Group #20273817 — subgroups

Subgroups — #20273817

Order 6784.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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	D₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810969	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810820	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810353	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#459	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	—	2	1	no	—
9	C₂×#11719923	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133216	—	2	1	no	—
12	#13133196	—	2	1	no	—
13	#13133194	—	2	2	no	—
14	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	D₄×Dic₅₃	—	2	1	no	—
16	C₄⋊#88322	C₂²	1	1	yes	characteristic
17	#13133138	—	2	1	no	—
18	#13133138	—	2	2	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₅₃	—	2	1	no	—
21	#127×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#102×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	—	2	1	no	—
28	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	—	2	1	no	—
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₂×Dic₁₀₆	—	2	2	no	—
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	—	4	2	no	—
34	C₂×Dic₁₀₆	—	4	2	no	—
35	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
36	D₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₅₃	—	2	2	no	—
42	#11719923	—	4	2	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₃	D₄	1	2	yes	—
46	#11719919	—	2	2	no	—
47	#11719919	—	2	1	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂³×C₅₃	—	2	1	no	—
52	C₂³×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	2	1	no	—
54	D₄×C₅₃	—	4	2	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	4	1	no	—
57	D₄×C₅₃	—	4	1	no	—
58	D₄×C₅₃	—	4	1	no	—
59	D₄×C₅₃	SD₁₆	1	2	yes	complemented
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	—	4	1	no	—
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	4	4	no	—
70	Dic₁₀₆	—	2	1	no	—
71	Dic₁₀₆	—	4	4	no	—
72	Dic₁₀₆	—	4	1	no	—
73	Dic₁₀₆	—	4	1	no	—
74	Dic₁₀₆	D₈	1	2	yes	complemented
75	C₈×C₅₃	—	4	1	no	—
76	#88322	—	4	1	no	—
77	#88322	—	4	1	no	—
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	4	1	no	—
80	C₂²×C₅₃	—	4	1	no	—
81	C₂²×C₅₃	—	8	1	no	—
82	C₂²×C₅₃	—	8	1	no	—
83	C₂²×C₅₃	—	8	1	no	—
84	C₂²×C₅₃	#120	1	1	yes	characteristic
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	4	1	no	—
87	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₄×C₅₃	#137	1	1	yes	characteristic
91	Dic₅₃	—	2	2	no	—
92	Dic₅₃	—	4	2	no	—
93	Dic₅₃	—	4	1	no	—
94	#1627	—	53	1	no	maximal, Sylow
95	C₂×C₅₃	—	4	2	no	—
96	C₂×C₅₃	—	16	1	no	—
97	C₂×C₅₃	#447	1	1	yes	characteristic
98	C₂×C₅₃	#448	1	1	yes	characteristic
99	C₂×C₅₃	#454	1	1	yes	characteristic
100	D₄×Q₈	—	53	1	no	—
101	#486	—	53	1	no	—
102	#460	—	53	1	no	—
103	#459	D₅₃	1	1	yes	characteristic
104	#332	—	53	1	no	—
105	Q₈⋊C₈	—	53	1	no	—
106	D₄⋊C₈	—	53	1	no	—
107	C₅₃	#1627	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	106	1	no	—
109	C₂×SD₁₆	—	106	1	no	—
110	C₂×D₈	—	2	1	no	—
111	C₄⋊Q₈	—	53	1	no	—
112	C₄⋊Q₈	—	106	1	no	—
113	#127	C₂×D₅₃	1	1	yes	characteristic
114	C₂²⋊Q₈	—	106	2	no	—
115	C₂²⋊Q₈	—	106	1	no	—
116	C₄×Q₈	—	53	1	no	—
117	C₄×D₄	—	106	1	no	—
118	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	53	1	no	—
120	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
121	#102	—	2	1	no	—
122	#102	—	106	1	no	—
123	#102	—	106	2	no	—
124	C₄×C₈	—	53	1	no	—
125	C₂×Q₈	—	53	1	no	—
126	C₂×Q₈	—	106	2	no	—
127	C₂×Q₈	—	106	1	no	—
128	C₂×Q₈	—	212	2	no	—
129	C₂×Q₈	—	212	2	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	106	2	no	—
136	SD₁₆	—	212	2	no	—
137	D₈	—	4	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	106	1	no	—
140	C₂×C₈	—	106	1	no	—
141	C₄⋊C₄	—	53	1	no	—
142	C₄⋊C₄	—	53	2	no	—
143	C₄⋊C₄	—	106	2	no	—
144	C₄⋊C₄	—	106	1	no	—
145	C₄⋊C₄	—	106	1	no	—
146	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂²⋊C₄	—	2	1	no	—
148	C₂²⋊C₄	—	106	2	no	—
149	C₄²	—	106	1	no	—
150	C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	C₂³	—	2	1	no	—
152	C₂³	—	4	1	no	—
153	Q₈	—	53	2	no	—
154	Q₈	—	106	1	no	—
155	Q₈	—	212	4	no	—
156	Q₈	—	212	1	no	—
157	Q₈	—	212	1	no	—
158	D₄	—	2	1	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	4	1	no	—
164	D₄	#11719923	1	2	yes	complemented
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	106	1	no	—
169	C₂×C₄	—	106	1	no	—
170	C₂×C₄	—	106	2	no	—
171	C₂×C₄	—	212	4	no	—
172	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₈	—	4	1	no	—
176	C₈	—	212	1	no	—
177	C₈	—	212	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	8	1	no	—
182	C₂²	—	8	1	no	—
183	C₂²	—	8	1	no	—
184	C₂²	#13133185	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	106	2	no	—
188	C₄	—	212	2	no	—
189	C₄	—	212	1	no	—
190	C₄	D₈×D₅₃	1	1	yes	characteristic
191	C₄	C₂×#11719923	1	1	yes	characteristic
192	C₄	#13133241	1	1	yes	characteristic
193	C₄	#13133247	1	1	yes	characteristic
194	C₂	—	4	2	no	—
195	C₂	—	16	1	no	—
196	C₂	#16810755	1	1	yes	characteristic
197	C₂	#16810761	1	1	yes	characteristic
198	C₂	#16810794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, characteristic, complemented

