Group #30760577 — subgroups

Subgroups — #30760577

Order 17152.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22561145	C₂	1	1	yes	maximal, characteristic
2	#22560337	C₂	1	1	yes	maximal, characteristic
3	#22560215	C₂	1	2	yes	maximal
4	#22558093	C₂	1	2	yes	maximal
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14523968	—	2	2	no	—
9	#16863266	C₂²	1	2	yes	—
10	#16863159	—	2	2	no	—
11	#16863159	—	2	2	no	—
12	#16863150	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	1	no	—
17	C₄×Dic₁₃₄	—	2	2	no	—
18	C₄×Dic₁₃₄	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
23	C₂×Dic₆₇⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₆₇	—	2	1	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523968	—	4	4	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂×Dic₁₃₄	—	2	2	no	—
35	C₂×Dic₁₃₄	—	2	1	no	—
36	C₂×Dic₁₃₄	D₈	1	2	yes	—
37	C₂²×Dic₆₇	—	2	1	no	—
38	C₂²×Dic₆₇	—	2	1	no	—
39	C₂²×Dic₆₇	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	1	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₄⋊Dic₆₇	—	2	2	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₄⋊Dic₆₇	—	4	2	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	8	2	no	—
48	Dic₆₇⋊C₄	—	4	2	no	—
49	C₄×Dic₆₇	—	4	1	no	—
50	C₄×Dic₆₇	—	4	1	no	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	#41	1	2	yes	—
62	C₂³×C₆₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	#123	1	2	yes	—
65	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#126	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#135	1	2	yes	—
70	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#137	1	2	yes	—
72	C₂×C₄×C₆₇	#137	1	2	yes	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	—	4	1	no	—
75	C₂×Dic₆₇	—	4	1	no	—
76	C₂×Dic₆₇	—	4	1	no	—
77	C₂×Dic₆₇	—	4	2	no	—
78	C₂×Dic₆₇	—	8	2	no	—
79	C₂×Dic₆₇	—	8	2	no	—
80	C₂×Dic₆₇	—	8	2	no	—
81	Dic₁₃₄	—	2	4	no	—
82	Dic₁₃₄	—	4	2	no	—
83	C₈×C₆₇	—	4	2	no	—
84	C₈×C₆₇	—	8	2	no	—
85	C₆₇⋊C₈	—	8	1	no	—
86	C₂²×C₆₇	#397	1	1	yes	characteristic
87	C₂²×C₆₇	#464	1	2	yes	—
88	C₂²×C₆₇	#466	1	1	yes	characteristic
89	C₂²×C₆₇	#470	1	1	yes	characteristic
90	C₂²×C₆₇	#471	1	2	yes	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	#480	1	2	yes	—
93	C₄×C₆₇	#485	1	2	yes	—
94	C₄×C₆₇	#485	1	2	yes	—
95	C₄×C₆₇	#488	1	1	yes	characteristic
96	C₄×C₆₇	#491	1	1	yes	characteristic
97	Dic₆₇	—	4	2	no	—
98	Dic₆₇	—	8	1	no	—
99	Dic₆₇	—	16	2	no	—
100	#10623	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	#1682	1	2	yes	—
102	C₂×C₆₇	#1704	1	2	yes	—
103	C₂×C₆₇	#2074	1	1	yes	characteristic
104	C₂×C₆₇	#2075	1	1	yes	characteristic
105	C₂×C₆₇	#2079	1	1	yes	characteristic
106	#1984	—	67	1	no	—
107	#1834	—	67	1	no	—
108	#1803	—	67	2	no	—
109	#1296	—	67	2	no	—
110	#1277	D₆₇	1	1	yes	characteristic
