Group #22571610 — subgroups

Subgroups — #22571610

Order 8576.

354 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16863930	C₂	1	2	yes	maximal, complemented
2	#16863704	C₂	1	2	yes	maximal, complemented
3	#16863697	C₂	1	2	yes	maximal, complemented
4	#16863654	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863649	C₂	1	1	yes	maximal, characteristic
6	#16863642	C₂	1	2	yes	maximal, complemented
7	#16863619	C₂	1	2	yes	maximal, complemented
8	#16863612	C₂	1	2	yes	maximal, complemented
9	#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
11	C₂×#11761617	C₂²	1	2	yes	—
12	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
13	#14524037	—	2	2	no	—
14	#14524035	C₂²	1	2	yes	—
15	#14524031	C₂²	1	2	yes	—
16	#14524026	—	2	2	no	—
17	#14524022	C₂²	1	2	yes	—
18	#14524017	C₂²	1	2	yes	—
19	C₄×Dic₁₃₄	C₂²	1	2	yes	—
20	C₄×D₂₆₈	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₆₇	—	2	2	no	—
22	#14524000	—	2	2	no	—
23	#14523977	C₂²	1	2	yes	—
24	#14523975	C₂²	1	1	yes	characteristic
25	#14523975	C₂²	1	1	yes	characteristic
26	#14523968	C₂²	1	2	yes	—
27	#14523968	C₂²	1	2	yes	—
28	#14523965	C₂²	1	2	yes	—
29	#14523965	C₂²	1	2	yes	—
30	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×Dic₁₃₄	C₂³	1	2	yes	—
36	C₂×D₂₆₈	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₆₇	—	2	2	no	—
38	C₂×C₄×D₆₇	—	2	2	no	—
39	Dic₂₆₈	—	4	1	no	—
40	#11761617	—	4	2	no	—
41	D₅₃₆	—	4	1	no	—
42	C₄⋊Dic₆₇	C₂³	1	2	yes	—
43	C₄⋊Dic₆₇	C₂³	1	2	yes	—
44	Dic₆₇⋊C₄	—	2	2	no	—
45	Dic₆₇⋊C₄	—	2	2	no	—
46	Dic₆₇⋊C₄	—	2	2	no	—
47	Dic₆₇⋊C₄	—	2	2	no	—
48	C₂²⋊Dic₆₇	—	2	2	no	—
49	(C₂×D₆₇)⋊C₄	—	2	2	no	—
50	(C₂×D₆₇)⋊C₄	—	2	2	no	—
51	(C₂×D₆₇)⋊C₄	—	2	2	no	—
52	(C₂×D₆₇)⋊C₄	—	2	2	no	—
53	C₄×Dic₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	D₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	M₁₆×C₆₇	—	4	1	no	—
57	M₁₆×C₆₇	—	4	1	no	—
58	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄²×C₆₇	D₄	1	1	yes	characteristic
62	C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₆₇	—	4	4	no	—
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	2	2	no	—
76	C₂×Dic₆₇	—	2	2	no	—
77	D₂₆₈	—	2	2	no	—
78	D₂₆₈	—	2	2	no	—
79	C₄×D₆₇	—	4	4	no	—
80	Dic₁₃₄	—	2	2	no	—
81	Dic₁₃₄	—	2	2	no	—
82	C₈×C₆₇	—	2	1	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₆₇	—	4	2	no	—
89	C₂×D₆₇	—	4	2	no	—
90	C₂×D₆₇	—	4	2	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₆₇	—	4	2	no	—
97	Dic₆₇	—	4	2	no	—
98	Dic₆₇	—	4	2	no	—
99	C₂×C₆₇	—	4	1	no	—
100	C₂×C₆₇	#537	1	1	yes	characteristic
101	C₂×C₆₇	#575	1	2	yes	—
102	D₆₇	—	8	2	no	—
103	#3256	—	67	1	no	maximal, Sylow
104	C₆₇	#3256	1	1	yes	characteristic, complemented, Sylow
105	#529	—	67	2	no	—
106	#479	—	67	2	no	—
