Group #30755498 — subgroups

Subgroups — #30755498

Order 17152.

466 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22561198	C₂	1	2	yes	maximal
3	#22558965	C₂	1	1	yes	maximal, characteristic
4	#22558889	C₂	1	2	yes	maximal
5	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₄	1	4	yes	—
7	C₂×#14524043	C₂²	1	2	yes	—
8	#16863776	—	2	2	no	—
9	#16863776	—	2	1	no	—
10	#16863776	—	2	1	no	—
11	#16863420	C₂²	1	2	yes	—
12	#16863164	—	2	4	no	—
13	#16863147	—	2	2	no	—
14	#16863144	—	2	4	no	—
15	#431×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#322×C₆₇	—	2	1	no	—
19	#322×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	C₂×C₄	1	4	yes	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	4	no	—
23	C₂×Dic₂₆₈	—	4	8	no	—
24	C₂×#11761612	—	2	2	no	—
25	#14524043	—	2	2	no	—
26	#14524043	—	2	2	no	—
27	#14524043	—	2	2	no	—
28	#14524043	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
37	C₂²⋊C₈×C₆₇	—	2	1	no	—
38	#97×C₆₇	—	2	1	no	—
39	C₄×C₈×C₆₇	—	2	1	no	—
40	C₄×C₈×C₆₇	—	2	1	no	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×Dic₁₃₄	—	2	4	no	—
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	8	no	—
47	C₂²×Dic₆₇	—	2	4	no	—
48	Dic₂₆₈	—	4	8	no	—
49	Dic₂₆₈	—	4	8	no	—
50	#11761612	—	4	4	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	#41	1	1	yes	characteristic
69	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#137	1	4	yes	—
79	C₂×Dic₆₇	—	4	4	no	—
80	C₂×Dic₆₇	—	8	4	no	—
81	Dic₁₃₄	—	4	8	no	—
82	Dic₁₃₄	—	8	4	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	4	2	no	—
85	C₈×C₆₇	—	4	2	no	—
86	C₈×C₆₇	#100	1	4	yes	—
87	C₆₇⋊C₈	—	8	2	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	#390	1	1	yes	characteristic
91	C₂²×C₆₇	#441	1	2	yes	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	C₂×#100	1	2	yes	—
95	C₄×C₆₇	#470	1	2	yes	—
96	C₄×C₆₇	#473	1	2	yes	—
97	C₄×C₆₇	#497	1	2	yes	—
98	Dic₆₇	—	8	4	no	—
99	#9807	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	#1965	1	1	yes	characteristic
102	C₂×C₆₇	#1971	1	1	yes	characteristic
103	C₂×C₆₇	#1979	1	1	yes	characteristic
104	C₂×#494	—	67	1	no	—
105	#1997	—	67	2	no	—
106	#1566	—	67	1	no	—
107	#1548	—	67	2	no	—
108	#1451	D₆₇	1	1	yes	characteristic
109	C₆₇	#9807	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₁₆	—	67	4	no	—
111	C₂×C₄⋊Q₈	—	67	2	no	—
112	#494	—	134	2	no	—
113	#494	—	134	1	no	—
114	#494	—	134	1	no	—
115	#431	C₂×D₆₇	1	1	yes	characteristic
116	#423	—	67	2	no	—
117	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	#343	—	134	4	no	—
120	#334	—	134	2	no	—
121	#333	—	134	4	no	—
122	#322	—	2	1	no	—
123	#322	—	2	1	no	—
124	C₂²×Q₈	—	67	4	no	—
125	C₂×Q₁₆	—	268	8	no	—
126	C₂×Q₁₆	—	134	4	no	—
127	C₂×Q₁₆	—	134	4	no	—
128	C₂×M₁₆	—	134	2	no	—
129	C₂×M₁₆	D₂₆₈	1	2	yes	—
130	C₂²×C₈	C₄×D₆₇	1	2	yes	—
131	C₄⋊Q₈	—	134	2	no	—
132	C₄⋊Q₈	—	134	2	no	—
133	C₄⋊Q₈	—	134	2	no	—
134	C₄⋊Q₈	—	134	2	no	—
135	C₂×C₄⋊C₄	—	134	2	no	—
136	C₂×C₄⋊C₄	—	134	2	no	—
137	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
138	C₄⋊C₈	—	134	2	no	—
139	C₄⋊C₈	—	134	2	no	—
140	C₄⋊C₈	D₂₆₈	1	2	yes	—
141	C₂²⋊C₈	—	2	1	no	—
142	#97	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	D₂₆₈	1	2	yes	—
146	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×Q₈	—	268	8	no	—
149	C₂×Q₈	—	134	4	no	—
150	C₂×Q₈	—	134	4	no	—
151	C₂²×C₄	—	134	4	no	—
152	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
154	C₂²×C₄	#11761628	1	1	yes	characteristic
155	Q₁₆	—	268	8	no	—
156	Q₁₆	—	268	8	no	—
157	M₁₆	—	2	4	no	—
158	M₁₆	—	268	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	268	2	no	—
163	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
164	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
165	C₄⋊C₄	—	268	2	no	—
166	C₄⋊C₄	—	268	2	no	—
167	C₄⋊C₄	—	268	2	no	—
168	C₄⋊C₄	—	268	2	no	—
169	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
170	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
171	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
172	C₄²	#11761628	1	1	yes	characteristic
173	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
174	Q₈	—	268	8	no	—
175	Q₈	—	536	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	268	4	no	—
179	C₂×C₄	—	536	4	no	—
180	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
182	C₂×C₄	#14524017	1	1	yes	characteristic
183	C₂×C₄	#14524017	1	1	yes	characteristic
184	C₂×C₄	#14524031	1	1	yes	characteristic
185	C₂×C₄	#14524040	1	1	yes	characteristic
186	C₂×C₄	#14524076	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	—	4	2	no	—
190	C₈	—	536	2	no	—
191	C₈	#14523959	1	4	yes	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#16863298	1	1	yes	characteristic
195	C₂²	#16863504	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	536	4	no	—
199	C₄	C₂×#14523959	1	2	yes	—
200	C₄	#16863654	1	2	yes	—
201	C₄	#16863671	1	2	yes	—
202	C₄	#16863790	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#22561049	1	1	yes	characteristic
205	C₂	#22561080	1	1	yes	characteristic
206	C₂	#22561124	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

