Group #22561548 — subgroups

Subgroups — #22561548

Order 8576.

250 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 91 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×#14524035	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523977	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523975	C₂	1	1	yes	maximal, characteristic
4	#16863294	C₂	1	1	yes	maximal, characteristic
5	#16863159	C₂	1	1	yes	maximal, characteristic
6	#16863159	C₂	1	1	yes	maximal, characteristic
7	#336×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524035	—	2	2	no	—
9	#14524035	—	2	2	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	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₇⋊C₄	—	2	1	no	—
15	C₂×Dic₆₇⋊C₄	—	2	1	no	—
16	C₂×Dic₆₇⋊C₄	—	2	1	no	—
17	#14523977	—	2	2	no	—
18	#14523975	—	2	2	no	—
19	C₆₇⋊#95	—	2	1	no	—
20	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₆₇	—	2	1	no	—
24	C₂²×Dic₆₇	—	2	1	no	—
25	C₂²×Dic₆₇	—	2	1	no	—
26	C₂²×Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	D₄	1	2	yes	—
33	C₄⋊Dic₆₇	D₄	1	2	yes	—
34	C₄⋊Dic₆₇	Q₈	1	2	yes	—
35	Dic₆₇⋊C₄	—	2	4	no	—
36	Dic₆₇⋊C₄	—	2	4	no	—
37	Dic₆₇⋊C₄	—	4	2	no	—
38	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	Q₈	1	2	yes	—
46	C₂×C₈×C₆₇	Q₈	1	2	yes	—
47	C₄²×C₆₇	—	2	1	no	—
48	C₄²×C₆₇	—	2	1	no	—
49	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	2	1	no	—
55	C₂×C₄×C₆₇	—	2	1	no	—
56	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
62	C₂×Dic₆₇	—	2	2	no	—
63	C₂×Dic₆₇	—	2	2	no	—
64	C₂×Dic₆₇	—	2	2	no	—
65	C₂×Dic₆₇	—	4	1	no	—
66	C₂×Dic₆₇	—	4	1	no	—
67	C₂×Dic₆₇	—	4	1	no	—
68	C₂×Dic₆₇	—	4	1	no	—
69	C₂×Dic₆₇	—	4	1	no	—
70	C₂×Dic₆₇	—	4	1	no	—
71	C₂×Dic₆₇	—	4	1	no	—
72	C₂×Dic₆₇	—	4	1	no	—
73	C₂×Dic₆₇	—	4	1	no	—
74	C₈×C₆₇	—	2	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₂²×C₆₇	#120	1	1	yes	characteristic
77	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₆₇	#135	1	1	yes	characteristic
80	C₂²×C₆₇	#135	1	1	yes	characteristic
81	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²×C₆₇	#137	1	1	yes	characteristic
83	C₄×C₆₇	—	4	1	no	—
84	C₄×C₆₇	—	4	1	no	—
85	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
89	Dic₆₇	—	4	2	no	—
90	Dic₆₇	—	4	2	no	—
91	Dic₆₇	—	4	2	no	—
92	Dic₆₇	—	8	1	no	—
93	C₂×C₆₇	#395	1	1	yes	characteristic
94	C₂×C₆₇	#449	1	1	yes	characteristic
95	C₂×C₆₇	#452	1	1	yes	characteristic
96	C₂×C₆₇	#478	1	1	yes	characteristic
97	C₂×C₆₇	#479	1	1	yes	characteristic
98	C₂×C₆₇	#499	1	1	yes	characteristic
99	C₂×C₆₇	#501	1	1	yes	characteristic
100	#2059	—	67	1	no	maximal, Sylow
101	C₆₇	#2059	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	67	1	no	—
103	C₂×#107	—	67	1	no	—
104	C₂×#106	—	67	1	no	—
105	#389	—	67	1	no	—
106	#340	—	67	1	no	—
107	#340	—	67	1	no	—
108	#336	D₆₇	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
111	#125	—	134	2	no	—
112	#125	—	134	2	no	—
113	C₂×C₄⋊C₄	—	67	1	no	—
114	C₂×C₄⋊C₄	—	67	1	no	—
115	C₂×C₄⋊C₄	—	67	1	no	—
116	C₂×C₄⋊C₄	—	67	1	no	—
117	C₂×C₄⋊C₄	—	134	1	no	—
118	C₂×C₄⋊C₄	—	134	1	no	—
119	C₂×C₄⋊C₄	—	134	1	no	—
120	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
121	#107	—	134	2	no	—
122	#106	—	134	2	no	—
123	#95	—	134	1	no	—
124	C₂²×C₄	—	134	1	no	—
125	C₂²×C₄	—	134	1	no	—
126	C₂²×C₄	—	134	1	no	—
127	C₂²×C₄	—	134	1	no	—
128	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
129	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	Dic₁₃₄	1	2	yes	—
136	C₂×C₈	Dic₁₃₄	1	2	yes	—
137	C₄⋊C₄	—	268	2	no	—
138	C₄⋊C₄	—	67	2	no	—
139	C₄⋊C₄	—	67	2	no	—
140	C₄⋊C₄	—	67	2	no	—
141	C₄⋊C₄	—	134	4	no	—
142	C₄⋊C₄	—	134	4	no	—
143	C₄⋊C₄	—	134	1	no	—
144	C₄⋊C₄	—	134	1	no	—
145	C₄⋊C₄	—	134	1	no	—
146	C₄⋊C₄	—	134	1	no	—
147	C₄⋊C₄	—	134	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	268	1	no	—
158	C₂×C₄	—	268	1	no	—
159	C₂×C₄	—	268	1	no	—
160	C₂×C₄	—	268	1	no	—
161	C₂×C₄	—	268	1	no	—
162	C₂×C₄	—	268	1	no	—
163	C₂×C₄	—	268	1	no	—
164	C₂×C₄	—	268	1	no	—
165	C₂×C₄	—	268	1	no	—
166	C₂×C₄	—	134	2	no	—
167	C₂×C₄	—	134	2	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
174	C₂×C₄	#11761627	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₂²	#14524013	1	1	yes	characteristic
178	C₂²	#14524023	1	1	yes	characteristic
179	C₂²	#14524043	1	1	yes	characteristic
180	C₂²	#14524065	1	1	yes	characteristic
181	C₂²	#14524065	1	1	yes	characteristic
182	C₂²	#14524071	1	1	yes	characteristic
183	C₂²	#14524076	1	1	yes	characteristic
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	268	2	no	—
187	C₄	—	268	2	no	—
188	C₄	—	268	2	no	—
189	C₄	—	536	1	no	—
190	C₄	#14524020	1	1	yes	characteristic
191	C₄	#14524020	1	1	yes	characteristic
192	C₄	#14524042	1	1	yes	characteristic
193	C₄	#14524042	1	1	yes	characteristic
194	C₂	#16863317	1	1	yes	characteristic
195	C₂	#16863543	1	1	yes	characteristic
196	C₂	#16863560	1	1	yes	characteristic
197	C₂	#16863697	1	1	yes	characteristic
198	C₂	#16863704	1	1	yes	characteristic
199	C₂	#16863798	1	1	yes	characteristic
200	C₂	#16863805	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

