Group #20276834 — subgroups

Subgroups — #20276834

Order 6784.

148 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 43 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	C₂×#13133158	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811041	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810894	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810890	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810452	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810437	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133245	—	2	1	no	—
9	#13133243	—	2	1	no	—
10	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
11	#13133223	C₂²	1	1	yes	characteristic
12	#13133158	—	2	1	no	—
13	#13133158	—	2	1	no	—
14	#13133156	—	2	1	no	—
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	C₁₆×D₅₃	—	2	1	no	—
17	#13133153	C₂²	1	1	yes	characteristic
18	#13133134	—	2	1	no	—
19	#13133133	—	2	1	no	—
20	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
21	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	#108×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₃	—	2	1	no	—
24	#11719945	D₄	1	1	yes	characteristic
25	C₂×D₂₁₂	—	2	1	no	—
26	D₈×C₅₃	—	2	1	no	—
27	D₈×C₅₃	—	2	1	no	—
28	M₁₆×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	#11719923	—	4	1	no	—
31	#11719922	—	4	1	no	—
32	#11719917	—	2	1	no	—
33	#11719917	D₄	1	1	yes	characteristic
34	D₄₂₄	—	2	1	no	—
35	D₄₂₄	—	2	1	no	—
36	D₄₂₄	—	4	1	no	—
37	#11719913	—	4	1	no	—
38	#11719912	—	2	1	no	—
39	C₈×D₅₃	—	2	1	no	—
40	C₁₆×C₅₃	—	2	1	no	—
41	#11719908	D₄	1	1	yes	characteristic
42	#11719908	D₄	1	1	yes	characteristic
43	C₂³×C₅₃	—	4	1	no	—
44	C₂²×D₅₃	—	4	1	no	—
45	D₄×C₅₃	—	4	1	no	—
46	D₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	2	1	no	—
49	D₂₁₂	—	4	1	no	—
50	D₂₁₂	—	4	1	no	—
51	D₂₁₂	D₈	1	1	yes	characteristic
52	C₄×D₅₃	—	2	1	no	—
53	Dic₁₀₆	D₈	1	1	yes	characteristic
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₈×C₅₃	#41	1	1	yes	characteristic
57	#88322	—	2	1	no	—
58	#88322	—	4	1	no	—
59	C₂²×C₅₃	—	8	1	no	—
60	C₂²×C₅₃	—	16	1	no	—
61	C₂²×C₅₃	#135	1	1	yes	characteristic
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	8	1	no	—
64	C₂×D₅₃	—	16	1	no	—
65	C₄×C₅₃	#121	1	1	yes	characteristic
66	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	Dic₅₃	—	2	1	no	—
68	#2221	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	—	16	1	no	—
71	C₂×C₅₃	#466	1	1	yes	characteristic
72	D₅₃	—	4	1	no	—
73	D₅₃	—	16	1	no	—
74	C₂×D₁₆	—	53	1	no	—
75	#504	—	53	1	no	—
76	#472	—	53	1	no	—
77	#472	—	53	1	no	—
78	#368	—	53	1	no	—
79	#361	—	53	1	no	—
80	#361	D₅₃	1	1	yes	characteristic, complemented
81	C₅₃	#2221	1	1	yes	characteristic, complemented, Sylow
82	AGL(1,ℤ/8)	—	106	1	no	—
83	AGL(1,ℤ/8)	—	106	1	no	—
84	C₂×D₈	—	53	1	no	—
85	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
86	#131	—	53	1	no	—
87	D₁₆	—	106	1	no	—
88	D₁₆	—	106	1	no	—
89	M₃₂	—	106	1	no	—
90	M₃₂	C₂×D₅₃	1	1	yes	characteristic
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	106	1	no	—
93	#108	—	53	1	no	—
94	#108	C₂×D₅₃	1	1	yes	characteristic
95	#100	—	106	1	no	—
96	#100	—	106	1	no	—
97	#41	—	53	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	106	1	no	—
100	SD₁₆	—	212	1	no	—
101	SD₁₆	—	212	1	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	106	1	no	—
105	D₈	—	106	1	no	—
106	D₈	—	212	1	no	—
107	D₈	—	212	1	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	53	1	no	—
110	M₁₆	—	106	1	no	—
111	M₁₆	—	106	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	53	1	no	—
116	C₁₆	—	53	1	no	—
117	C₂³	—	4	1	no	—
118	C₂³	—	212	1	no	—
119	Q₈	—	53	1	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	53	1	no	—
123	D₄	—	106	1	no	—
124	D₄	—	212	1	no	—
125	D₄	—	212	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₈	—	4	1	no	—
129	C₈	—	106	1	no	—
130	C₈	—	212	1	no	—
131	C₈	#11719945	1	1	yes	characteristic
132	C₈	D₄×D₅₃	1	1	yes	characteristic
133	C₂²	—	8	1	no	—
134	C₂²	—	16	1	no	—
135	C₂²	—	848	1	no	—
136	C₂²	—	106	1	no	—
137	C₂²	—	424	1	no	—
138	C₂²	#13133238	1	1	yes	characteristic
139	C₄	—	106	1	no	—
140	C₄	#13133190	1	1	yes	characteristic
141	C₄	D₈×D₅₃	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	16	1	no	—
144	C₂	—	848	1	no	—
145	C₂	—	212	1	no	—
146	C₂	#16810858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133158

