Group #22561600 — subgroups

Subgroups — #22561600

Order 8576.

248 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14523976	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523967	C₂	1	1	yes	maximal, characteristic
4	#16863303	C₂	1	1	yes	maximal, characteristic
5	#16863295	C₂	1	1	yes	maximal, characteristic
6	#16863159	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₇	—	2	1	no	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	#14523976	—	2	2	no	—
14	Dic₆₇⋊C₈	—	2	2	no	—
15	#14523967	—	2	2	no	—
16	C₆₇⋊#95	—	2	1	no	—
17	C₆₇⋊#95	—	2	1	no	—
18	C₆₇⋊#95	—	2	1	no	—
19	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	#103×C₆₇	—	2	2	no	—
23	C₂²×Dic₆₇	—	2	1	no	—
24	C₂²×Dic₆₇	—	2	1	no	—
25	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
26	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄×Dic₆₇	—	2	1	no	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₂×Q₈×C₆₇	—	2	1	no	—
38	C₂×Q₈×C₆₇	—	2	1	no	—
39	C₂×Q₈×C₆₇	—	2	1	no	—
40	C₂×Q₈×C₆₇	—	2	1	no	—
41	C₂×Q₈×C₆₇	—	2	1	no	—
42	C₂×Q₈×C₆₇	Q₈	1	2	yes	—
43	C₂²×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
51	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₆₇	—	2	2	no	—
53	Q₈×C₆₇	—	2	2	no	—
54	Q₈×C₆₇	—	4	1	no	—
55	Q₈×C₆₇	—	4	1	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	4	1	no	—
59	C₂×C₄×C₆₇	—	4	1	no	—
60	C₂×C₄×C₆₇	—	4	1	no	—
61	C₂×C₄×C₆₇	—	4	1	no	—
62	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	1	no	—
69	C₂×Dic₆₇	—	2	1	no	—
70	C₂×Dic₆₇	—	2	1	no	—
71	C₂×Dic₆₇	—	2	1	no	—
72	C₂×Dic₆₇	—	2	1	no	—
73	C₂×Dic₆₇	—	4	1	no	—
74	C₂×Dic₆₇	—	4	1	no	—
75	C₂×Dic₆₇	—	4	1	no	—
76	C₂×Dic₆₇	—	4	1	no	—
77	C₂×Dic₆₇	—	4	1	no	—
78	C₂×Dic₆₇	—	4	1	no	—
79	C₂×Dic₆₇	Q₁₆	1	2	yes	—
80	C₈×C₆₇	—	4	1	no	—
81	C₆₇⋊C₈	—	2	2	no	—
82	C₂²×C₆₇	#121	1	1	yes	characteristic
83	C₂²×C₆₇	#121	1	1	yes	characteristic
84	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₆₇	#135	1	1	yes	characteristic
87	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂²×C₆₇	#137	1	1	yes	characteristic
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	—	4	2	no	—
91	C₄×C₆₇	#121	1	1	yes	characteristic
92	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₆₇	#126	1	1	yes	characteristic
95	Dic₆₇	—	2	2	no	—
96	Dic₆₇	—	4	1	no	—
97	Dic₆₇	—	8	1	no	—
98	Dic₆₇	—	8	1	no	—
99	C₂×C₆₇	#396	1	1	yes	characteristic
100	C₂×C₆₇	#462	1	1	yes	characteristic
101	C₂×C₆₇	#463	1	1	yes	characteristic
102	C₂×C₆₇	#467	1	1	yes	characteristic
103	C₂×C₆₇	#468	1	1	yes	characteristic
104	C₂×C₆₇	#477	1	1	yes	characteristic
105	C₂×C₆₇	#479	1	1	yes	characteristic
106	#2068	—	67	1	no	maximal, Sylow
107	C₆₇	#2068	1	1	yes	characteristic, complemented, Sylow
108	C₂×#107	—	67	1	no	—
109	C₂×C₄⋊C₈	—	67	1	no	—
