Group #22561069 — subgroups

Subgroups — #22561069

Order 8576.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#11761618	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523976	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523967	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523966	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#16863302	C₂	1	1	yes	maximal, characteristic
7	#391×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂×#11761618	—	2	2	no	—
10	C₂×#11761618	—	2	2	no	—
11	C₂×#11761618	C₄	1	4	yes	complemented
12	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	#14523976	—	2	2	no	—
16	#14523967	—	2	2	no	—
17	#14523966	—	2	2	no	—
18	C₄×C₆₇⋊C₈	—	2	2	no	—
19	C₆₇⋊#95	—	2	2	no	—
20	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#95×C₆₇	—	2	2	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₁₃₄	—	2	1	no	—
26	C₂×Dic₁₃₄	—	2	1	no	—
27	C₂×Dic₁₃₄	—	2	1	no	—
28	C₂×Dic₁₃₄	—	2	1	no	—
29	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
30	C₂²×Dic₆₇	—	2	1	no	—
31	C₂²×Dic₆₇	—	2	1	no	—
32	#11761618	—	2	8	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₄⋊Dic₆₇	—	2	1	no	—
40	C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×Q₈×C₆₇	—	2	1	no	—
42	C₂×Q₈×C₆₇	—	2	1	no	—
43	C₂×Q₈×C₆₇	—	2	1	no	—
44	C₂×Q₈×C₆₇	—	2	1	no	—
45	C₂×Q₈×C₆₇	—	2	1	no	—
46	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₄²×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	—	2	1	no	—
55	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₆₇	—	2	2	no	—
57	Q₈×C₆₇	—	2	2	no	—
58	Q₈×C₆₇	—	4	1	no	—
59	Q₈×C₆₇	—	4	1	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	4	1	no	—
64	C₂×C₄×C₆₇	—	4	1	no	—
65	C₂×C₄×C₆₇	—	4	1	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	Q₁₆	1	4	yes	complemented
68	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	4	1	no	—
76	C₂×Dic₆₇	—	4	1	no	—
77	C₂×Dic₆₇	—	4	1	no	—
78	C₂×Dic₆₇	—	4	2	no	—
79	Dic₁₃₄	—	2	2	no	—
80	Dic₁₃₄	—	2	2	no	—
81	Dic₁₃₄	—	4	1	no	—
82	Dic₁₃₄	—	4	1	no	—
83	C₆₇⋊C₈	—	2	2	no	—
84	C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₆₇	#121	1	1	yes	characteristic
87	C₂²×C₆₇	#127	1	1	yes	characteristic
88	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
89	C₂²×C₆₇	#135	1	2	yes	—
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	—	8	1	no	—
93	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₄×C₆₇	#121	1	1	yes	characteristic
96	C₄×C₆₇	#124	1	1	yes	characteristic
97	Dic₆₇	—	4	2	no	—
98	Dic₆₇	—	8	1	no	—
99	C₂×C₆₇	#390	1	1	yes	characteristic
100	C₂×C₆₇	C₄×Q₁₆	1	2	yes	—
101	C₂×C₆₇	#467	1	2	yes	—
102	C₂×C₆₇	#494	1	1	yes	characteristic
103	C₂×C₆₇	#495	1	1	yes	characteristic
104	#1969	—	67	1	no	maximal, Sylow
105	C₆₇	#1969	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₁₆	—	67	1	no	—
107	C₂×#107	—	67	1	no	—
108	C₂×#103	—	67	1	no	—
109	C₂×#103	—	67	1	no	—
110	C₂×C₄×C₈	—	67	1	no	—
111	#391	—	67	1	no	—
112	#391	D₆₇	1	1	yes	characteristic
113	C₂²×Q₈	—	67	1	no	—
114	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
115	C₂×Q₁₆	—	67	4	no	—
116	C₂×Q₁₆	—	134	2	no	—
117	C₂×Q₁₆	—	134	2	no	—
118	C₂²×C₈	—	67	1	no	—
119	C₂²×C₈	—	67	1	no	—
120	C₂×C₄⋊C₄	—	67	1	no	—
121	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
123	#107	—	134	2	no	—
124	#103	—	134	2	no	—
125	#103	—	134	2	no	—
126	C₄×C₈	—	134	2	no	—
127	#95	—	2	2	no	—
128	#95	—	134	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	67	2	no	—
135	C₂×Q₈	—	134	1	no	—
136	C₂×Q₈	—	134	1	no	—
137	C₂×Q₈	—	134	1	no	—
138	C₂×Q₈	—	134	1	no	—
139	C₂×Q₈	—	134	1	no	—
140	C₂×Q₈	C₄×D₆₇	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	134	1	no	—
144	C₂²×C₄	—	134	1	no	—
145	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
146	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
147	Q₁₆	—	134	8	no	—
148	C₂×C₈	—	67	2	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	—	67	2	no	—
151	C₂×C₈	—	67	2	no	—
152	C₂×C₈	—	134	1	no	—
153	C₂×C₈	—	134	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	134	1	no	—
157	C₄⋊C₄	—	134	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	268	1	no	—
166	Q₈	—	268	1	no	—
167	Q₈	—	134	2	no	—
168	Q₈	—	134	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	268	1	no	—
177	C₂×C₄	—	268	1	no	—
178	C₂×C₄	—	268	1	no	—
179	C₂×C₄	—	268	2	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	#11761618	1	4	yes	complemented
182	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
183	C₂×C₄	D₄×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	#11761627	1	1	yes	characteristic
187	C₂×C₄	#11761628	1	1	yes	characteristic
188	C₈	—	67	4	no	—
189	C₈	—	134	2	no	—
190	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂²	#14524015	1	1	yes	characteristic
192	C₂²	#14524039	1	1	yes	characteristic
193	C₂²	C₂×#11761618	1	2	yes	—
194	C₂²	#14524063	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	—	8	1	no	—
198	C₄	—	268	2	no	—
199	C₄	—	536	1	no	—
200	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
201	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
202	C₄	#14524016	1	1	yes	characteristic
203	C₄	#14524029	1	1	yes	characteristic
204	C₂	#16863296	1	1	yes	characteristic
205	C₂	C₄×#11761618	1	2	yes	—
206	C₂	#16863635	1	2	yes	—
207	C₂	#16863775	1	1	yes	characteristic
208	C₂	#16863778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761618

