Group #30774318 — subgroups

Subgroups — #30774318

Order 17152.

566 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂×#16863773	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863210	C₂	1	4	yes	maximal, complemented
3	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₃₆	—	2	1	no	—
6	C₂²×D₅₃₆	C₂²	1	2	yes	—
7	C₂×#14524040	—	2	1	no	—
8	#16863773	—	2	2	no	—
9	#16863773	C₄	1	4	yes	—
10	C₂×#14523979	C₂²	1	2	yes	—
11	C₄×#11761612	—	2	1	no	—
12	#16863210	—	2	8	no	—
13	#16863161	—	2	1	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₆₇	—	2	2	no	—
17	C₂²×D₂₆₈	—	2	2	no	—
18	C₂²×D₂₆₈	—	4	2	no	—
19	C₂×D₅₃₆	—	2	4	no	—
20	C₂×D₅₃₆	—	2	4	no	—
21	C₂×D₅₃₆	—	2	2	no	—
22	C₂×D₅₃₆	—	2	4	no	—
23	C₂×D₅₃₆	—	4	4	no	—
24	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
25	C₂×#11761612	—	2	2	no	—
26	#14524040	—	2	4	no	—
27	#14524040	—	4	1	no	—
28	#14524040	—	4	1	no	—
29	#14523979	—	2	2	no	—
30	#14523979	D₄	1	4	yes	complemented
31	#14523954	—	4	1	no	—
32	C₄×C₆₇⋊C₈	—	4	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	4	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂³×D₆₇	—	4	2	no	—
41	C₂×D₂₆₈	—	2	4	no	—
42	C₂×D₂₆₈	—	4	4	no	—
43	C₂×D₂₆₈	—	4	4	no	—
44	C₂×D₂₆₈	—	4	4	no	—
45	C₂×D₂₆₈	—	4	4	no	—
46	C₂×D₂₆₈	—	4	2	no	—
47	C₂×D₂₆₈	—	8	8	no	—
48	D₅₃₆	—	2	8	no	—
49	D₅₃₆	—	4	8	no	—
50	D₅₃₆	—	4	2	no	—
51	D₅₃₆	—	4	2	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₆₇	D₈	1	1	yes	characteristic
65	C₄²×C₆₇	D₈	1	1	yes	characteristic
66	C₁₆×C₆₇	—	4	2	no	—
67	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×D₆₇	—	4	4	no	—
69	C₂²×D₆₇	—	8	8	no	—
70	C₂²×D₆₇	—	8	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	#102	1	2	yes	—
73	C₂×C₄×C₆₇	#102	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
78	D₂₆₈	—	4	4	no	—
79	D₂₆₈	—	4	4	no	—
80	D₂₆₈	—	8	8	no	—
81	D₂₆₈	—	8	8	no	—
82	D₂₆₈	—	8	4	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	C₄×D₄	1	2	yes	—
85	C₈×C₆₇	#124	1	1	yes	characteristic
86	C₈×C₆₇	#127	1	1	yes	characteristic
87	C₆₇⋊C₈	—	4	4	no	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
90	C₂²×C₆₇	#460	1	2	yes	—
91	C₂×D₆₇	—	8	8	no	—
92	C₂×D₆₇	—	8	4	no	—
93	C₂×D₆₇	—	16	8	no	—
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	#361	1	4	yes	complemented
96	C₄×C₆₇	#390	1	1	yes	characteristic
97	C₄×C₆₇	#418	1	1	yes	characteristic
98	C₄×C₆₇	#459	1	2	yes	—
99	#13257	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	#1982	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#361	1	2	yes	—
103	D₆₇	—	16	4	no	—
104	C₂×#493	—	67	1	no	—
105	#2150	D₆₇	1	1	yes	characteristic, complemented
106	C₂×#361	—	67	4	no	—
107	C₄×#108	—	67	1	no	—
108	C₆₇	#13257	1	1	yes	characteristic, complemented, Sylow
109	C₂²×D₈	—	67	2	no	—
110	C₂²×D₈	—	134	1	no	—
111	C₂×#127	—	134	1	no	—