111	C₆₇	#10623	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	—	67	1	no	—
113	C₂×C₄⋊C₈	—	67	1	no	—
114	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
115	C₂×#103	—	134	2	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#384	—	67	2	no	—
118	#340	—	134	2	no	—
119	#340	—	134	2	no	—
120	#336	—	134	2	no	—
121	C₂²×Q₈	—	67	1	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	134	1	no	—
124	C₂²×C₈	D₂₆₈	1	2	yes	—
125	C₄×Q₈	—	134	2	no	—
126	C₄×Q₈	—	134	2	no	—
127	C₂×C₄⋊C₄	—	67	1	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄⋊C₄	—	134	2	no	—
130	C₂×C₄⋊C₄	—	134	1	no	—
131	C₂×C₄⋊C₄	—	134	2	no	—
132	C₂×C₄²	—	134	1	no	—
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	134	2	no	—
137	#103	—	268	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	67	2	no	—
140	C₂×Q₈	—	134	2	no	—
141	C₂×Q₈	—	134	2	no	—
142	C₂×Q₈	—	134	1	no	—
143	C₂²×C₄	—	268	2	no	—
144	C₂²×C₄	—	134	1	no	—
145	C₂²×C₄	—	134	1	no	—
146	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
148	C₂²×C₄	#11761629	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	268	1	no	—
156	C₂×C₈	—	268	2	no	—
157	C₄⋊C₄	—	268	2	no	—
158	C₄⋊C₄	—	268	2	no	—
159	C₄⋊C₄	—	268	2	no	—
160	C₄⋊C₄	—	268	2	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	536	2	no	—
163	C₄⋊C₄	—	134	2	no	—
164	C₄²	—	268	1	no	—
165	C₄²	—	268	1	no	—
166	C₄²	#11761627	1	2	yes	—
167	C₄²	#11761628	1	2	yes	—
168	C₂³	#14524018	1	1	yes	characteristic
169	Q₈	—	268	2	no	—
170	Q₈	—	134	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	268	1	no	—
173	C₂×C₄	—	268	1	no	—
174	C₂×C₄	—	268	1	no	—
175	C₂×C₄	—	268	2	no	—
176	C₂×C₄	—	536	2	no	—
177	C₂×C₄	—	536	2	no	—
178	C₂×C₄	—	536	2	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	#14524025	1	2	yes	—
181	C₂×C₄	#14524027	1	1	yes	characteristic
182	C₂×C₄	#14524031	1	1	yes	characteristic
183	C₂×C₄	#14524038	1	2	yes	—
184	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524065	1	2	yes	—
186	C₂×C₄	#14524066	1	1	yes	characteristic
187	C₂×C₄	#14524072	1	2	yes	—
188	C₂×C₄	#14524076	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	2	no	—
191	C₈	—	536	1	no	—
192	C₂²	#16863323	1	1	yes	characteristic
193	C₂²	#16863619	1	2	yes	—
194	C₂²	#16863633	1	1	yes	characteristic
195	C₂²	#16863655	1	1	yes	characteristic
196	C₂²	#16863660	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	268	2	no	—
199	C₄	—	536	1	no	—
200	C₄	—	1072	2	no	—
201	C₄	#16863708	1	2	yes	—
202	C₄	#16863736	1	2	yes	—
203	C₄	#16863738	1	2	yes	—
204	C₄	#16863751	1	1	yes	characteristic
205	C₄	#16863765	1	1	yes	characteristic
206	C₂	#22559654	1	2	yes	—
207	C₂	#22559792	1	2	yes	—
208	C₂	#22561629	1	1	yes	characteristic
209	C₂	#22561634	1	1	yes	characteristic
210	C₂	#22561656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561145