110	C₂×#103	—	67	1	no	—
111	C₂×#103	D₆₇	1	1	yes	characteristic
112	#391	—	67	1	no	—
113	#389	—	67	1	no	—
114	#340	—	67	1	no	—
115	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
116	C₂²×C₈	—	67	1	no	—
117	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	67	1	no	—
119	C₂×C₄⋊C₄	—	67	1	no	—
120	C₂×C₄⋊C₄	—	134	1	no	—
121	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
122	C₂×C₄²	—	67	1	no	—
123	#107	—	134	2	no	—
124	C₄⋊C₈	—	134	2	no	—
125	#103	—	2	2	no	—
126	#103	—	134	2	no	—
127	#95	—	134	1	no	—
128	#95	—	134	1	no	—
129	#95	—	134	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	Dic₁₃₄	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	67	1	no	—
139	C₂²×C₄	—	67	1	no	—
140	C₂²×C₄	—	134	1	no	—
141	C₂²×C₄	—	134	1	no	—
142	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	67	2	no	—
147	C₂×C₈	—	134	1	no	—
148	C₂×C₈	—	134	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	268	2	no	—
151	C₄⋊C₄	—	134	1	no	—
152	C₄⋊C₄	—	134	1	no	—
153	C₄⋊C₄	—	134	1	no	—
154	C₄⋊C₄	—	134	1	no	—
155	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
156	C₄²	—	134	1	no	—
157	C₄²	—	134	1	no	—
158	C₂³	D₄×D₆₇	1	1	yes	characteristic
159	Q₈	—	2	2	no	—
160	Q₈	—	2	2	no	—
161	Q₈	—	4	1	no	—
162	Q₈	—	4	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	268	1	no	—
170	C₂×C₄	—	268	1	no	—
171	C₂×C₄	—	268	1	no	—
172	C₂×C₄	—	268	1	no	—
173	C₂×C₄	—	268	1	no	—
174	C₂×C₄	—	268	1	no	—
175	C₂×C₄	—	67	2	no	—
176	C₂×C₄	—	134	1	no	—
177	C₂×C₄	—	134	1	no	—
178	C₂×C₄	—	134	1	no	—
179	C₂×C₄	—	134	1	no	—
180	C₂×C₄	—	134	1	no	—
181	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
184	C₂×C₄	#11761627	1	1	yes	characteristic
185	C₂×C₄	#11761627	1	1	yes	characteristic
186	C₂×C₄	#11761628	1	1	yes	characteristic
187	C₈	—	4	1	no	—
188	C₈	—	134	2	no	—
189	C₂²	#14524015	1	1	yes	characteristic
190	C₂²	#14524016	1	1	yes	characteristic
191	C₂²	#14524020	1	1	yes	characteristic
192	C₂²	Q₁₆×D₆₇	1	1	yes	characteristic
193	C₂²	#14524064	1	1	yes	characteristic
194	C₂²	#14524070	1	1	yes	characteristic
195	C₂²	#14524075	1	1	yes	characteristic
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	268	1	no	—
199	C₄	—	536	1	no	—
200	C₄	—	536	1	no	—
201	C₄	—	134	2	no	—
202	C₄	#14524014	1	1	yes	characteristic
203	C₄	#14524020	1	1	yes	characteristic
204	C₄	#14524022	1	1	yes	characteristic
205	C₄	#14524036	1	1	yes	characteristic
206	C₂	#16863319	1	1	yes	characteristic
207	C₂	#16863604	1	1	yes	characteristic
208	C₂	#16863613	1	1	yes	characteristic
209	C₂	#16863636	1	1	yes	characteristic
210	C₂	#16863645	1	1	yes	characteristic
211	C₂	#16863689	1	1	yes	characteristic
212	C₂	#16863701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523976

