Group #22561118 — subgroups

Subgroups — #22561118

Order 8576.

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

normal subgroups only

This group has 316 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×#11761613	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523978	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523960	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14523959	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863352	C₂	1	1	yes	maximal, characteristic, complemented
7	#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761613	—	2	1	no	—
10	C₂×#11761613	—	2	1	no	—
11	C₂×#11761613	—	2	2	no	—
12	C₂×#11761613	C₄	1	4	yes	complemented
13	C₂×#11761612	C₂²	1	1	yes	characteristic
14	C₂×#11761612	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	#14523978	—	2	2	no	—
17	#14523963	—	2	2	no	—
18	#14523960	—	2	2	no	—
19	#14523959	—	2	2	no	—
20	#14523954	—	2	1	no	—
21	C₄×C₆₇⋊C₈	—	2	1	no	—
22	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#117×C₆₇	C₂²	1	1	yes	characteristic
25	#102×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	#11761613	—	2	4	no	—
34	#11761613	—	2	4	no	—
35	#11761612	—	2	1	no	—
36	#11761612	—	2	1	no	—
37	#11761612	—	2	1	no	—
38	#11761612	D₄	1	2	yes	—
39	C₂×C₆₇⋊C₈	—	2	1	no	—
40	C₂×C₆₇⋊C₈	—	2	1	no	—
41	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
45	C₂⁴×C₆₇	—	2	1	no	—
46	C₂×D₄×C₆₇	—	2	1	no	—
47	C₂×D₄×C₆₇	—	2	1	no	—
48	C₂×D₄×C₆₇	—	2	1	no	—
49	C₂×D₄×C₆₇	—	2	1	no	—
50	C₂×D₄×C₆₇	—	2	1	no	—
51	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	1	no	—
54	M₁₆×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
57	C₂²⋊C₄×C₆₇	—	2	1	no	—
58	C₄²×C₆₇	—	2	1	no	—
59	C₂³×C₆₇	—	2	2	no	—
60	C₂³×C₆₇	—	4	1	no	—
61	C₂³×C₆₇	—	4	1	no	—
62	C₂³×C₆₇	—	8	2	no	—
63	C₂³×C₆₇	#41	1	1	yes	characteristic
64	C₂²×D₆₇	—	2	2	no	—
65	C₂²×D₆₇	—	4	1	no	—
66	C₂²×D₆₇	—	4	1	no	—
67	C₂²×D₆₇	—	8	2	no	—
68	D₄×C₆₇	—	2	2	no	—
69	D₄×C₆₇	—	2	2	no	—
70	D₄×C₆₇	—	4	1	no	—
71	D₄×C₆₇	—	4	1	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
79	D₂₆₈	—	2	2	no	—
80	D₂₆₈	—	2	2	no	—
81	D₂₆₈	—	4	1	no	—
82	D₂₆₈	—	4	1	no	—
83	C₈×C₆₇	—	4	1	no	—
84	C₆₇⋊C₈	—	2	2	no	—
85	C₆₇⋊C₈	—	4	1	no	—
86	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	—	4	2	no	—
90	C₂²×C₆₇	—	8	2	no	—
91	C₂²×C₆₇	—	8	2	no	—
92	C₂²×C₆₇	—	8	2	no	—
93	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#121	1	1	yes	characteristic
95	C₂²×C₆₇	#124	1	1	yes	characteristic
96	C₂×D₆₇	—	4	2	no	—
97	C₂×D₆₇	—	8	2	no	—
98	C₂×D₆₇	—	8	2	no	—
99	C₂×D₆₇	—	8	2	no	—
100	C₄×C₆₇	—	4	2	no	—
101	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₄×C₆₇	#121	1	1	yes	characteristic
104	C₄×C₆₇	#127	1	1	yes	characteristic
105	C₂×C₆₇	—	2	1	no	—
106	C₂×C₆₇	—	2	1	no	—
107	C₂×C₆₇	—	8	2	no	—
108	C₂×C₆₇	#390	1	1	yes	characteristic
109	C₂×C₆₇	#472	1	2	yes	—
110	D₆₇	—	8	2	no	—
111	#1978	—	67	1	no	maximal, Sylow