C₂×#14523977

C₂

yes

maximal, characteristic

C₂×#14523975

C₂

yes

maximal, characteristic

#16863294

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#336×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

#14524035

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×Dic₆₇⋊C₄

—

C₂×Dic₆₇⋊C₄

—

#14523977

—

#14523975

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

Q₈

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

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

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

100

—

101

C₆₇

#2059

yes

characteristic, complemented, Sylow

102

C₂×#125

—

103

C₂×#107

—

104

C₂×#106

—

105

#389

—

106

#340

—

107

#340

—

108

#336

D₆₇

yes

characteristic

109

C₂²×C₈

C₂×D₆₇

yes

characteristic

110

C₂²×C₈

C₂×D₆₇

yes

characteristic

111

#125

—

134

—

112

#125

—

134

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

134

—

118

C₂×C₄⋊C₄

—

134

—

119

C₂×C₄⋊C₄

—

134

—

120

C₂×C₄²

C₂×D₆₇

yes

characteristic

121

#107

—

134

—

122

#106

—

134

—

123

#95

—

134

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

—

134

—

126

C₂²×C₄

—

134

—

127

C₂²×C₄

—

134

—

128

C₂²×C₄

D₂₆₈

yes

characteristic

129

C₂²×C₄

D₂₆₈

yes

characteristic

130

C₂²×C₄

C₂²×D₆₇

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

Dic₁₃₄

yes

—

136

C₂×C₈

Dic₁₃₄

yes

—

137

C₄⋊C₄

—

268

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

134

—

142

C₄⋊C₄

—

134

—

143

C₄⋊C₄

—

134

—

144

C₄⋊C₄

—

134

—

145

C₄⋊C₄

—

134

—

146

C₄⋊C₄

—

134

—

147

C₄⋊C₄

—

134

—

148

C₄²

—

149

C₄²

—

150

C₂³

C₂×D₂₆₈

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

268

—

158

C₂×C₄

—

268

—

159

C₂×C₄

—

268

—

160

C₂×C₄

—

268

—

161

C₂×C₄

—

268

—

162

C₂×C₄

—

268

—

163

C₂×C₄

—

268

—

164

C₂×C₄

—

268

—

165

C₂×C₄

—

268

—

166

C₂×C₄

—

134

—

167

C₂×C₄

—

134

—

168

C₂×C₄

—

134

—

169

C₂×C₄

D₄×D₆₇

yes

characteristic

170

C₂×C₄

D₄×D₆₇

yes

characteristic

171

C₂×C₄

Q₈×D₆₇

yes

characteristic

172

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

173

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

174

C₂×C₄

#11761627

yes

characteristic

175

C₈

—

176

C₈

—

177

C₂²

#14524013

yes

characteristic

178

C₂²

#14524023

yes

characteristic

179

C₂²

#14524043

yes

characteristic

180

C₂²

#14524065

yes

characteristic

181

C₂²

#14524065

yes

characteristic

182

C₂²

#14524071

yes

characteristic

183

C₂²

#14524076

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

268

—

187

C₄

—

268

—

188

C₄

—

268

—

189

C₄

—

536

—

190

C₄

#14524020

yes

characteristic

191

C₄

#14524020

yes

characteristic

192

C₄

#14524042

yes

characteristic

193

C₄

#14524042

yes

characteristic

194

C₂

#16863317

yes

characteristic

195

C₂

#16863543

yes

characteristic

196

C₂

#16863560

yes

characteristic

197

C₂

#16863697

yes

characteristic

198

C₂

#16863704

yes

characteristic

199

C₂

#16863798

yes

characteristic

200

C₂

#16863805

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#2059
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
101. C₆₇	#2059	semidirect

H ≅ G/N

type

101. C₆₇

#2059

semidirect

Subgroups — #22561548

Conjugacy classes of subgroups

Complements