Group #30869527 — subgroups

Subgroups — #30869527

Order 17152.

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

normal subgroups only

This group has 586 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	#22570733	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	2	yes	maximal
3	#22561190	C₂	1	2	yes	maximal
4	#22560809	C₂	1	8	yes	maximal
5	#22560350	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₂²	1	2	yes	—
8	#16863805	C₄	1	4	yes	complemented
9	#16863805	C₄	1	4	yes	complemented
10	#16863526	—	2	2	no	—
11	#16863526	—	2	2	no	—
12	C₈×Dic₁₃₄	—	2	2	no	—
13	C₈×Dic₁₃₄	—	2	2	no	—
14	#16863434	C₂²	1	4	yes	—
15	#16863422	C₂²	1	4	yes	—
16	#16863420	C₂²	1	2	yes	—
17	#16863351	C₂²	1	4	yes	—
18	#16863161	C₂²	1	4	yes	—
19	#16863156	C₂²	1	8	yes	—
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	C₂²	1	1	yes	characteristic
23	#336×C₆₇	C₂²	1	4	yes	—
24	C₂×#11761612	C₂³	1	4	yes	—
25	#14524043	C₂×C₄	1	2	yes	—
26	#14524043	C₂×C₄	1	2	yes	—
27	#14524035	C₂×C₄	1	4	yes	—
28	#14524022	—	2	2	no	—
29	#14524022	—	2	2	no	—
30	C₄×Dic₁₃₄	—	2	2	no	—
31	C₄×Dic₁₃₄	—	2	2	no	—
32	#14524002	C₂³	1	4	yes	—
33	#14523977	C₂×C₄	1	8	yes	—
34	#14523975	C₂×C₄	1	4	yes	—
35	#14523975	C₂×C₄	1	4	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	Dic₆₇⋊C₈	—	2	4	no	—
39	Dic₆₇⋊C₈	—	2	4	no	—
40	#14523953	—	2	4	no	—
41	C₈×Dic₆₇	—	2	4	no	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	#97×C₆₇	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	#95×C₆₇	—	2	2	no	—
51	C₂×Dic₁₃₄	—	2	2	no	—
52	C₂×Dic₁₃₄	—	2	2	no	—
53	#11761612	—	2	8	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₄⋊Dic₆₇	—	2	2	no	—
56	C₄⋊Dic₆₇	—	2	2	no	—
57	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
58	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
59	Dic₆₇⋊C₄	—	2	4	no	—
60	Dic₆₇⋊C₄	—	2	4	no	—
61	C₂²⋊Dic₆₇	—	2	4	no	—
62	C₄×Dic₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	#41	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
84	C₂×Dic₆₇	—	2	4	no	—
85	C₂×Dic₆₇	—	2	4	no	—
86	Dic₁₃₄	—	4	4	no	—
87	Dic₁₃₄	—	4	4	no	—
88	C₈×C₆₇	—	2	4	no	—
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₆₇⋊C₈	—	4	4	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#541	1	2	yes	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	8	2	no	—
98	C₄×C₆₇	#519	1	2	yes	—
99	C₄×C₆₇	#536	1	2	yes	—
100	Dic₆₇	—	4	4	no	—
101	Dic₆₇	—	4	4	no	—
102	#22257	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	#2327	1	1	yes	characteristic
105	C₂×C₆₇	#2955	1	2	yes	—
106	#3162	—	67	1	no	—
107	#2960	—	67	2	no	—
108	#1995	—	67	2	no	—
109	#1922	—	67	8	no	—
110	#1838	—	67	1	no	—
111	#1746	D₆₇	1	1	yes	characteristic
112	C₆₇	#22257	1	1	yes	characteristic, complemented, Sylow
113	#533	—	67	2	no	—