C₂

yes

maximal, characteristic, complemented

#16811041

C₂

yes

maximal, characteristic, complemented

#16810894

C₂

yes

maximal, characteristic, complemented

#16810890

C₂

yes

maximal, characteristic, complemented

#16810452

C₂

yes

maximal, characteristic, complemented

#16810437

C₂

yes

maximal, characteristic, complemented

C₅₃×#361

C₂

yes

maximal, characteristic, complemented

#13133245

—

#13133243

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133223

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₁₆×D₅₃

—

#13133153

C₂²

yes

characteristic

#13133134

—

#13133133

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

#11719945

D₄

yes

characteristic

C₂×D₂₁₂

—

D₈×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

—

—

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

D₄

yes

characteristic

#11719908

D₄

yes

characteristic

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

D₈

yes

characteristic

C₄×D₅₃

—

Dic₁₀₆

D₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#135

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2221

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#466

yes

characteristic

D₅₃

—

D₅₃

—

C₂×D₁₆

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2221

yes

characteristic, complemented, Sylow

AGL(1,ℤ/8)

—

106

—

AGL(1,ℤ/8)

—

106

—

C₂×D₈

—

C₂×D₈

C₂×D₅₃

yes

characteristic

#131

—

D₁₆

—

106

—

D₁₆

—

106

—

M₃₂

—

106

—

M₃₂

C₂×D₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#108

—

#108

C₂×D₅₃

yes

characteristic

#100

—

106

—

#100

—

106

—

#41

—

C₂×D₄

—

C₂×D₄

—

106

—

100

SD₁₆

—

212

—

101

SD₁₆

—

212

—

102

D₈

—

103

D₈

—

104

D₈

—

106

—

105

D₈

—

106

—

106

D₈

—

212

—

107

D₈

—

212

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

106

—

111

M₁₆

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

C₂²×D₅₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₂³

—

118

C₂³

—

212

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

106

—

124

D₄

—

212

—

125

D₄

—

212

—

126

C₂×C₄

—

106

—

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

106

—

130

C₈

—

212

—

131

C₈

#11719945

yes

characteristic

132

C₈

D₄×D₅₃

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

848

—

136

C₂²

—

106

—

137

C₂²

—

424

—

138

C₂²

#13133238

yes

characteristic

139

C₄

—

106

—

140

C₄

#13133190

yes

characteristic

141

C₄

D₈×D₅₃

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

848

—

145

C₂

—

212

—

146

C₂

#16810858

yes

characteristic

	subgroup	G/H
center	C₂	#16810858
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#361	C₂
socle	C₂×C₅₃	#466
hypercenter	#361	D₅₃
hypocenter	C₅₃	#2221
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#16810858

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#361

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
1. C₂×#13133158	C₂	semidirect
2. #16811041	C₂	semidirect
3. #16810894	C₂	semidirect
4. #16810890	C₂	semidirect
5. #16810452	C₂	semidirect
6. #16810437	C₂	semidirect
7. C₅₃×#361	C₂	semidirect
80. #361	D₅₃	semidirect
81. C₅₃	#2221	semidirect

H ≅ G/N

type

1. C₂×#13133158

C₂

semidirect

2. #16811041

C₂

semidirect

3. #16810894

C₂

semidirect

4. #16810890

C₂

semidirect

5. #16810452

C₂

semidirect

6. #16810437

C₂

semidirect

7. C₅₃×#361

C₂

semidirect

80. #361

D₅₃

semidirect

81. C₅₃

#2221

semidirect

Subgroups — #20276834

Conjugacy classes of subgroups

Complements