Group #30748821 — subgroups

Subgroups — #30748821

Order 17152.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22568921	C₂	1	1	yes	maximal, characteristic
2	#22558970	C₂	1	2	yes	maximal, complemented
3	#22558844	C₂	1	1	yes	maximal, characteristic
4	#22558119	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524050	C₄	1	2	yes	—
7	#16863474	—	2	2	no	—
8	D₄×C₆₇⋊C₈	—	2	2	no	—
9	#16863453	C₂²	1	2	yes	—
10	#16863420	C₂²	1	1	yes	characteristic
11	#16863363	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863148	—	2	2	no	—
14	#16863146	—	2	2	no	—
15	#16863136	—	2	2	no	—
16	#532×C₆₇	C₂²	1	1	yes	characteristic
17	#432×C₆₇	C₂²	1	2	yes	—
18	#329×C₆₇	—	2	2	no	—
19	#14524050	—	4	4	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
29	#14523954	—	2	2	no	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₆₇	Q₈	1	1	yes	characteristic
34	#127×C₆₇	Q₈	1	1	yes	characteristic
35	#124×C₆₇	D₄	1	2	yes	—
36	#121×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₂²⋊C₈×C₆₇	—	2	2	no	—
41	#97×C₆₇	—	2	2	no	—
42	#11761612	—	4	4	no	—
43	#11761612	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₂×C₆₇⋊C₈	—	4	4	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
50	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
52	#41×C₆₇	—	4	2	no	—
53	#41×C₆₇	—	4	2	no	—
54	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	#41	1	2	yes	—
61	C₂×C₈×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂²⋊C₄×C₆₇	—	2	4	no	—
64	C₄²×C₆₇	C₂×D₄	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	#117	1	1	yes	characteristic
69	Q₈×C₆₇	—	4	2	no	—
70	D₄×C₆₇	—	4	2	no	—
71	D₄×C₆₇	—	4	4	no	—
72	D₄×C₆₇	—	4	4	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#121	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₆₇	—	8	2	no	—
84	C₆₇⋊C₈	—	2	1	no	—
85	C₆₇⋊C₈	—	2	2	no	—
86	C₆₇⋊C₈	—	4	2	no	—
87	C₆₇⋊C₈	—	8	2	no	—
88	C₆₇⋊C₈	#100	1	2	yes	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	—	2	2	no	—
91	C₂²×C₆₇	—	4	2	no	—
92	C₂²×C₆₇	—	4	4	no	—
93	C₂²×C₆₇	#380	1	1	yes	characteristic
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	4	2	no	—
98	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
99	C₄×C₆₇	#413	1	1	yes	characteristic
100	#8630	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	4	1	no	—
102	C₂×C₆₇	—	8	2	no	—
103	C₂×C₆₇	#1800	1	1	yes	characteristic
104	C₂×C₆₇	#1814	1	2	yes	—
105	#2931	—	67	1	no	—
106	#1568	—	67	2	no	—
107	#1536	—	67	1	no	—
108	#1536	D₆₇	1	1	yes	characteristic
109	#1307	—	67	2	no	—
110	C₆₇	#8630	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	—	67	2	no	—
112	#532	C₂×D₆₇	1	1	yes	characteristic
113	#436	—	134	2	no	—
114	C₈×D₄	—	134	2	no	—
115	#432	—	67	2	no	—
116	#432	C₂×D₆₇	1	2	yes	—
117	#423	—	67	1	no	—
118	#408	—	67	2	no	—
119	C₂×C₄×C₈	—	67	1	no	—
120	#335	—	134	2	no	—
121	#334	—	134	2	no	—
122	#329	—	2	2	no	—
123	#329	—	134	2	no	—
124	C₂×#41	C₄×D₆₇	1	2	yes	—
125	#131	—	268	4	no	—
126	C₂×M₁₆	—	67	2	no	—
127	C₂×M₁₆	—	134	2	no	—
128	C₂²×C₈	—	67	1	no	—
129	C₂²×C₈	—	67	1	no	—
130	C₂²×C₈	—	134	2	no	—
131	C₄⋊Q₈	Dic₁₃₄	1	1	yes	characteristic
132	#127	Dic₁₃₄	1	1	yes	characteristic
133	#124	C₂²⋊D₆₇	1	2	yes	—
134	#121	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	134	2	no	—
139	C₄⋊C₈	—	134	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	134	2	no	—
142	C₂²⋊C₈	—	134	4	no	—
143	#97	—	2	2	no	—
144	#97	—	134	2	no	—
145	C₄×C₈	—	67	2	no	—
146	C₄×C₈	—	67	2	no	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	Dic₆₇⋊C₄	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₄	#11761627	1	2	yes	—
156	M₁₆	—	268	4	no	—
157	M₁₆	—	268	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	268	4	no	—
160	C₂×C₈	—	268	2	no	—
161	C₂×C₈	—	268	1	no	—
162	C₂×C₈	—	67	1	no	—
163	C₂×C₈	—	67	2	no	—
164	C₂×C₈	—	67	1	no	—
165	C₂×C₈	—	134	2	no	—
166	C₂×C₈	—	134	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	D₄×D₆₇	1	2	yes	—
170	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#14523999	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
186	C₂×C₄	#14524015	1	2	yes	—
187	C₂×C₄	#14524020	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	268	2	no	—
190	C₈	—	536	2	no	—
191	C₈	—	67	2	no	—
192	C₈	—	134	1	no	—
193	C₈	—	134	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	#16863254	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₆₇	1	1	yes	characteristic
204	C₄	#16863377	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#22560199	1	1	yes	characteristic
208	C₂	#22560261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22568921