#22561198

C₂

yes

maximal

#22558965

C₂

yes

maximal, characteristic

#22558889

C₂

yes

maximal

C₆₇×#1451

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₄

yes

—

C₂×#14524043

C₂²

yes

—

—

—

—

C₂²

yes

—

#16863164

—

#16863147

—

#16863144

—

#431×C₆₇

C₂²

yes

characteristic

C₄×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

—

#322×C₆₇

—

C₂²×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂×M₁₆×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

D₄

yes

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#100

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#441

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#100

yes

—

C₄×C₆₇

#470

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

#497

yes

—

Dic₆₇

—

#9807

—

maximal, Sylow

100

C₂×C₆₇

—

101

C₂×C₆₇

#1965

yes

characteristic

102

C₂×C₆₇

#1971

yes

characteristic

103

C₂×C₆₇

#1979

yes

characteristic

104

C₂×#494

—

105

#1997

—

106

#1566

—

107

#1548

—

108

#1451

D₆₇

yes

characteristic

109

C₆₇

#9807

yes

characteristic, complemented, Sylow

110

C₂²×Q₁₆

—

111

C₂×C₄⋊Q₈

—

112

#494

—

134

—

113

#494

—

134

—

114

#494

—

134

—

115

#431

C₂×D₆₇

yes

characteristic

116

#423

—

117

C₄×M₁₆

C₂×D₆₇

yes

characteristic

118

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

119

#343

—

134

—

120

#334

—

134

—

121

#333

—

134

—

122

#322

—

123

#322

—

124

C₂²×Q₈

—

125

C₂×Q₁₆

—

268

—

126

C₂×Q₁₆

—

134

—

127

C₂×Q₁₆

—

134

—

128

C₂×M₁₆

—

134

—

129

C₂×M₁₆

D₂₆₈

yes

—

130

C₂²×C₈

C₄×D₆₇

yes

—

131

C₄⋊Q₈

—

134

—

132

C₄⋊Q₈

—

134

—

133

C₄⋊Q₈

—

134

—

134

C₄⋊Q₈

—

134

—

135

C₂×C₄⋊C₄

—

134

—

136

C₂×C₄⋊C₄

—

134

—

137

C₂×C₄²

C₂²×D₆₇

yes

characteristic

138

C₄⋊C₈

—

134

—

139

C₄⋊C₈

—

134

—

140

C₄⋊C₈

D₂₆₈

yes

—

141

C₂²⋊C₈

—

142

#97

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

D₂₆₈

yes

—

146

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

147

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

148

C₂×Q₈

—

268

—

149

C₂×Q₈

—

134

—

150

C₂×Q₈

—

134

—

151

C₂²×C₄

—

134

—

152

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

153

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

154

C₂²×C₄

#11761628

yes

characteristic

155

Q₁₆

—

268

—

156

Q₁₆

—

268

—

157

M₁₆

—

158

M₁₆

—

268

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

268

—

163

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

164

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

165

C₄⋊C₄

—

268

—

166

C₄⋊C₄

—

268

—

167

C₄⋊C₄

—

268

—

168

C₄⋊C₄

—

268

—

169

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

170

C₄²

C₂×D₂₆₈

yes

characteristic

171

C₄²

C₂×D₂₆₈

yes

characteristic

172

C₄²

#11761628

yes

characteristic

173

C₂³

C₄×D₂₆₈

yes

characteristic

174

Q₈

—

268

—

175

Q₈

—

536

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

536

—

180

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

181

C₂×C₄

C₄×D₂₆₈

yes

characteristic

182

C₂×C₄

#14524017

yes

characteristic

183

C₂×C₄

#14524017

yes

characteristic

184

C₂×C₄

#14524031

yes

characteristic

185

C₂×C₄

#14524040

yes

characteristic

186

C₂×C₄

#14524076

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

536

—

191

C₈

#14523959

yes

—

192

C₂²

—

193

C₂²

—

194

C₂²

#16863298

yes

characteristic

195

C₂²

#16863504

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

536

—

199

C₄

C₂×#14523959

yes

—

200

C₄

#16863654

yes

—

201

C₄

#16863671

yes

—

202

C₄

#16863790

yes

—

203

C₂

—

204

C₂

#22561049

yes

characteristic

205

C₂

#22561080

yes

characteristic

206

C₂

#22561124

yes

characteristic

	subgroup	G/H
center	C₂²	#16863298
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#9807
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

subgroup

G/H

center

C₂²

#16863298

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1451

C₂

socle

C₂²×C₆₇

#390

prime radical

C₂³×C₆₇

C₄×D₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

N	H ≅ G/N	type
109. C₆₇	#9807	semidirect

H ≅ G/N

type

109. C₆₇

#9807

semidirect

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

subgroup	index
C₂×#16863776	2
#22561198	2
#22558965	2
#22558889	2
C₆₇×#1451	2
#9807	67

subgroup

index

C₆₇×#1451

Subgroups — #30755498

Conjugacy classes of subgroups

Complements

Maximal subgroups