Group #31009208 — subgroups

Subgroups — #31009208

Order 17152.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#22569076	C₂	1	2	yes	maximal
3	#22562079	C₂	1	4	yes	maximal, complemented
4	#22562067	C₂	1	4	yes	maximal, complemented
5	#22561928	C₂	1	2	yes	maximal
6	#22561805	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
9	#16863873	—	2	2	no	—
10	C₂×Dic₅₃₆	C₄	1	8	yes	complemented
11	C₂×Dic₅₃₆	C₂²	1	2	yes	—
12	C₂×Dic₅₃₆	C₂²	1	2	yes	—
13	C₂×#14523983	C₂²	1	1	yes	characteristic
14	#16863502	—	2	4	no	—
15	#16863490	C₂²	1	8	yes	—
16	#16863404	—	2	2	no	—
17	#16863351	C₂²	1	1	yes	characteristic
18	#16863199	C₂²	1	8	yes	—
19	#16863171	C₂²	1	2	yes	—
20	C₁₆×Dic₆₇	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#427×C₆₇	C₂²	1	2	yes	—
23	#367×C₆₇	C₂²	1	2	yes	—
24	#366×C₆₇	C₂²	1	2	yes	—
25	C₂²×Dic₁₃₄	—	2	2	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	—	2	4	no	—
28	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	—
29	C₂×Dic₂₆₈	C₂³	1	4	yes	—
30	C₂×#11761612	D₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	—	4	4	no	—
32	#14524008	—	2	8	no	—
33	#14524002	D₄	1	1	yes	characteristic
34	#14523999	—	4	4	no	—
35	Dic₅₃₆	C₂×C₄	1	8	yes	—
36	Dic₅₃₆	C₂×C₄	1	8	yes	—
37	#14523983	—	2	2	no	—
38	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
39	#14523966	—	2	8	no	—
40	#14523953	D₄	1	2	yes	—
41	C₈×Dic₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#117×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	C₂³	1	4	yes	—
48	#106×C₆₇	—	2	2	no	—
49	C₂×Dic₁₃₄	—	2	8	no	—
50	C₂×Dic₁₃₄	—	2	4	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	Dic₂₆₈	—	2	8	no	—
55	Dic₂₆₈	—	2	4	no	—
56	Dic₂₆₈	C₂²×C₄	1	8	yes	—
57	#11761612	—	2	2	no	—
58	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₆₇	D₈	1	2	yes	—
60	Dic₆₇⋊C₄	—	4	8	no	—
61	C₂²⋊Dic₆₇	D₈	1	2	yes	—
62	C₄×Dic₆₇	—	4	8	no	—
63	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₆₇⋊C₁₆	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	#41	1	2	yes	—
69	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂²⋊C₄×C₆₇	—	4	2	no	—
72	C₄²×C₆₇	—	4	2	no	—
73	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#135	1	2	yes	—
80	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₆₇	—	4	4	no	—
82	C₂×Dic₆₇	—	4	8	no	—
83	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
84	Dic₁₃₄	—	2	8	no	—
85	Dic₁₃₄	—	4	4	no	—
86	Dic₁₃₄	—	4	8	no	—
87	C₈×C₆₇	C₄×D₄	1	2	yes	—
88	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
90	C₆₇⋊C₈	—	2	2	no	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	C₄×D₈	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	8	4	no	—
95	C₄×C₆₇	C₄×D₈	1	2	yes	—
96	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
98	Dic₆₇	—	2	4	no	—
99	Dic₆₇	—	4	8	no	—
100	#33458	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	2	no	—
102	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₆₇	#3417	1	2	yes	—
104	C₂²×Q₃₂	—	67	1	no	—
105	#2950	—	67	2	no	—
106	#2176	—	67	4	no	—
107	C₄×Q₃₂	—	67	4	no	—
108	#2158	D₆₇	1	1	yes	characteristic
109	#2141	—	67	2	no	—
110	#2108	—	67	1	no	—
111	C₆₇	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	67	2	no	—
113	#519	—	134	2	no	—
114	C₂×Q₃₂	—	67	8	no	—
115	C₂×Q₃₂	—	67	2	no	—
116	C₂×Q₃₂	—	67	2	no	—
117	C₂×M₃₂	—	67	1	no	—
118	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
119	#441	—	134	4	no	—
120	C₄×Q₁₆	—	67	8	no	—
121	#427	C₂×D₆₇	1	2	yes	—
122	#419	—	134	2	no	—
123	#405	—	67	1	no	—
124	#367	C₂×D₆₇	1	2	yes	—
125	#366	C₂×D₆₇	1	2	yes	—
126	#358	—	67	8	no	—
127	#346	—	67	2	no	—
128	C₄×C₁₆	—	67	2	no	—
129	C₂²×Q₈	—	134	2	no	—
130	C₂×Q₁₆	—	67	8	no	—
131	C₂×Q₁₆	—	67	4	no	—
132	C₂×Q₁₆	—	134	2	no	—
133	C₂×Q₁₆	—	134	4	no	—
134	C₂×M₁₆	—	67	1	no	—
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄×Q₈	—	268	4	no	—
137	C₄×Q₈	—	134	8	no	—
138	#117	—	2	2	no	—
139	#117	—	268	4	no	—
140	#117	—	67	1	no	—
141	Q₃₂	—	67	8	no	—
142	Q₃₂	—	67	8	no	—
143	M₃₂	—	134	2	no	—
144	C₂×C₁₆	—	67	2	no	—
145	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
146	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
148	#107	C₂²×D₆₇	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	134	8	no	—
151	#97	—	67	2	no	—
152	C₄×C₈	—	67	2	no	—
153	C₂×Q₈	—	268	2	no	—
154	C₂×Q₈	—	268	4	no	—
155	C₂×Q₈	—	134	8	no	—
156	C₂×Q₈	—	134	4	no	—
157	C₂²×C₄	—	268	2	no	—
158	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
159	Q₁₆	—	67	8	no	—
160	Q₁₆	—	134	8	no	—
161	Q₁₆	—	134	4	no	—
162	M₁₆	—	134	2	no	—
163	C₂×C₈	—	67	2	no	—
164	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
165	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761629	1	2	yes	—
167	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	268	8	no	—
170	C₄⋊C₄	—	67	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	67	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	268	8	no	—
175	C₄²	—	67	2	no	—
176	C₁₆	—	134	2	no	—
177	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
178	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
179	C₂³	D₈×D₆₇	1	1	yes	characteristic
180	Q₈	—	268	4	no	—
181	Q₈	—	268	8	no	—
182	Q₈	—	134	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	268	4	no	—
185	C₂×C₄	—	268	8	no	—
186	C₂×C₄	—	67	4	no	—
187	C₂×C₄	#14524005	1	2	yes	—
188	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
189	C₂×C₄	#14524066	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
191	C₈	—	134	2	no	—
192	C₈	#14524005	1	2	yes	—
193	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
194	C₈	C₂×#11761629	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#16863478	1	2	yes	—
197	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	268	8	no	—
200	C₄	—	134	4	no	—
201	C₄	#16863478	1	2	yes	—
202	C₄	C₂×#14524005	1	1	yes	characteristic
203	C₄	C₂×#14524066	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#16863478	1	1	yes	characteristic
206	C₂	#22573359	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₃₆