C₂

yes

maximal, characteristic

C₂×Dic₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal, characteristic

#16863303

C₂

yes

maximal, characteristic

#16863295

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

C₂×#103×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523976

—

Dic₆₇⋊C₈

—

#14523967

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×Q₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#126

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#396

yes

characteristic

100

C₂×C₆₇

#462

yes

characteristic

101

C₂×C₆₇

#463

yes

characteristic

102

C₂×C₆₇

#467

yes

characteristic

103

C₂×C₆₇

#468

yes

characteristic

104

C₂×C₆₇

#477

yes

characteristic

105

C₂×C₆₇

yes

106

—

107

C₆₇

#2068

yes

characteristic, complemented, Sylow

108

C₂×#107

—

109

C₂×C₄⋊C₈

—

110

C₂×#103

—

111

C₂×#103

D₆₇

yes

characteristic

112

#391

—

113

#389

—

114

#340

—

115

C₂²×Q₈

C₂×D₆₇

yes

characteristic

116

C₂²×C₈

—

117

C₂²×C₈

C₂×D₆₇

yes

characteristic

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

134

—

121

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

122

C₂×C₄²

—

123

#107

—

134

—

124

C₄⋊C₈

—

134

—

125

#103

—

126

#103

—

134

—

127

#95

—

134

—

128

#95

—

134

—

129

#95

—

134

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

Dic₁₃₄

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

134

—

141

C₂²×C₄

—

134

—

142

C₂²×C₄

C₂²×D₆₇

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

134

—

148

C₂×C₈

—

134

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

268

—

151

C₄⋊C₄

—

134

—

152

C₄⋊C₄

—

134

—

153

C₄⋊C₄

—

134

—

154

C₄⋊C₄

—

134

—

155

C₄⋊C₄

C₂²⋊D₆₇

yes

—

156

C₄²

—

134

—

157

C₄²

—

134

—

158

C₂³

D₄×D₆₇

yes

characteristic

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

268

—

170

C₂×C₄

—

268

—

171

C₂×C₄

—

268

—

172

C₂×C₄

—

268

—

173

C₂×C₄

—

268

—

174

C₂×C₄

—

268

—

175

C₂×C₄

—

176

C₂×C₄

—

134

—

177

C₂×C₄

—

134

—

178

C₂×C₄

—

134

—

179

C₂×C₄

—

134

—

180

C₂×C₄

—

134

—

181

C₂×C₄

D₄×D₆₇

yes

characteristic

182

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

183

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

184

C₂×C₄

#11761627

yes

characteristic

185

C₂×C₄

#11761627

yes

characteristic

186

C₂×C₄

#11761628

yes

characteristic

187

C₈

—

188

C₈

—

134

—

189

C₂²

#14524015

yes

characteristic

190

C₂²

#14524016

yes

characteristic

191

C₂²

#14524020

yes

characteristic

192

C₂²

Q₁₆×D₆₇

yes

characteristic

193

C₂²

#14524064

yes

characteristic

194

C₂²

#14524070

yes

characteristic

195

C₂²

#14524075

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

268

—

199

C₄

—

536

—

200

C₄

—

536

—

201

C₄

—

134

—

202

C₄

#14524014

yes

characteristic

203

C₄

#14524020

yes

characteristic

204

C₄

#14524022

yes

characteristic

205

C₄

#14524036

yes

characteristic

206

C₂

#16863319

yes

characteristic

207

C₂

#16863604

yes

characteristic

208

C₂

#16863613

yes

characteristic

209

C₂

#16863636

yes

characteristic

210

C₂

#16863645

yes

characteristic

211

C₂

#16863689

yes

characteristic

212

C₂

#16863701

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#103×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#103	D₆₇
hypocenter	C₆₇	#2068
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#103×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
107. C₆₇	#2068	semidirect

H ≅ G/N

type

107. C₆₇

#2068

semidirect

Subgroups — #22561600

Conjugacy classes of subgroups

Complements