112	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
113	#493	—	67	4	no	—
114	#493	—	134	2	no	—
115	C₂×#108	—	67	2	no	—
116	C₄×M₁₆	—	134	1	no	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₄⋊C₁₆	—	2	2	no	—
119	#361	—	134	8	no	—
120	#341	—	134	1	no	—
121	C₂²×D₄	—	268	2	no	—
122	C₂²×D₄	—	134	2	no	—
123	C₂×D₈	—	268	4	no	—
124	C₂×D₈	—	67	4	no	—
125	C₂×D₈	—	134	4	no	—
126	C₂×D₈	—	134	4	no	—
127	C₂×D₈	—	134	2	no	—
128	C₂×D₈	—	134	4	no	—
129	C₂×M₁₆	—	134	2	no	—
130	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
132	#127	—	268	1	no	—
133	#127	—	268	1	no	—
134	#127	—	134	4	no	—
135	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
136	M₃₂	—	2	4	no	—
137	C₂×C₁₆	—	2	2	no	—
138	#108	—	67	4	no	—
139	#108	—	134	2	no	—
140	#97	—	268	1	no	—
141	C₄×C₈	—	268	1	no	—
142	C₄×C₈	C₄×D₆₇	1	2	yes	—
143	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
144	C₂⁴	—	268	2	no	—
145	C₂×D₄	—	268	4	no	—
146	C₂×D₄	—	268	4	no	—
147	C₂×D₄	—	268	4	no	—
148	C₂×D₄	—	268	2	no	—
149	C₂×D₄	—	268	4	no	—
150	C₂×D₄	—	536	8	no	—
151	C₂×D₄	—	134	4	no	—
152	C₂²×C₄	#11761617	1	2	yes	—
153	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
154	D₈	—	268	8	no	—
155	D₈	—	268	2	no	—
156	D₈	—	268	2	no	—
157	D₈	—	134	8	no	—
158	M₁₆	—	268	2	no	—
159	M₁₆	—	134	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	268	2	no	—
162	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₈	D₄×D₆₇	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761629	1	2	yes	—
167	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
168	C₄²	D₅₃₆	1	1	yes	characteristic
169	C₄²	D₅₃₆	1	1	yes	characteristic
170	C₁₆	—	4	2	no	—
171	C₂³	—	268	4	no	—
172	C₂³	—	536	8	no	—
173	C₂³	—	536	4	no	—
174	C₂³	C₂×#11761617	1	1	yes	characteristic
175	D₄	—	268	4	no	—
176	D₄	—	268	4	no	—
177	D₄	—	536	8	no	—
178	D₄	—	536	4	no	—
179	D₄	—	536	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#14523965	1	2	yes	—
182	C₂×C₄	#14523965	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#14524018	1	2	yes	—
185	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
186	C₂×C₄	#14524071	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	268	4	no	—
189	C₈	#14524005	1	2	yes	—
190	C₈	#14524032	1	1	yes	characteristic
191	C₈	#14524039	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	536	4	no	—
194	C₂²	—	536	8	no	—
195	C₂²	—	1072	8	no	—
196	C₂²	C₂×#14523965	1	1	yes	characteristic
197	C₂²	#16863597	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#16863210	1	4	yes	complemented
200	C₄	#16863297	1	1	yes	characteristic
201	C₄	#16863402	1	1	yes	characteristic
202	C₄	#16863592	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	1072	4	no	—
205	C₂	#22561135	1	1	yes	characteristic
206	C₂	C₂×#16863210	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863773