C₂

yes

maximal, characteristic

#22569076

C₂

yes

maximal

#22562079

C₂

yes

maximal, complemented

#22562067

C₂

yes

maximal, complemented

#22561928

C₂

yes

maximal

#22561805

C₂

yes

maximal, characteristic

C₆₇×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₂²

yes

—

#16863873

—

C₂×Dic₅₃₆

C₄

yes

complemented

C₂×Dic₅₃₆

C₂²

yes

—

C₂×Dic₅₃₆

C₂²

yes

—

C₂×#14523983

C₂²

yes

characteristic

#16863502

—

#16863490

C₂²

yes

—

#16863404

—

#16863351

C₂²

yes

characteristic

#16863199

C₂²

yes

—

#16863171

C₂²

yes

—

C₁₆×Dic₆₇

C₂²

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

#367×C₆₇

C₂²

yes

—

#366×C₆₇

C₂²

yes

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂×C₄

yes

—

C₂×Dic₂₆₈

C₂³

yes

—

C₂×#11761612

D₄

yes

characteristic

C₄×Dic₁₃₄

—

#14524008

—

#14524002

D₄

yes

characteristic

#14523999

—

Dic₅₃₆

C₂×C₄

yes

—

Dic₅₃₆

C₂×C₄

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₂³

yes

—

#14523966

—

#14523953

D₄

yes

—

C₈×Dic₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

#107×C₆₇

C₂³

yes

—

#106×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

C₂²×C₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₆₇

D₈

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

D₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×D₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