114	#501	—	67	4	no	—
115	#501	—	67	4	no	—
116	#446	—	134	2	no	—
117	#446	—	134	2	no	—
118	C₈×Q₈	—	134	2	no	—
119	C₈×Q₈	—	134	2	no	—
120	#427	—	67	4	no	—
121	#424	—	67	4	no	—
122	#423	—	67	2	no	—
123	#405	—	67	4	no	—
124	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
126	#376	C₂×D₆₇	1	1	yes	characteristic
127	#341	—	67	4	no	—
128	#339	—	67	8	no	—
129	#336	C₂×D₆₇	1	4	yes	—
130	C₂×M₁₆	—	67	4	no	—
131	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
133	C₄⋊Q₈	—	67	2	no	—
134	C₄⋊Q₈	—	67	2	no	—
135	#125	—	67	4	no	—
136	C₂²⋊Q₈	—	134	2	no	—
137	C₂²⋊Q₈	—	134	2	no	—
138	C₄×Q₈	—	134	2	no	—
139	C₄×Q₈	—	134	2	no	—
140	#117	—	67	4	no	—
141	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
142	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
143	#107	—	67	8	no	—
144	#106	—	67	4	no	—
145	#106	—	67	4	no	—
146	C₄⋊C₈	—	134	4	no	—
147	C₄⋊C₈	—	134	4	no	—
148	C₄⋊C₈	—	134	2	no	—
149	C₄⋊C₈	—	134	2	no	—
150	#97	—	134	4	no	—
151	#97	C₄×D₆₇	1	4	yes	—
152	C₄×C₈	—	134	4	no	—
153	C₄×C₈	D₂₆₈	1	1	yes	characteristic
154	C₄×C₈	D₂₆₈	1	2	yes	—
155	C₄×C₈	D₂₆₈	1	1	yes	characteristic
156	#95	—	2	2	no	—
157	C₂×Q₈	—	134	2	no	—
158	C₂×Q₈	—	134	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	#11761628	1	2	yes	—
161	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
162	M₁₆	—	134	8	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	134	4	no	—
166	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
167	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
168	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
169	C₄⋊C₄	—	67	4	no	—
170	C₄⋊C₄	—	67	4	no	—
171	C₄⋊C₄	—	134	4	no	—
172	C₄⋊C₄	—	134	4	no	—
173	C₄⋊C₄	—	134	2	no	—
174	C₄⋊C₄	—	134	2	no	—
175	C₂²⋊C₄	—	134	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	134	4	no	—
178	C₄²	C₂×C₄×D₆₇	1	2	yes	—
179	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
180	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
181	C₂³	C₂×#11761628	1	1	yes	characteristic
182	Q₈	—	268	4	no	—
183	Q₈	—	268	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	134	4	no	—
188	C₂×C₄	—	134	4	no	—
189	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
190	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
191	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
193	C₂×C₄	#14524090	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	268	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
200	C₂²	#16864008	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	268	4	no	—
205	C₄	—	268	4	no	—
206	C₄	#16863873	1	2	yes	—
207	C₄	#16863972	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	#22562928	1	1	yes	characteristic
210	C₂	#22569113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570733