C₂

yes

maximal, characteristic

C₂×#16863210

C₂

yes

maximal, complemented

C₄×#14523979

C₂

yes

maximal, characteristic, complemented

C₆₇×#2150

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₃₆

—

C₂²×D₅₃₆

C₂²

yes

—

C₂×#14524040

—

#16863773

—

#16863773

C₄

yes

—

C₂×#14523979

C₂²

yes

—

C₄×#11761612

—

#16863210

—

#16863161

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

complemented

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

C₄×D₄

yes

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#102

yes

characteristic

C₂²×C₆₇

#460

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#13257

—

maximal, Sylow

100

C₂×C₆₇

—

101

C₂×C₆₇

#1982

yes

characteristic

102

C₂×C₆₇

C₂×#361

yes

—

103

D₆₇

—

104

C₂×#493

—

105

#2150

D₆₇

yes

characteristic, complemented

106

C₂×#361

—

107

C₄×#108

—

108

C₆₇

#13257

yes

characteristic, complemented, Sylow

109

C₂²×D₈

—

110

C₂²×D₈

—

134

—

111

C₂×#127

—

134

—

112

C₂×M₃₂

C₂×D₆₇

yes

—

113

#493

—

114

#493

—

134

—

115

C₂×#108

—

116

C₄×M₁₆

—

134

—

117

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

118

C₄⋊C₁₆

—

119

#361

—

134

—

120

#341

—

134

—

121

C₂²×D₄

—

268

—

122

C₂²×D₄

—

134

—

123

C₂×D₈

—

268

—

124

C₂×D₈

—

125

C₂×D₈

—

134

—

126

C₂×D₈

—

134

—

127

C₂×D₈

—

134

—

128

C₂×D₈

—

134

—

129

C₂×M₁₆

—

134

—

130

C₂²×C₈

D₂₆₈

yes

characteristic

131

C₂²×C₈

C₂²×D₆₇

yes

characteristic

132

#127

—

268

—

133

#127

—

268

—

134

#127

—

134

—

135

C₂×C₄²

D₂₆₈

yes

characteristic

136

M₃₂

—

137

C₂×C₁₆

—

138

#108

—

139

#108

—

134

—

140

#97

—

268

—

141

C₄×C₈

—

268

—

142

C₄×C₈

C₄×D₆₇

yes

—

143

C₄×C₈

C₂²⋊D₆₇

yes

—

144

C₂⁴

—

268

—

145

C₂×D₄

—

268

—

146

C₂×D₄

—

268

—

147

C₂×D₄

—

268

—

148

C₂×D₄

—

268

—

149

C₂×D₄

—

268

—

150

C₂×D₄

—

536

—

151

C₂×D₄

—

134

—

152

C₂²×C₄

#11761617

yes

—

153

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

154

D₈

—

268

—

155

D₈

—

268

—

156

D₈

—

268

—

157

D₈

—

134

—

158

M₁₆

—

268

—

159

M₁₆

—

134

—

160

C₂×C₈

—

161

C₂×C₈

—

268

—

162

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

163

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

164

C₂×C₈

D₄×D₆₇

yes

—

165

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

166

C₂×C₈

#11761629

yes

—

167

C₄²

(C₂×D₆₇)⋊C₄

yes

—

168

C₄²

D₅₃₆

yes

characteristic

169

C₄²

D₅₃₆

yes

characteristic

170

C₁₆

—

171

C₂³

—

268

—

172

C₂³

—

536

—

173

C₂³

—

536

—

174

C₂³

C₂×#11761617

yes

characteristic

175

D₄

—

268

—

176

D₄

—

268

—

177

D₄

—

536

—

178

D₄

—

536

—

179

D₄

—

536

—

180

C₂×C₄

—

181

C₂×C₄

#14523965

yes

—

182

C₂×C₄

#14523965

yes

—

183

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

184

C₂×C₄

#14524018

yes

—

185

C₂×C₄

C₂×D₅₃₆

yes

characteristic

186

C₂×C₄

#14524071

yes

—

187

C₈

—

188

C₈

—

268

—

189

C₈

#14524005

yes

—

190

C₈

#14524032

yes

characteristic

191

C₈

#14524039

yes

characteristic

192

C₂²

—

193

C₂²

—

536

—

194

C₂²

—

536

—

195

C₂²

—

1072

—

196

C₂²

C₂×#14523965

yes

characteristic

197

C₂²

#16863597

yes

—

198

C₄

—

199

C₄

#16863210

yes

complemented

200

C₄

#16863297

yes

characteristic

201

C₄

#16863402

yes

characteristic

202

C₄

#16863592

yes

—

203

C₂

—

204

C₂

—

1072

—

205

C₂

#22561135

yes

characteristic

206

C₂

C₂×#16863210

yes

—

	subgroup	G/H
center	C₂²	C₂×#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂×#16863773	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#13257
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#14523965

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

C₂²×C₆₇

C₂

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₂×#16863210	C₂	semidirect
3. C₄×#14523979	C₂	semidirect
4. C₆₇×#2150	C₂	semidirect
30. #14523979	D₄	semidirect
95. C₄×C₆₇	#361	semidirect
105. #2150	D₆₇	semidirect
108. C₆₇	#13257	semidirect
199. C₄	#16863210	semidirect

H ≅ G/N

type

2. C₂×#16863210

C₂

semidirect

3. C₄×#14523979

C₂

semidirect

4. C₆₇×#2150

C₂

semidirect

30. #14523979

D₄

semidirect

95. C₄×C₆₇

#361

semidirect

105. #2150

D₆₇

semidirect

108. C₆₇

#13257

semidirect

199. C₄

#16863210

semidirect

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

subgroup	index
C₂×#16863773	2
C₂×#16863210	2
C₄×#14523979	2
C₆₇×#2150	2
#13257	67

subgroup

index

C₆₇×#2150

Subgroups — #30774318

Conjugacy classes of subgroups

Complements

Maximal subgroups