Group #20273908 — subgroups

Subgroups — #20273908

Order 6784.

250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810969	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810903	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810833	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133150	—	2	1	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133140	—	2	1	no	—
14	#13133138	—	2	1	no	—
15	#13133138	—	2	2	no	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₂×#41×C₅₃	—	2	1	no	—
18	#127×C₅₃	—	2	1	no	—
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	#121×C₅₃	—	2	1	no	—
21	#121×C₅₃	—	2	2	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	—	2	1	no	—
24	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	#41×C₅₃	—	4	2	no	—
26	#41×C₅₃	—	4	2	no	—
27	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×D₄×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	—	2	2	no	—
30	C₂×D₄×C₅₃	—	2	1	no	—
31	C₂×D₄×C₅₃	—	2	1	no	—
32	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂×D₄×C₅₃	D₄	1	2	yes	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂×Dic₁₀₆	—	2	1	no	—
37	C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂²⋊C₄×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#11719925	—	4	2	no	—
45	C₄⋊Dic₅₃	—	2	1	no	—
46	C₄⋊Dic₅₃	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂³×C₅₃	—	2	1	no	—
51	C₂³×C₅₃	—	2	2	no	—
52	Q₈×C₅₃	—	2	1	no	—
53	Q₈×C₅₃	D₈	1	2	yes	—
54	D₄×C₅₃	—	2	1	no	—
55	D₄×C₅₃	—	2	1	no	—
56	D₄×C₅₃	—	4	2	no	—
57	D₄×C₅₃	—	4	4	no	—
58	D₄×C₅₃	—	4	1	no	—
59	D₄×C₅₃	—	4	1	no	—
60	D₄×C₅₃	—	4	1	no	—
61	D₄×C₅₃	—	4	1	no	—
62	D₄×C₅₃	—	8	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₅₃	—	4	1	no	—
73	Dic₁₀₆	—	4	1	no	—
74	Dic₁₀₆	—	4	1	no	—
75	#88322	—	4	1	no	—
76	#88322	—	4	1	no	—
77	#88322	—	4	1	no	—
78	C₂²×C₅₃	—	4	2	no	—
79	C₂²×C₅₃	—	4	1	no	—
80	C₂²×C₅₃	—	4	1	no	—
81	C₂²×C₅₃	—	4	2	no	—
82	C₂²×C₅₃	—	4	2	no	—
83	C₂²×C₅₃	—	4	1	no	—
84	C₂²×C₅₃	#120	1	1	yes	characteristic
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	4	1	no	—
88	C₄×C₅₃	—	4	1	no	—
89	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₄×C₅₃	#135	1	1	yes	characteristic
91	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₄×C₅₃	#137	1	1	yes	characteristic
93	Dic₅₃	—	8	1	no	—
94	#1641	—	53	1	no	maximal, Sylow
95	C₂×C₅₃	—	8	1	no	—
96	C₂×C₅₃	—	8	2	no	—
97	C₂×C₅₃	#447	1	1	yes	characteristic
98	C₂×C₅₃	#452	1	1	yes	characteristic
99	C₂×C₅₃	#455	1	1	yes	characteristic
100	#550	D₅₃	1	1	yes	characteristic
101	#486	—	53	1	no	—
102	#474	—	53	1	no	—
103	#462	—	53	1	no	—
104	#331	—	53	1	no	—
105	Q₈⋊C₈	—	53	1	no	—
106	D₄⋊C₈	—	53	1	no	—
107	C₅₃	#1641	1	1	yes	characteristic, complemented, Sylow
108	C₂×#41	—	2	1	no	—
109	C₂×Q₁₆	—	106	1	no	—
110	C₄⋊Q₈	—	53	1	no	—
111	#127	—	2	1	no	—
112	#127	C₂×D₅₃	1	1	yes	characteristic
113	#121	—	2	2	no	—
114	#121	—	2	1	no	—
115	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
116	C₄×D₄	—	2	1	no	—
117	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
118	#107	—	106	1	no	—
119	C₄⋊C₈	—	53	1	no	—
120	C₄⋊C₈	—	53	1	no	—
121	#103	—	106	1	no	—
122	#102	—	106	1	no	—
123	#102	—	106	2	no	—
124	C₄×C₈	—	53	1	no	—
125	#41	—	4	2	no	—
126	#41	—	4	2	no	—
127	C₂×Q₈	—	106	1	no	—
128	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	Q₁₆	—	212	2	no	—
138	C₂×C₈	—	106	1	no	—
139	C₂×C₈	—	106	1	no	—
140	C₂×C₈	—	106	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	106	1	no	—
143	C₄⋊C₄	—	106	1	no	—
144	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
145	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	C₂²×D₅₃	1	1	yes	characteristic
150	C₂³	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	Q₈	—	2	1	no	—
153	Q₈	—	212	1	no	—
154	Q₈	—	212	1	no	—
155	Q₈	#11719922	1	2	yes	—
156	D₄	—	2	1	no	—
157	D₄	—	2	1	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	4	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	4	1	no	—
164	D₄	—	8	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	212	1	no	—
172	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₈	—	212	1	no	—
176	C₈	—	212	1	no	—
177	C₈	—	212	1	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	#13133186	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	424	1	no	—
190	C₄	C₂×#11719922	1	1	yes	characteristic
191	C₄	#13133237	1	1	yes	characteristic
192	C₄	#13133243	1	1	yes	characteristic
193	C₄	#13133247	1	1	yes	characteristic
194	C₂	—	8	1	no	—
195	C₂	—	8	2	no	—
196	C₂	#16810757	1	1	yes	characteristic
197	C₂	#16810785	1	1	yes	characteristic
198	C₂	#16810801	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810969