112	C₆₇	#1978	1	1	yes	characteristic, complemented, Sylow
113	C₂²×D₈	—	67	1	no	—
114	C₂×#108	—	67	1	no	—
115	#418	—	67	1	no	—
116	#418	D₆₇	1	1	yes	characteristic, complemented
117	C₂×#100	—	67	1	no	—
118	C₂×#100	—	67	1	no	—
119	#405	—	67	1	no	—
120	C₂²×D₄	—	67	1	no	—
121	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
122	C₂×D₈	—	67	4	no	—
123	C₂×D₈	—	134	1	no	—
124	C₂×D₈	—	134	1	no	—
125	C₂×D₈	—	134	2	no	—
126	C₂×M₁₆	—	67	1	no	—
127	C₂×M₁₆	—	67	1	no	—
128	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
129	C₂²×C₈	—	67	1	no	—
130	#117	C₂×D₆₇	1	1	yes	characteristic
131	#108	—	134	2	no	—
132	#102	—	2	2	no	—
133	#102	—	134	2	no	—
134	#100	—	134	2	no	—
135	#100	—	134	2	no	—
136	#97	—	134	1	no	—
137	C₄×C₈	—	134	1	no	—
138	C₂⁴	—	2	1	no	—
139	C₂⁴	—	134	1	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	67	2	no	—
146	C₂×D₄	—	134	1	no	—
147	C₂×D₄	—	134	1	no	—
148	C₂×D₄	—	134	1	no	—
149	C₂×D₄	—	134	1	no	—
150	C₂×D₄	—	134	1	no	—
151	C₂×D₄	C₄×D₆₇	1	2	yes	—
152	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
153	D₈	—	134	4	no	—
154	D₈	—	134	4	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	67	2	no	—
158	M₁₆	—	134	1	no	—
159	M₁₆	—	134	1	no	—
160	M₁₆	—	134	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	67	2	no	—
163	C₂×C₈	—	67	1	no	—
164	C₂×C₈	—	67	1	no	—
165	C₂×C₈	—	67	2	no	—
166	C₂×C₈	—	134	1	no	—
167	C₂×C₈	—	134	1	no	—
168	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	—	4	1	no	—
174	C₂³	—	8	2	no	—
175	C₂³	—	268	1	no	—
176	C₂³	—	268	1	no	—
177	C₂³	—	536	2	no	—
178	C₂³	—	134	2	no	—
179	C₂³	#11761627	1	1	yes	characteristic
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	4	1	no	—
183	D₄	—	4	1	no	—
184	D₄	—	268	1	no	—
185	D₄	—	268	1	no	—
186	D₄	—	134	2	no	—
187	D₄	—	134	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
190	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
191	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
192	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₄	#11761628	1	1	yes	characteristic
195	C₈	—	4	1	no	—
196	C₈	—	268	1	no	—
197	C₈	—	67	2	no	—
198	C₈	—	67	2	no	—
199	C₈	—	134	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	8	2	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	268	2	no	—
206	C₂²	—	536	2	no	—
207	C₂²	—	536	2	no	—
208	C₂²	—	536	2	no	—
209	C₂²	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
210	C₂²	#14524015	1	1	yes	characteristic
211	C₂²	#14524029	1	1	yes	characteristic
212	C₄	—	4	2	no	—
213	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
214	C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₄	#14524016	1	1	yes	characteristic
216	C₄	#14524039	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	8	2	no	—
220	C₂	—	536	2	no	—
221	C₂	#16863296	1	1	yes	characteristic
222	C₂	#16863664	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761613