107	#478	—	67	2	no	—
108	#470	—	67	1	no	—
109	#469	—	67	1	no	—
110	#468	—	67	2	no	—
111	#464	—	67	2	no	—
112	#463	—	67	2	no	—
113	#423	D₆₇	1	1	yes	characteristic, complemented
114	C₂×Q₁₆	—	67	1	no	—
115	C₂×SD₁₆	—	67	2	no	—
116	C₂×D₈	—	67	1	no	—
117	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
118	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
119	#126	—	134	2	no	—
120	#125	—	67	2	no	—
121	#124	—	67	2	no	—
122	#123	—	134	2	no	—
123	C₂²⋊Q₈	—	67	2	no	—
124	#121	—	67	2	no	—
125	C₄×Q₈	—	67	2	no	—
126	C₄×D₄	—	67	2	no	—
127	C₄×D₄	—	134	2	no	—
128	#117	—	134	2	no	—
129	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
130	#107	—	67	2	no	—
131	#106	—	67	1	no	—
132	#106	—	67	1	no	—
133	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
134	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
135	#103	—	67	2	no	—
136	#103	—	67	2	no	—
137	#102	—	67	2	no	—
138	#102	—	67	2	no	—
139	C₂×Q₈	—	67	2	no	—
140	C₂×D₄	—	67	2	no	—
141	C₂×D₄	—	134	2	no	—
142	C₂²×C₄	—	134	2	no	—
143	C₂²×C₄	D₂₆₈	1	2	yes	—
144	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
145	Q₁₆	—	268	1	no	—
146	SD₁₆	—	268	2	no	—
147	D₈	—	268	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	4	1	no	—
150	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₆₇	1	2	yes	—
152	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	C₄⋊C₄	—	67	2	no	—
154	C₄⋊C₄	—	67	2	no	—
155	C₄⋊C₄	—	134	2	no	—
156	C₄⋊C₄	—	134	2	no	—
157	C₄⋊C₄	—	134	2	no	—
158	C₄⋊C₄	—	134	2	no	—
159	C₂²⋊C₄	—	134	2	no	—
160	C₂²⋊C₄	—	134	2	no	—
161	C₂²⋊C₄	—	134	2	no	—
162	C₂²⋊C₄	—	134	2	no	—
163	C₂²⋊C₄	—	134	2	no	—
164	C₄²	—	134	2	no	—
165	C₄²	D₂₆₈	1	1	yes	characteristic
166	C₄²	D₂₆₈	1	1	yes	characteristic
167	C₄²	C₂²×D₆₇	1	2	yes	—
168	C₂³	—	134	2	no	—
169	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
170	Q₈	—	134	2	no	—
171	Q₈	—	134	2	no	—
172	D₄	—	268	4	no	—
173	D₄	—	134	2	no	—
174	D₄	—	134	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	268	4	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	—	134	2	no	—
182	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
183	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
184	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
185	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	268	2	no	—
192	C₂²	—	268	2	no	—
193	C₂²	—	268	2	no	—
194	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	268	2	no	—
199	C₄	—	268	2	no	—
200	C₄	—	268	2	no	—
201	C₄	#14524088	1	1	yes	characteristic
202	C₄	#14524091	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	536	2	no	—
205	C₂	#16863977	1	1	yes	characteristic
206	C₂	#16864196	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863930

