Group #22561113 — subgroups

Subgroups — #22561113

Order 8576.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#11761615	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523978	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863405	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863402	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523959	C₂	1	1	yes	maximal, characteristic
6	#16863354	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761615	—	2	1	no	—
10	C₂×#11761615	—	2	1	no	—
11	C₂×#11761615	—	2	2	no	—
12	C₂×#11761615	C₄	1	4	yes	complemented
13	C₂×#11761612	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#14524002	C₂²	1	1	yes	characteristic
16	#14523978	—	2	2	no	—
17	#14523967	—	2	2	no	—
18	#14523965	—	2	2	no	—
19	#14523959	—	2	2	no	—
20	#14523953	—	2	1	no	—
21	C₈×Dic₆₇	—	2	1	no	—
22	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#101×C₆₇	—	2	2	no	—
26	C₂³×D₆₇	—	2	1	no	—
27	C₂×D₂₆₈	—	2	1	no	—
28	C₂×D₂₆₈	—	2	1	no	—
29	C₂×D₂₆₈	—	2	1	no	—
30	C₂×D₂₆₈	—	2	1	no	—
31	C₂×D₂₆₈	—	2	1	no	—
32	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
33	#11761615	—	2	4	no	—
34	#11761615	—	2	4	no	—
35	#11761612	—	2	1	no	—
36	#11761612	—	2	1	no	—
37	C₂×C₆₇⋊C₈	—	2	1	no	—
38	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
42	C₄⋊Dic₆₇	D₄	1	2	yes	—
43	C₂²⋊Dic₆₇	—	2	1	no	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₂×Q₈×C₆₇	—	2	1	no	—
46	C₂×Q₈×C₆₇	—	2	1	no	—
47	C₂×Q₈×C₆₇	—	2	1	no	—
48	C₂×Q₈×C₆₇	—	2	1	no	—
49	C₂×Q₈×C₆₇	—	2	1	no	—
50	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	1	no	—
54	M₁₆×C₆₇	—	2	1	no	—
55	M₁₆×C₆₇	—	2	1	no	—
56	M₁₆×C₆₇	D₄	1	2	yes	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂³×C₆₇	#41	1	1	yes	characteristic
60	C₂²×D₆₇	—	2	2	no	—
61	C₂²×D₆₇	—	4	1	no	—
62	C₂²×D₆₇	—	4	1	no	—
63	C₂²×D₆₇	—	8	2	no	—
64	Q₈×C₆₇	—	2	2	no	—
65	Q₈×C₆₇	—	2	2	no	—
66	Q₈×C₆₇	—	4	1	no	—
67	Q₈×C₆₇	—	4	1	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	—	4	1	no	—
71	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂×Dic₆₇	—	2	2	no	—
78	D₂₆₈	—	2	2	no	—
79	D₂₆₈	—	2	2	no	—
80	D₂₆₈	—	4	1	no	—
81	D₂₆₈	—	4	1	no	—
82	C₈×C₆₇	—	2	2	no	—
83	C₈×C₆₇	—	4	1	no	—
84	C₆₇⋊C₈	—	4	1	no	—
85	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₆₇	#121	1	1	yes	characteristic
90	C₂²×C₆₇	#124	1	1	yes	characteristic
91	C₂×D₆₇	—	4	2	no	—
92	C₂×D₆₇	—	8	2	no	—
93	C₂×D₆₇	—	8	2	no	—
94	C₂×D₆₇	—	8	2	no	—
95	C₄×C₆₇	—	4	2	no	—
96	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₄×C₆₇	#121	1	1	yes	characteristic
99	C₄×C₆₇	#127	1	1	yes	characteristic
100	Dic₆₇	—	4	2	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	#390	1	1	yes	characteristic
104	C₂×C₆₇	#471	1	2	yes	—
105	D₆₇	—	8	2	no	—
106	#1977	—	67	1	no	maximal, Sylow
107	C₆₇	#1977	1	1	yes	characteristic, complemented, Sylow
108	C₂²×SD₁₆	—	67	1	no	—
109	C₂×#108	—	67	1	no	—
110	#419	—	67	1	no	—
111	#418	—	67	1	no	—
112	C₂×#101	D₆₇	1	1	yes	characteristic, complemented
113	C₂×#100	—	67	1	no	—
114	#405	—	67	1	no	—
115	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
116	C₂²×D₄	—	67	1	no	—
117	C₂×SD₁₆	—	67	4	no	—
118	C₂×SD₁₆	—	134	1	no	—
119	C₂×SD₁₆	—	134	1	no	—
120	C₂×SD₁₆	—	134	2	no	—
121	C₂×M₁₆	—	67	1	no	—
122	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
123	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	—	67	1	no	—
125	#117	—	67	1	no	—
126	#108	—	134	2	no	—
127	#103	—	134	2	no	—
128	#102	—	134	2	no	—
129	#101	—	2	2	no	—
130	#100	—	134	2	no	—
131	#97	—	134	1	no	—
132	C₄×C₈	—	134	1	no	—
133	C₂⁴	—	134	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₄×D₆₇	1	2	yes	—
140	C₂×D₄	—	67	2	no	—
141	C₂×D₄	—	134	1	no	—
142	C₂×D₄	—	134	1	no	—
143	C₂×D₄	—	134	1	no	—
144	C₂×D₄	—	134	1	no	—
145	C₂×D₄	—	134	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
148	SD₁₆	—	134	4	no	—
149	SD₁₆	—	134	4	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	—	2	1	no	—
152	M₁₆	—	2	1	no	—
153	M₁₆	—	134	1	no	—
154	M₁₆	—	134	1	no	—
155	M₁₆	C₂²⋊D₆₇	1	2	yes	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	67	2	no	—
159	C₂×C₈	—	67	1	no	—
160	C₂×C₈	—	67	1	no	—
161	C₂×C₈	—	67	2	no	—
162	C₂×C₈	—	134	1	no	—
163	C₄⋊C₄	—	67	2	no	—
164	C₂²⋊C₄	—	134	1	no	—
165	C₄²	—	134	1	no	—
166	C₂³	—	268	1	no	—
167	C₂³	—	268	1	no	—
168	C₂³	—	536	2	no	—
169	C₂³	—	134	2	no	—
170	C₂³	#11761628	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	Q₈	—	4	1	no	—
175	D₄	—	268	1	no	—
176	D₄	—	268	1	no	—
177	D₄	—	134	2	no	—
178	D₄	—	134	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	134	2	no	—
183	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
184	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
185	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
186	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂×C₄	#11761627	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	4	1	no	—
191	C₈	—	268	1	no	—
192	C₈	—	67	2	no	—
193	C₈	—	67	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	268	2	no	—
196	C₂²	—	536	2	no	—
197	C₂²	—	536	2	no	—
198	C₂²	—	536	2	no	—
199	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂²	#14524016	1	1	yes	characteristic
201	C₂²	#14524029	1	1	yes	characteristic
202	C₄	—	4	2	no	—
203	C₄	—	268	2	no	—
204	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
205	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
206	C₄	#14524015	1	1	yes	characteristic
207	C₄	#14524039	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	—	536	2	no	—
211	C₂	#16863296	1	1	yes	characteristic
212	C₂	#16863662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761615