C₂

yes

maximal, characteristic

C₂×#14523978

C₂

yes

maximal, characteristic, complemented

#16863400

C₂

yes

maximal, characteristic, complemented

C₂×#14523960

C₂

yes

maximal, characteristic, complemented

C₂×#14523959

C₂

yes

maximal, characteristic, complemented

#16863352

C₂

yes

maximal, characteristic, complemented

#418×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₈

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

#102×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂×C₆₇⋊C₈

—

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

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×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

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

100

C₄×C₆₇

—

101

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

102

C₄×C₆₇

C₄×D₄

yes

characteristic

103

C₄×C₆₇

#121

yes

characteristic

104

C₄×C₆₇

#127

yes

characteristic

105

C₂×C₆₇

—

106

C₂×C₆₇

—

107

C₂×C₆₇

—

108

C₂×C₆₇

#390

yes

characteristic

109

C₂×C₆₇

#472

yes

—

110

D₆₇

—

111

#1978

—

maximal, Sylow

112

C₆₇

#1978

yes

characteristic, complemented, Sylow

113

C₂²×D₈

—

114

C₂×#108

—

115

#418

—

116

#418

D₆₇

yes

characteristic, complemented

117

C₂×#100

—

118

C₂×#100

—

119

#405

—

120

C₂²×D₄

—

121

C₂²×D₄

C₂×D₆₇

yes

characteristic

122

C₂×D₈

—

123

C₂×D₈

—

134

—

124

C₂×D₈

—

134

—

125

C₂×D₈

—

134

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

128

C₂×M₁₆

C₂×D₆₇

yes

characteristic

129

C₂²×C₈

—

130

#117

C₂×D₆₇

yes

characteristic

131

#108

—

134

—

132

#102

—

133

#102

—

134

—

134

#100

—

134

—

135

#100

—

134

—

136

#97

—

134

—

137

C₄×C₈

—

134

—

138

C₂⁴

—

139

C₂⁴

—

134

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

134

—

147

C₂×D₄

—

134

—

148

C₂×D₄

—

134

—

149

C₂×D₄

—

134

—

150

C₂×D₄

—

134

—

151

C₂×D₄

C₄×D₆₇

yes

—

152

C₂²×C₄

C₂²×D₆₇

yes

characteristic

153

D₈

—

134

—

154

D₈

—

134

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

134

—

159

M₁₆

—

134

—

160

M₁₆

—

134

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

134

—

167

C₂×C₈

—

134

—

168

C₄⋊C₄

C₂²⋊D₆₇

yes

—

169

C₂²⋊C₄

—

170

C₄²

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

268

—

176

C₂³

—

268

—

177

C₂³

—

536

—

178

C₂³

—

134

—

179

C₂³

#11761627

yes

characteristic

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

268

—

185

D₄

—

268

—

186

D₄

—

134

—

187

D₄

—

134

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

190

C₂×C₄

D₄×D₆₇

yes

characteristic

191

C₂×C₄

D₄×D₆₇

yes

characteristic

192

C₂×C₄

D₄×D₆₇

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

194

C₂×C₄

#11761628

yes

characteristic

195

C₈

—

196

C₈

—

268

—

197

C₈

—

198

C₈

—

199

C₈

—

134

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

268

—

206

C₂²

—

536

—

207

C₂²

—

536

—

208

C₂²

—

536

—

209

C₂²

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

210

C₂²

#14524015

yes

characteristic

211

C₂²

#14524029

yes

characteristic

212

C₄

—

213

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

214

C₄

C₄×C₂²⋊D₆₇

yes

characteristic

215

C₄

#14524016

yes

characteristic

216

C₄

#14524039

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

—

536

—

221

C₂

#16863296

yes

characteristic

222

C₂

#16863664

yes

—

	subgroup	G/H
center	C₂²	#14524015
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#418×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#418	D₆₇
hypocenter	C₆₇	#1978
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

subgroup

G/H

center

C₂²

#14524015

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#418×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

N	H ≅ G/N	type
2. C₂×#14523978	C₂	semidirect
3. #16863400	C₂	semidirect
4. C₂×#14523960	C₂	semidirect
5. C₂×#14523959	C₂	semidirect
6. #16863352	C₂	semidirect
7. #418×C₆₇	C₂	semidirect
12. C₂×#11761613	C₄	semidirect
112. C₆₇	#1978	semidirect
116. #418	D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523978

C₂

semidirect

3. #16863400

C₂

semidirect

4. C₂×#14523960

C₂

semidirect

5. C₂×#14523959

C₂

semidirect

6. #16863352

C₂

semidirect

7. #418×C₆₇

C₂

semidirect

12. C₂×#11761613

C₄

semidirect

112. C₆₇

#1978

semidirect

116. #418

D₆₇

semidirect

Subgroups — #22561118

Conjugacy classes of subgroups

Complements