#88322	D₄	1	1	yes	characteristic
28	D₄₂₄	—	2	1	no	—
29	D₄₂₄	—	2	1	no	—
30	#11719912	—	2	1	no	—
31	#11719912	—	2	1	no	—
32	#11719912	—	4	1	no	—
33	C₈×D₅₃	—	4	1	no	—
34	C₁₆×C₅₃	—	2	1	no	—
35	#11719908	—	2	1	no	—
36	C₂²×D₅₃	—	4	1	no	—
37	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
41	D₂₁₂	—	4	1	no	—
42	D₂₁₂	—	4	1	no	—
43	D₂₁₂	—	8	2	no	—
44	C₄×D₅₃	—	2	1	no	—
45	C₄×D₅₃	—	2	1	no	—
46	C₈×C₅₃	—	4	1	no	—
47	C₈×C₅₃	#41	1	1	yes	characteristic
48	C₈×C₅₃	#41	1	1	yes	characteristic
49	#88322	—	2	1	no	—
50	#88322	—	4	1	no	—
51	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×D₅₃	—	2	1	no	—
53	C₂×D₅₃	—	4	1	no	—
54	C₂×D₅₃	—	8	1	no	—
55	C₂×D₅₃	—	8	1	no	—
56	C₂×D₅₃	—	8	1	no	—
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	#123	1	1	yes	characteristic
59	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₅₃	—	2	1	no	—
61	#2238	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	#481	1	1	yes	characteristic
64	D₅₃	—	4	1	no	—
65	D₅₃	—	16	1	no	—
66	#469	—	53	1	no	—
67	#430	—	53	1	no	—
68	#365	—	53	1	no	—
69	#365	D₅₃	1	1	yes	characteristic, complemented
70	#361	—	53	1	no	—
71	#361	—	53	1	no	—
72	#349	—	53	1	no	—
73	C₅₃	#2238	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₈	—	53	1	no	—
75	C₂×M₁₆	—	53	1	no	—
76	C₂×M₁₆	—	106	1	no	—
77	#121	—	106	1	no	—
78	M₃₂	—	53	1	no	—
79	M₃₂	C₂×D₅₃	1	1	yes	characteristic
80	#108	—	53	1	no	—
81	#108	—	106	1	no	—
82	#108	C₂×D₅₃	1	1	yes	characteristic
83	#106	C₂×D₅₃	1	1	yes	characteristic
84	#102	—	106	1	no	—
85	C₂×D₄	—	106	1	no	—
86	C₂×D₄	—	212	1	no	—
87	C₂²×C₄	—	53	1	no	—
88	D₈	—	106	1	no	—
89	D₈	—	106	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	106	1	no	—
92	M₁₆	—	106	1	no	—
93	M₁₆	—	106	1	no	—
94	M₁₆	—	212	1	no	—
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	—	212	1	no	—
97	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	212	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	106	1	no	—
102	C₂³	—	53	1	no	—
103	C₂³	—	212	1	no	—
104	D₄	—	424	2	no	—
105	D₄	—	212	1	no	—
106	D₄	—	212	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	106	1	no	—
110	C₂×C₄	—	106	1	no	—
111	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	106	1	no	—
114	C₈	—	212	1	no	—
115	C₈	#11719945	1	1	yes	characteristic
116	C₈	#11719949	1	1	yes	characteristic
117	C₂²	—	106	1	no	—
118	C₂²	—	424	1	no	—
119	C₂²	—	424	1	no	—
120	C₂²	—	424	1	no	—
121	C₂²	—	212	1	no	—
122	C₂²	SD₁₆×D₅₃	1	1	yes	characteristic
123	C₄	—	8	1	no	—
124	C₄	—	106	1	no	—
125	C₄	#13133202	1	1	yes	characteristic
126	C₄	#13133244	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	848	1	no	—
129	C₂	—	212	1	no	—
130	C₂	#16810943	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810877

C₂

yes

maximal, characteristic

#16810670

C₂

yes

maximal, characteristic, complemented

#16810445

C₂

yes

maximal, characteristic, complemented

#16810437

C₂

yes

maximal, characteristic, complemented

#16810435

C₂

yes

maximal, characteristic, complemented

#16810404

C₂

yes

maximal, characteristic, complemented

C₅₃×#365

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×#11719912

C₂²

yes

characteristic

M₁₆×D₅₃

—

#13133192

—

#13133157

C₂²

yes

characteristic

#13133153

C₂²

yes

characteristic

#13133152

—

#13133137

—

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

D₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719921

—

#11719917

—

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

#11719912

—

#11719912

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

C₂²×D₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2238

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#481

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#361

—

#361

—

#349

—

C₅₃

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

#121

—

106

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

characteristic

#108

—

#108

—

106

—

#108

C₂×D₅₃

yes

characteristic

#106

C₂×D₅₃

yes

characteristic

#102

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

D₈

—

106

—

D₈

—

106

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

212

—

100

C₁₆

—

101

C₁₆

—

106

—

102

C₂³

—

103

C₂³

—

212

—

104

D₄

—

424

—

105

D₄

—

212

—

106

D₄

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

D₄×D₅₃

yes

characteristic

112

C₈

—

113

C₈

—

106

—

114

C₈

—

212

—

115

C₈

#11719945

yes

characteristic

116

C₈

#11719949

yes

characteristic

117

C₂²

—

106

—

118

C₂²

—

424

—

119

C₂²

—

424

—

120

C₂²

—

424

—

121

C₂²

—

212

—

122

C₂²

SD₁₆×D₅₃

yes

characteristic

123

C₄

—

124

C₄

—

106

—

125

C₄

#13133202

yes

characteristic

126

C₄

#13133244

yes

characteristic

127

C₂

—

128

C₂

—

848

—

129

C₂

—

212

—

130

C₂

#16810943

yes

characteristic

	subgroup	G/H
center	C₂	#16810943
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#365	C₂
socle	C₂×C₅₃	#481
hypercenter	#365	D₅₃
hypocenter	C₅₃	#2238
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16810943

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#365

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
2. #16810670	C₂	semidirect
3. #16810445	C₂	semidirect
4. #16810437	C₂	semidirect
5. #16810435	C₂	semidirect
6. #16810404	C₂	semidirect
7. C₅₃×#365	C₂	semidirect
69. #365	D₅₃	semidirect
73. C₅₃	#2238	semidirect

H ≅ G/N

type

2. #16810670

C₂

semidirect

3. #16810445

C₂

semidirect

4. #16810437

C₂

semidirect

5. #16810435

C₂

semidirect

6. #16810404

C₂

semidirect

7. C₅₃×#365

C₂

semidirect

69. #365

D₅₃

semidirect

73. C₅₃

#2238

semidirect

Subgroups — #20276931

Conjugacy classes of subgroups

Complements