C₂

yes

maximal, characteristic

#22558970

C₂

yes

maximal, complemented

#22558844

C₂

yes

maximal, characteristic

#22558119

C₂

yes

maximal, complemented

C₆₇×#1536

C₂

yes

maximal, characteristic

C₂×#14524050

C₄

yes

—

#16863474

—

D₄×C₆₇⋊C₈

—

#16863453

C₂²

yes

—

#16863420

C₂²

yes

characteristic

#16863363

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

#532×C₆₇

C₂²

yes

characteristic

#432×C₆₇

C₂²

yes

—

#329×C₆₇

—

#14524050

—

C₂×#11761612

—

C₂×#11761612

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×#41×C₆₇

C₂×C₄

yes

—

C₄⋊Q₈×C₆₇

Q₈

yes

characteristic

#127×C₆₇

Q₈

yes

characteristic

#124×C₆₇

D₄

yes

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₄⋊C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#100

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

100

—

101

C₂×C₆₇

—

102

C₂×C₆₇

—

103

C₂×C₆₇

#1800

yes

characteristic

104

C₂×C₆₇

#1814

yes

—

105

#2931

—

106

#1568

—

107

#1536

—

108

#1536

D₆₇

yes

characteristic

109

#1307

—

110

C₆₇

#8630

yes

characteristic, complemented, Sylow

111

C₂×#131

—

112

#532

C₂×D₆₇

yes

characteristic

113

#436

—

134

—

114

C₈×D₄

—

134

—

115

#432

—

116

#432

C₂×D₆₇

yes

—

117

#423

—

118

#408

—

119

C₂×C₄×C₈

—

120

#335

—

134

—

121

#334

—

134

—

122

#329

—

123

#329

—

134

—

124

C₂×#41

C₄×D₆₇

yes

—

125

#131

—

268

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

134

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

134

—

131

C₄⋊Q₈

Dic₁₃₄

yes

characteristic

132

#127

Dic₁₃₄

yes

characteristic

133

#124

C₂²⋊D₆₇

yes

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

C₂²×D₆₇

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

134

—

139

C₄⋊C₈

—

134

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

134

—

142

C₂²⋊C₈

—

134

—

143

#97

—

144

#97

—

134

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#41

—

148

#41

—

149

C₂×Q₈

Dic₆₇⋊C₄

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

Dic₆₇⋊C₄

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

155

C₂²×C₄

#11761627

yes

—

156

M₁₆

—

268

—

157

M₁₆

—

268

—

158

C₂×C₈

—

159

C₂×C₈

—

268

—

160

C₂×C₈

—

268

—

161

C₂×C₈

—

268

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

134

—

166

C₂×C₈

—

134

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

D₄×D₆₇

yes

—

170

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

171

C₄²

C₂×Dic₁₃₄

yes

characteristic

172

C₂³

—

173

C₂³

#14523999

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

185

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

186

C₂×C₄

#14524015

yes

—

187

C₂×C₄

#14524020

yes

—

188

C₈

—

189

C₈

—

268

—

190

C₈

—

536

—

191

C₈

—

192

C₈

—

134

—

193

C₈

—

134

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#16863254

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₆₇

yes

characteristic

204

C₄

#16863377

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#22560199

yes

characteristic

208

C₂

#22560261

yes

—

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1536	C₂
socle	C₂²×C₆₇	#380
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1536	D₆₇
hypocenter	C₆₇	#8630
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1536

C₂

socle

C₂²×C₆₇

#380

prime radical

#532×C₆₇

C₂²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #22558970	C₂	semidirect
4. #22558119	C₂	semidirect
110. C₆₇	#8630	semidirect

H ≅ G/N

type

2. #22558970

C₂

semidirect

4. #22558119

C₂

semidirect

110. C₆₇

#8630

semidirect

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

subgroup	index
#22568921	2
#22558970	2
#22558844	2
#22558119	2
C₆₇×#1536	2
#8630	67

subgroup

index

C₆₇×#1536

Subgroups — #30748821

Conjugacy classes of subgroups

Complements

Maximal subgroups