C₂²×C₄

yes

—

C₁₆×C₆₇

C₂²×C₄

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₈

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₄×D₄

yes

—

C₈×C₆₇

C₂³×C₄

yes

characteristic

C₈×C₆₇

C₂×#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₈

yes

—

C₂²×C₆₇

C₂²×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×D₈

yes

—

C₄×C₆₇

C₂×C₄×D₄

yes

characteristic

C₄×C₆₇

C₂×#135

yes

characteristic

Dic₆₇

—

Dic₆₇

—

100

#33458

—

maximal, Sylow

101

C₂×C₆₇

—

102

C₂×C₆₇

C₂×C₄×D₈

yes

characteristic

103

C₂×C₆₇

#3417

yes

—

104

C₂²×Q₃₂

—

105

#2950

—

106

#2176

—

107

C₄×Q₃₂

—

108

#2158

D₆₇

yes

characteristic

109

#2141

—

110

#2108

—

111

C₆₇

#33458

yes

characteristic, complemented, Sylow

112

C₂²×Q₁₆

—

113

#519

—

134

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂×M₃₂

—

118

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

119

#441

—

134

—

120

C₄×Q₁₆

—

121

#427

C₂×D₆₇

yes

—

122

#419

—

134

—

123

#405

—

124

#367

C₂×D₆₇

yes

—

125

#366

C₂×D₆₇

yes

—

126

#358

—

127

#346

—

128

C₄×C₁₆

—

129

C₂²×Q₈

—

134

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

134

—

133

C₂×Q₁₆

—

134

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₆₇

yes

characteristic

136

C₄×Q₈

—

268

—

137

C₄×Q₈

—

134

—

138

#117

—

139

#117

—

268

—

140

#117

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

134

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₆₇

yes

—

146

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

147

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

148

#107

C₂²×D₆₇

yes

—

149

#106

—

150

#103

—

134

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

268

—

154

C₂×Q₈

—

268

—

155

C₂×Q₈

—

134

—

156

C₂×Q₈

—

134

—

157

C₂²×C₄

—

268

—

158

C₂²×C₄

D₄×D₆₇

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

134

—

161

Q₁₆

—

134

—

162

M₁₆

—

134

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₆₇

yes

—

165

C₂×C₈

D₄×D₆₇

yes

characteristic

166

C₂×C₈

#11761629

yes

—

167

C₂×C₈

C₂³×D₆₇

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

268

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

268

—

175

C₄²

—

176

C₁₆

—

134

—

177

C₁₆

C₂×C₄×D₆₇

yes

—

178

C₁₆

C₂×C₄×D₆₇

yes

—

179

C₂³

D₈×D₆₇

yes

characteristic

180

Q₈

—

268

—

181

Q₈

—

268

—

182

Q₈

—

134

—

183

C₂×C₄

—

184

C₂×C₄

—

268

—

185

C₂×C₄

—

268

—

186

C₂×C₄

—

187

C₂×C₄

#14524005

yes

—

188

C₂×C₄

D₈×D₆₇

yes

characteristic

189

C₂×C₄

#14524066

yes

—

190

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

191

C₈

—

134

—

192

C₈

#14524005

yes

—

193

C₈

C₂²×C₄×D₆₇

yes

characteristic

194

C₈

C₂×#11761629

yes

characteristic

195

C₂²

—

196

C₂²

#16863478

yes

—

197

C₂²

C₂×D₈×D₆₇

yes

characteristic

198

C₄

—

199

C₄

—

268

—

200

C₄

—

134

—

201

C₄

#16863478

yes

—

202

C₄

C₂×#14524005

yes

characteristic

203

C₄

C₂×#14524066

yes

characteristic

204

C₂

—

205

C₂

C₂×#16863478

yes

characteristic

206

C₂

#22573359

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33458
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₆₇

commutator

C₈×C₆₇

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₆₇

Fitting

C₆₇×#2158

C₂

socle

C₂²×C₆₇

C₂²×D₈

prime radical

C₂³×C₆₇

C₂×D₈

cosocle

C₂×C₈×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #22562079	C₂	semidirect
4. #22562067	C₂	semidirect
10. C₂×Dic₅₃₆	C₄	semidirect
111. C₆₇	#33458	semidirect

H ≅ G/N

type

3. #22562079

C₂

semidirect

4. #22562067

C₂

semidirect

10. C₂×Dic₅₃₆

C₄

semidirect

111. C₆₇

#33458

semidirect

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

subgroup	index
C₂²×Dic₅₃₆	2
#22569076	2
#22562079	2
#22562067	2
#22561928	2
#22561805	2
C₆₇×#2158	2
#33458	67

subgroup

index

C₂²×Dic₅₃₆

C₆₇×#2158

Subgroups — #31009208

Conjugacy classes of subgroups

Complements

Maximal subgroups