#16810820

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

#16810353

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#459

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂×#11719923

—

#13133217

C₂²

yes

—

—

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

D₄×Dic₅₃

—

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×D₈×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

SD₁₆

yes

complemented

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

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

D₈

yes

complemented

C₈×C₅₃

—

#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

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1627

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

100

D₄×Q₈

—

101

#486

—

102

#460

—

103

#459

D₅₃

yes

characteristic

104

#332

—

105

Q₈⋊C₈

—

106

D₄⋊C₈

—

107

C₅₃

#1627

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

—

106

—

109

C₂×SD₁₆

—

106

—

110

C₂×D₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

106

—

113

#127

C₂×D₅₃

yes

characteristic

114

C₂²⋊Q₈

—

106

—

115

C₂²⋊Q₈

—

106

—

116

C₄×Q₈

—

117

C₄×D₄

—

106

—

118

C₄×D₄

C₂×D₅₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

C₂×D₅₃

yes

characteristic

121

#102

—

122

#102

—

106

—

123

#102

—

106

—

124

C₄×C₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

106

—

127

C₂×Q₈

—

106

—

128

C₂×Q₈

—

212

—

129

C₂×Q₈

—

212

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

134

C₂²×C₄

—

135

C₂²×C₄

—

106

—

136

SD₁₆

—

212

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

106

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

106

—

144

C₄⋊C₄

—

106

—

145

C₄⋊C₄

—

106

—

146

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

106

—

149

C₄²

—

106

—

150

C₄²

C₂²×D₅₃

yes

characteristic

151

C₂³

—

152

C₂³

—

153

Q₈

—

154

Q₈

—

106

—

155

Q₈

—

212

—

156

Q₈

—

212

—

157

Q₈

—

212

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

#11719923

yes

complemented

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

106

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

212

—

172

C₂×C₄

D₄×D₅₃

yes

characteristic

173

C₂×C₄

D₄×D₅₃

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

175

C₈

—

176

C₈

—

212

—

177

C₈

—

212

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#13133185

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

106

—

188

C₄

—

212

—

189

C₄

—

212

—

190

C₄

D₈×D₅₃

yes

characteristic

191

C₄

C₂×#11719923

yes

characteristic

192

C₄

#13133241

yes

characteristic

193

C₄

#13133247

yes

characteristic

194

C₂

—

195

C₂

—

196

C₂

#16810755

yes

characteristic

197

C₂

#16810761

yes

characteristic

198

C₂

#16810794

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#459	C₂
socle	C₂²×C₅₃	#120
hypercenter	#459	D₅₃
hypocenter	C₅₃	#1627
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#459

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. D₄×Dic₁₀₆	C₂	semidirect
2. #16810969	C₂	semidirect
3. #16810820	C₂	semidirect
4. #16810368	C₂	semidirect
5. #16810353	C₂	semidirect
6. D₄⋊#88322	C₂	semidirect
59. D₄×C₅₃	SD₁₆	semidirect
74. Dic₁₀₆	D₈	semidirect
107. C₅₃	#1627	semidirect
164. D₄	#11719923	semidirect

H ≅ G/N

type

1. D₄×Dic₁₀₆

C₂

semidirect

2. #16810969

C₂

semidirect

3. #16810820

C₂

semidirect

4. #16810368

C₂

semidirect

5. #16810353

C₂

semidirect

6. D₄⋊#88322

C₂

semidirect

59. D₄×C₅₃

SD₁₆

semidirect

74. Dic₁₀₆

D₈

semidirect

107. C₅₃

#1627

semidirect

164. D₄

#11719923

semidirect

Subgroups — #20273817

Conjugacy classes of subgroups

Complements