C₂

yes

maximal, characteristic

#22560337

C₂

yes

maximal, characteristic

#22560215

C₂

yes

maximal

#22558093

C₂

yes

maximal

C₆₇×#1277

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₂²

yes

characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×#14523968

—

#16863266

C₂²

yes

—

#16863159

—

#16863159

—

#16863150

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

D₄

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

D₄

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₈

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#397

yes

characteristic

C₂²×C₆₇

#464

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

#480

yes

—

C₄×C₆₇

#485

yes

—

C₄×C₆₇

#485

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

100

#10623

—

maximal, Sylow

101

C₂×C₆₇

#1682

yes

—

102

C₂×C₆₇

#1704

yes

—

103

C₂×C₆₇

#2074

yes

characteristic

104

C₂×C₆₇

#2075

yes

characteristic

105

C₂×C₆₇

#2079

yes

characteristic

106

#1984

—

107

#1834

—

108

#1803

—

109

#1296

—

110

#1277

D₆₇

yes

characteristic

111

C₆₇

#10623

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

—

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

115

C₂×#103

—

134

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

#384

—

118

#340

—

134

—

119

#340

—

134

—

120

#336

—

134

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

134

—

124

C₂²×C₈

D₂₆₈

yes

—

125

C₄×Q₈

—

134

—

126

C₄×Q₈

—

134

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

134

—

129

C₂×C₄⋊C₄

—

134

—

130

C₂×C₄⋊C₄

—

134

—

131

C₂×C₄⋊C₄

—

134

—

132

C₂×C₄²

—

134

—

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

134

—

137

#103

—

268

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

134

—

141

C₂×Q₈

—

134

—

142

C₂×Q₈

—

134

—

143

C₂²×C₄

—

268

—

144

C₂²×C₄

—

134

—

145

C₂²×C₄

—

134

—

146

C₂²×C₄

D₄×D₆₇

yes

characteristic

147

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

148

C₂²×C₄

#11761629

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

268

—

156

C₂×C₈

—

268

—

157

C₄⋊C₄

—

268

—

158

C₄⋊C₄

—

268

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

268

—

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

536

—

163

C₄⋊C₄

—

134

—

164

C₄²

—

268

—

165

C₄²

—

268

—

166

C₄²

#11761627

yes

—

167

C₄²

#11761628

yes

—

168

C₂³

#14524018

yes

characteristic

169

Q₈

—

268

—

170

Q₈

—

134

—

171

C₂×C₄

—

172

C₂×C₄

—

268

—

173

C₂×C₄

—

268

—

174

C₂×C₄

—

268

—

175

C₂×C₄

—

268

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

536

—

178

C₂×C₄

—

536

—

179

C₂×C₄

—

134

—

180

C₂×C₄

#14524025

yes

—

181

C₂×C₄

#14524027

yes

characteristic

182

C₂×C₄

#14524031

yes

characteristic

183

C₂×C₄

#14524038

yes

—

184

C₂×C₄

D₈×D₆₇

yes

characteristic

185

C₂×C₄

#14524065

yes

—

186

C₂×C₄

#14524066

yes

characteristic

187

C₂×C₄

#14524072

yes

—

188

C₂×C₄

#14524076

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

536

—

192

C₂²

#16863323

yes

characteristic

193

C₂²

#16863619

yes

—

194

C₂²

#16863633

yes

characteristic

195

C₂²

#16863655

yes

characteristic

196

C₂²

#16863660

yes

—

197

C₄

—

198

C₄

—

268

—

199

C₄

—

536

—

200

C₄

—

1072

—

201

C₄

#16863708

yes

—

202

C₄

#16863736

yes

—

203

C₄

#16863738

yes

—

204

C₄

#16863751

yes

characteristic

205

C₄

#16863765

yes

characteristic

206

C₂

#22559654

yes

—

207

C₂

#22559792

yes

—

208

C₂

#22561629

yes

characteristic

209

C₂

#22561634

yes

characteristic

210

C₂

#22561656

yes

characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#10623
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#14524018

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1277

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
111. C₆₇	#10623	semidirect

H ≅ G/N

type

111. C₆₇

#10623

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22561145	2
#22560337	2
#22560215	2
#22558093	2
C₆₇×#1277	2
#10623	67

subgroup

index

C₆₇×#1277

Subgroups — #30760577

Conjugacy classes of subgroups

Complements

Maximal subgroups