C₂

yes

maximal, complemented

#16863704

C₂

yes

maximal, complemented

#16863697

C₂

yes

maximal, complemented

#16863654

C₂

yes

maximal, characteristic, complemented

#16863649

C₂

yes

maximal, characteristic

#16863642

C₂

yes

maximal, complemented

#16863619

C₂

yes

maximal, complemented

#16863612

C₂

yes

maximal, complemented

#423×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761617

C₂²

yes

—

C₂×D₅₃₆

C₂²

yes

characteristic

#14524037

—

#14524035

C₂²

yes

—

#14524031

C₂²

yes

—

#14524026

—

#14524022

C₂²

yes

—

#14524017

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×D₂₆₈

C₂²

yes

—

C₄×C₂²⋊D₆₇

—

#14524000

—

#14523977

C₂²

yes

—

#14523975

C₂²

yes

characteristic

#14523975

C₂²

yes

characteristic

#14523968

C₂²

yes

—

#14523968

C₂²

yes

—

#14523965

C₂²

yes

—

#14523965

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

Dic₂₆₈

—

#11761617

—

D₅₃₆

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

#537

yes

characteristic

101

C₂×C₆₇

#575

yes

—

102

D₆₇

—

103

#3256

—

maximal, Sylow

104

C₆₇

#3256

yes

characteristic, complemented, Sylow

105

#529

—

106

#479

—

107

#478

—

108

#470

—

109

#469

—

110

#468

—

111

#464

—

112

#463

—

113

#423

D₆₇

yes

characteristic, complemented

114

C₂×Q₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₆₇

yes

characteristic

118

C₂×M₁₆

C₂×D₆₇

yes

characteristic

119

#126

—

134

—

120

#125

—

121

#124

—

122

#123

—

134

—

123

C₂²⋊Q₈

—

124

#121

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

C₄×D₄

—

134

—

128

#117

—

134

—

129

C₂×C₄²

C₂×D₆₇

yes

characteristic

130

#107

—

131

#106

—

132

#106

—

133

C₄⋊C₈

C₂×D₆₇

yes

—

134

C₄⋊C₈

C₂×D₆₇

yes

—

135

#103

—

136

#103

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

—

134

—

142

C₂²×C₄

—

134

—

143

C₂²×C₄

D₂₆₈

yes

—

144

C₂²×C₄

C₂²×D₆₇

yes

characteristic

145

Q₁₆

—

268

—

146

SD₁₆

—

268

—

147

D₈

—

268

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

C₂²×D₆₇

yes

characteristic

151

C₂×C₈

C₂²×D₆₇

yes

—

152

C₂×C₈

C₂²×D₆₇

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

134

—

156

C₄⋊C₄

—

134

—

157

C₄⋊C₄

—

134

—

158

C₄⋊C₄

—

134

—

159

C₂²⋊C₄

—

134

—

160

C₂²⋊C₄

—

134

—

161

C₂²⋊C₄

—

134

—

162

C₂²⋊C₄

—

134

—

163

C₂²⋊C₄

—

134

—

164

C₄²

—

134

—

165

C₄²

D₂₆₈

yes

characteristic

166

C₄²

D₂₆₈

yes

characteristic

167

C₄²

C₂²×D₆₇

yes

—

168

C₂³

—

134

—

169

C₂³

C₂×D₂₆₈

yes

characteristic

170

Q₈

—

134

—

171

Q₈

—

134

—

172

D₄

—

268

—

173

D₄

—

134

—

174

D₄

—

134

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

134

—

180

C₂×C₄

—

134

—

181

C₂×C₄

—

134

—

182

C₂×C₄

C₂×D₂₆₈

yes

—

183

C₂×C₄

C₂×D₂₆₈

yes

characteristic

184

C₂×C₄

C₂×D₂₆₈

yes

—

185

C₂×C₄

C₂³×D₆₇

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

268

—

192

C₂²

—

268

—

193

C₂²

—

268

—

194

C₂²

C₂²×D₂₆₈

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

268

—

199

C₄

—

268

—

200

C₄

—

268

—

201

C₄

#14524088

yes

characteristic

202

C₄

#14524091

yes

characteristic

203

C₂

—

204

C₂

—

536

—

205

C₂

#16863977

yes

characteristic

206

C₂

#16864196

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#423	D₆₇
hypocenter	C₆₇	#3256
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#423×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #16863930	C₂	semidirect
2. #16863704	C₂	semidirect
3. #16863697	C₂	semidirect
4. #16863654	C₂	semidirect
6. #16863642	C₂	semidirect
7. #16863619	C₂	semidirect
8. #16863612	C₂	semidirect
9. #423×C₆₇	C₂	semidirect
104. C₆₇	#3256	semidirect
113. #423	D₆₇	semidirect

H ≅ G/N

type

1. #16863930

C₂

semidirect

2. #16863704

C₂

semidirect

3. #16863697

C₂

semidirect

4. #16863654

C₂

semidirect

6. #16863642

C₂

semidirect

7. #16863619

C₂

semidirect

8. #16863612

C₂

semidirect

9. #423×C₆₇

C₂

semidirect

104. C₆₇

#3256

semidirect

113. #423

D₆₇

semidirect

Subgroups — #22571610

Conjugacy classes of subgroups

Complements