C₂

yes

maximal, characteristic, complemented

#16810903

C₂

yes

maximal, characteristic, complemented

#16810833

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#550

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#41×C₅₃

—

#127×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

#121×C₅₃

—

#121×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

D₈

yes

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1641

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

100

D₅₃

yes

characteristic

101

#486

—

102

#474

—

103

#462

—

104

#331

—

105

Q₈⋊C₈

—

106

D₄⋊C₈

—

107

C₅₃

#1641

yes

characteristic, complemented, Sylow

108

C₂×#41

—

109

C₂×Q₁₆

—

106

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

C₂×D₅₃

yes

characteristic

113

#121

—

114

#121

—

115

C₄×Q₈

C₂×D₅₃

yes

characteristic

116

C₄×D₄

—

117

C₄×D₄

C₂×D₅₃

yes

characteristic

118

#107

—

106

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

106

—

122

#102

—

106

—

123

#102

—

106

—

124

C₄×C₈

—

125

#41

—

126

#41

—

127

C₂×Q₈

—

106

—

128

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

134

C₂×D₄

C₂²⋊D₅₃

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

Q₁₆

—

212

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

106

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

106

—

143

C₄⋊C₄

—

106

—

144

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

145

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

C₂²×D₅₃

yes

characteristic

150

C₂³

—

151

C₂³

—

152

Q₈

—

153

Q₈

—

212

—

154

Q₈

—

212

—

155

Q₈

#11719922

yes

—

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

212

—

172

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

173

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

175

C₈

—

212

—

176

C₈

—

212

—

177

C₈

—

212

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#13133186

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

424

—

190

C₄

C₂×#11719922

yes

characteristic

191

C₄

#13133237

yes

characteristic

192

C₄

#13133243

yes

characteristic

193

C₄

#13133247

yes

characteristic

194

C₂

—

195

C₂

—

196

C₂

#16810757

yes

characteristic

197

C₂

#16810785

yes

characteristic

198

C₂

#16810801

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#550	C₂
socle	C₂²×C₅₃	#120
hypercenter	#550	D₅₃
hypocenter	C₅₃	#1641
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #550

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#550

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #550

N	H ≅ G/N	type
1. #16810969	C₂	semidirect
2. #16810903	C₂	semidirect
3. #16810833	C₂	semidirect
4. #16810367	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
6. D₄⋊#88322	C₂	semidirect
107. C₅₃	#1641	semidirect

H ≅ G/N

type

1. #16810969

C₂

semidirect

2. #16810903

C₂

semidirect

3. #16810833

C₂

semidirect

4. #16810367

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

6. D₄⋊#88322

C₂

semidirect

107. C₅₃

#1641

semidirect

Subgroups — #20273908

Conjugacy classes of subgroups

Complements