C₂

yes

maximal, characteristic

C₂×#14523976

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal, characteristic

C₂×#14523966

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863302

C₂

yes

maximal, characteristic

#391×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

C₄×C₆₇⋊C₈

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊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

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

Q₁₆

yes

complemented

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

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

#135

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#390

yes

characteristic

100

C₂×C₆₇

C₄×Q₁₆

yes

—

101

C₂×C₆₇

#467

yes

—

102

C₂×C₆₇

#494

yes

characteristic

103

C₂×C₆₇

yes

104

—

105

C₆₇

#1969

yes

characteristic, complemented, Sylow

106

C₂²×Q₁₆

—

107

C₂×#107

—

108

C₂×#103

—

109

C₂×#103

—

110

C₂×C₄×C₈

—

111

#391

—

112

#391

D₆₇

yes

characteristic

113

C₂²×Q₈

—

114

C₂²×Q₈

C₂×D₆₇

yes

characteristic

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

134

—

117

C₂×Q₁₆

—

134

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

122

C₂×C₄²

C₂×D₆₇

yes

characteristic

123

#107

—

134

—

124

#103

—

134

—

125

#103

—

134

—

126

C₄×C₈

—

134

—

127

#95

—

128

#95

—

134

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

134

—

136

C₂×Q₈

—

134

—

137

C₂×Q₈

—

134

—

138

C₂×Q₈

—

134

—

139

C₂×Q₈

—

134

—

140

C₂×Q₈

C₄×D₆₇

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

134

—

144

C₂²×C₄

—

134

—

145

C₂²×C₄

C₂²⋊D₆₇

yes

—

146

C₂²×C₄

C₂²×D₆₇

yes

characteristic

147

Q₁₆

—

134

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

134

—

153

C₂×C₈

—

134

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

134

—

157

C₄⋊C₄

—

134

—

158

C₄²

—

159

C₄²

—

160

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

268

—

166

Q₈

—

268

—

167

Q₈

—

134

—

168

Q₈

—

134

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

268

—

177

C₂×C₄

—

268

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

268

—

180

C₂×C₄

—

134

—

181

C₂×C₄

#11761618

yes

complemented

182

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

183

C₂×C₄

D₄×D₆₇

yes

characteristic

184

C₂×C₄

D₄×D₆₇

yes

characteristic

185

C₂×C₄

D₄×D₆₇

yes

characteristic

186

C₂×C₄

#11761627

yes

characteristic

187

C₂×C₄

#11761628

yes

characteristic

188

C₈

—

189

C₈

—

134

—

190

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

191

C₂²

#14524015

yes

characteristic

192

C₂²

#14524039

yes

characteristic

193

C₂²

C₂×#11761618

yes

—

194

C₂²

#14524063

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

268

—

199

C₄

—

536

—

200

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

201

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

202

C₄

#14524016

yes

characteristic

203

C₄

#14524029

yes

characteristic

204

C₂

#16863296

yes

characteristic

205

C₂

C₄×#11761618

yes

—

206

C₂

#16863635

yes

—

207

C₂

#16863775

yes

characteristic

208

C₂

#16863778

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#391×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#391	D₆₇
hypocenter	C₆₇	#1969
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#391×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
11. C₂×#11761618	C₄	semidirect
67. C₂×C₄×C₆₇	Q₁₆	semidirect
105. C₆₇	#1969	semidirect
181. C₂×C₄	#11761618	semidirect

H ≅ G/N

type

11. C₂×#11761618

C₄

semidirect

67. C₂×C₄×C₆₇

Q₁₆

semidirect

105. C₆₇

#1969

semidirect

181. C₂×C₄

#11761618

semidirect

Subgroups — #22561069

Conjugacy classes of subgroups

Complements