C₂

yes

maximal, characteristic

C₂×#14523978

C₂

yes

maximal, characteristic, complemented

#16863405

C₂

yes

maximal, characteristic, complemented

#16863402

C₂

yes

maximal, characteristic

C₂×#14523959

C₂

yes

maximal, characteristic

#16863354

C₂

yes

maximal, characteristic, complemented

C₂×#101×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

—

—

—

—

C₈×Dic₆₇

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#101×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂×C₄

yes

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

100

Dic₆₇

—

101

C₂×C₆₇

—

102

C₂×C₆₇

—

103

C₂×C₆₇

#390

yes

characteristic

104

C₂×C₆₇

#471

yes

—

105

D₆₇

—

106

#1977

—

maximal, Sylow

107

C₆₇

#1977

yes

characteristic, complemented, Sylow

108

C₂²×SD₁₆

—

109

C₂×#108

—

110

#419

—

111

#418

—

112

C₂×#101

D₆₇

yes

characteristic, complemented

113

C₂×#100

—

114

#405

—

115

C₂²×Q₈

C₂×D₆₇

yes

characteristic

116

C₂²×D₄

—

117

C₂×SD₁₆

—

118

C₂×SD₁₆

—

134

—

119

C₂×SD₁₆

—

134

—

120

C₂×SD₁₆

—

134

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×D₆₇

yes

characteristic

123

C₂×M₁₆

C₂×D₆₇

yes

characteristic

124

C₂²×C₈

—

125

#117

—

126

#108

—

134

—

127

#103

—

134

—

128

#102

—

134

—

129

#101

—

130

#100

—

134

—

131

#97

—

134

—

132

C₄×C₈

—

134

—

133

C₂⁴

—

134

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

C₄×D₆₇

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

134

—

142

C₂×D₄

—

134

—

143

C₂×D₄

—

134

—

144

C₂×D₄

—

134

—

145

C₂×D₄

—

134

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂²×D₆₇

yes

characteristic

148

SD₁₆

—

134

—

149

SD₁₆

—

134

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

134

—

154

M₁₆

—

134

—

155

M₁₆

C₂²⋊D₆₇

yes

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

134

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

134

—

165

C₄²

—

134

—

166

C₂³

—

268

—

167

C₂³

—

268

—

168

C₂³

—

536

—

169

C₂³

—

134

—

170

C₂³

#11761628

yes

characteristic

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

D₄

—

268

—

176

D₄

—

268

—

177

D₄

—

134

—

178

D₄

—

134

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

134

—

183

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

184

C₂×C₄

D₄×D₆₇

yes

characteristic

185

C₂×C₄

D₄×D₆₇

yes

characteristic

186

C₂×C₄

D₄×D₆₇

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

188

C₂×C₄

#11761627

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

—

268

—

192

C₈

—

193

C₈

—

194

C₂²

—

195

C₂²

—

268

—

196

C₂²

—

536

—

197

C₂²

—

536

—

198

C₂²

—

536

—

199

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

200

C₂²

#14524016

yes

characteristic

201

C₂²

#14524029

yes

characteristic

202

C₄

—

203

C₄

—

268

—

204

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

205

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

206

C₄

#14524015

yes

characteristic

207

C₄

#14524039

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

—

536

—

211

C₂

#16863296

yes

characteristic

212

C₂

#16863662

yes

—

	subgroup	G/H
center	C₂²	#14524016
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#101×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₂×#101	D₆₇
hypocenter	C₆₇	#1977
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#14524016

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#101×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. C₂×#14523978	C₂	semidirect
3. #16863405	C₂	semidirect
6. #16863354	C₂	semidirect
7. C₂×#101×C₆₇	C₂	semidirect
12. C₂×#11761615	C₄	semidirect
107. C₆₇	#1977	semidirect
112. C₂×#101	D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523978

C₂

semidirect

3. #16863405

C₂

semidirect

6. #16863354

C₂

semidirect

7. C₂×#101×C₆₇

C₂

semidirect

12. C₂×#11761615

C₄

semidirect

107. C₆₇

#1977

semidirect

112. C₂×#101

D₆₇

semidirect

Subgroups — #22561113

Conjugacy classes of subgroups

Complements