C₂

yes

maximal, characteristic

#22569143

C₂

yes

maximal

#22561190

C₂

yes

maximal

#22560809

C₂

yes

maximal

#22560350

C₂

yes

maximal, characteristic

C₆₇×#1746

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863805

C₄

yes

complemented

#16863805

C₄

yes

complemented

#16863526

—

#16863526

—

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

—

#16863434

C₂²

yes

—

#16863422

C₂²

yes

—

#16863420

C₂²

yes

—

#16863351

C₂²

yes

—

#16863161

C₂²

yes

—

#16863156

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₂²

yes

—

C₂×#11761612

C₂³

yes

—

#14524043

C₂×C₄

yes

—

#14524043

C₂×C₄

yes

—

#14524035

C₂×C₄

yes

—

#14524022

—

#14524022

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

#14524002

C₂³

yes

—

#14523977

C₂×C₄

yes

—

#14523975

C₂×C₄

yes

—

#14523975

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

Dic₆₇⋊C₈

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#97×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

#95×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂²×C₄

yes

—

C₄⋊Dic₆₇

C₂²×C₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

C₂×#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#541

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#519

yes

—

C₄×C₆₇

#536

yes

—

100

Dic₆₇

—

101

Dic₆₇

—

102

#22257

—

maximal, Sylow

103

C₂×C₆₇

—

104

C₂×C₆₇

#2327

yes

characteristic

105

C₂×C₆₇

#2955

yes

—

106

#3162

—

107

#2960

—

108

#1995

—

109

#1922

—

110

#1838

—

111

#1746

D₆₇

yes

characteristic

112

C₆₇

#22257

yes

characteristic, complemented, Sylow

113

#533

—

114

#501

—

115

#501

—

116

#446

—

134

—

117

#446

—

134

—

118

C₈×Q₈

—

134

—

119

C₈×Q₈

—

134

—

120

#427

—

121

#424

—

122

#423

—

123

#405

—

124

C₂×#97

C₂×D₆₇

yes

characteristic

125

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

126

#376

C₂×D₆₇

yes

characteristic

127

#341

—

128

#339

—

129

#336

C₂×D₆₇

yes

—

130

C₂×M₁₆

—

131

C₂²×C₈

C₂²×D₆₇

yes

—

132

C₂²×C₈

C₂²×D₆₇

yes

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

#125

—

136

C₂²⋊Q₈

—

134

—

137

C₂²⋊Q₈

—

134

—

138

C₄×Q₈

—

134

—

139

C₄×Q₈

—

134

—

140

#117

—

141

C₂×C₄²

C₂²×D₆₇

yes

—

142

C₂×C₄²

C₂²×D₆₇

yes

characteristic

143

#107

—

144

#106

—

145

#106

—

146

C₄⋊C₈

—

134

—

147

C₄⋊C₈

—

134

—

148

C₄⋊C₈

—

134

—

149

C₄⋊C₈

—

134

—

150

#97

—

134

—

151

#97

C₄×D₆₇

yes

—

152

C₄×C₈

—

134

—

153

C₄×C₈

D₂₆₈

yes

characteristic

154

C₄×C₈

D₂₆₈

yes

—

155

C₄×C₈

D₂₆₈

yes

characteristic

156

#95

—

157

C₂×Q₈

—

134

—

158

C₂×Q₈

—

134

—

159

C₂²×C₄

—

160

C₂²×C₄

#11761628

yes

—

161

C₂²×C₄

C₂³×D₆₇

yes

characteristic

162

M₁₆

—

134

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

134

—

166

C₂×C₈

C₂×C₄×D₆₇

yes

—

167

C₂×C₈

C₂×D₂₆₈

yes

—

168

C₂×C₈

C₂×D₂₆₈

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

134

—

172

C₄⋊C₄

—

134

—

173

C₄⋊C₄

—

134

—

174

C₄⋊C₄

—

134

—

175

C₂²⋊C₄

—

134

—

176

C₄²

—

177

C₄²

—

134

—

178

C₄²

C₂×C₄×D₆₇

yes

—

179

C₄²

C₂×D₂₆₈

yes

characteristic

180

C₄²

C₂×D₂₆₈

yes

characteristic

181

C₂³

C₂×#11761628

yes

characteristic

182

Q₈

—

268

—

183

Q₈

—

268

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

134

—

188

C₂×C₄

—

134

—

189

C₂×C₄

C₄×D₂₆₈

yes

—

190

C₂×C₄

C₄×D₂₆₈

yes

—

191

C₂×C₄

C₂²×C₄×D₆₇

yes

characteristic

192

C₂×C₄

C₂²×D₂₆₈

yes

characteristic

193

C₂×C₄

#14524090

yes

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

268

—

198

C₂²

—

199

C₂²

C₂×C₄×D₂₆₈

yes

characteristic

200

C₂²

#16864008

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

268

—

205

C₄

—

268

—

206

C₄

#16863873

yes

—

207

C₄

#16863972

yes

—

208

C₂

—

209

C₂

#22562928

yes

characteristic

210

C₂

#22569113

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1746	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1746	D₆₇
hypocenter	C₆₇	#22257
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1746

subgroup

G/H

center

C₂×C₄

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#1746

C₂

socle

C₂²×C₆₇

C₂×C₄×D₄

prime radical

C₂³×C₆₇

C₂×#41

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1746

N	H ≅ G/N	type
8. #16863805	C₄	semidirect
9. #16863805	C₄	semidirect
112. C₆₇	#22257	semidirect

H ≅ G/N

type

8. #16863805

C₄

semidirect

9. #16863805

C₄

semidirect

112. C₆₇

#22257

semidirect

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

subgroup	index
#22570733	2
#22569143	2
#22561190	2
#22560809	2
#22560350	2
C₆₇×#1746	2
#22257	67

subgroup

index

C₆₇×#1746

Subgroups — #30869527

Conjugacy classes of subgroups

Complements

Maximal subgroups