Group #22573290 — subgroups

Subgroups — #22573290

Order 8576.

356 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16864119	C₂	1	2	yes	maximal, complemented
2	#16863790	C₂	1	2	yes	maximal, complemented
3	#16863780	C₂	1	1	yes	maximal, characteristic
4	#16863619	C₂	1	2	yes	maximal, complemented
5	#16863612	C₂	1	2	yes	maximal
6	#16863607	C₂	1	2	yes	maximal, complemented
7	#16863498	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₆₈	C₂²	1	2	yes	—
13	C₂×#11761617	C₂²	1	2	yes	—
14	C₂×#11761617	C₂²	1	2	yes	—
15	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
16	#14524043	C₂²	1	2	yes	—
17	#14524038	—	2	2	no	—
18	#14524038	—	2	2	no	—
19	#14524031	C₂²	1	2	yes	—
20	#14524031	C₂²	1	2	yes	—
21	#14524019	—	2	2	no	—
22	C₄×Dic₁₃₄	C₂²	1	2	yes	—
23	C₄×Dic₁₃₄	C₂²	1	2	yes	—
24	C₄×D₂₆₈	C₂²	1	2	yes	—
25	#14524001	—	2	2	no	—
26	#14523977	C₂²	1	1	yes	characteristic
27	#14523968	C₂²	1	2	yes	—
28	#14523968	C₂²	1	2	yes	—
29	#14523965	C₂²	1	2	yes	—
30	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#97×C₆₇	C₂²	1	2	yes	—
34	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₃₄	C₂³	1	2	yes	—
36	C₂×Dic₁₃₄	C₂³	1	2	yes	—
37	C₂×D₂₆₈	C₂³	1	2	yes	—
38	C₂×C₄×D₆₇	—	2	2	no	—
39	Dic₂₆₈	—	2	2	no	—
40	Dic₂₆₈	—	4	1	no	—
41	Dic₂₆₈	—	4	2	no	—
42	#11761617	—	2	4	no	—
43	#11761617	—	4	2	no	—
44	D₅₃₆	—	4	1	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₄⋊Dic₆₇	C₂³	1	2	yes	—
48	Dic₆₇⋊C₄	—	2	2	no	—
49	Dic₆₇⋊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	(C₂×D₆₇)⋊C₄	—	2	2	no	—
54	(C₂×D₆₇)⋊C₄	—	2	2	no	—
55	C₄×Dic₆₇	—	2	2	no	—
56	C₄×Dic₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
62	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂³	1	2	yes	—
64	C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₆₇	—	2	2	no	—
66	Q₈×C₆₇	—	4	1	no	—
67	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₆₇	—	2	2	no	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	—	2	2	no	—
75	D₂₆₈	—	2	2	no	—
76	D₂₆₈	—	2	2	no	—
77	C₄×D₆₇	—	4	4	no	—
78	Dic₁₃₄	—	2	2	no	—
79	Dic₁₃₄	—	2	2	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	#41	1	2	yes	—
85	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₆₇	—	4	2	no	—
87	C₂×D₆₇	—	4	2	no	—
88	C₂×D₆₇	—	4	2	no	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
94	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₆₇	—	4	2	no	—
96	Dic₆₇	—	4	2	no	—
97	Dic₆₇	—	4	2	no	—
98	C₂×C₆₇	#550	1	1	yes	characteristic
99	C₂×C₆₇	#575	1	1	yes	characteristic
100	C₂×C₆₇	#578	1	1	yes	characteristic
101	D₆₇	—	8	2	no	—
102	#3402	—	67	1	no	maximal, Sylow
103	C₆₇	#3402	1	1	yes	characteristic, complemented, Sylow
104	#556	—	67	2	no	—
105	#497	—	67	2	no	—
106	#495	—	67	1	no	—
107	#464	—	67	2	no	—
108	#463	—	67	2	no	—
109	#462	—	67	2	no	—
110	#446	D₆₇	1	1	yes	characteristic, complemented
111	#440	—	67	2	no	—
112	C₄×Q₁₆	—	67	1	no	—
113	C₂×Q₁₆	—	67	1	no	—
114	C₂×Q₁₆	—	67	2	no	—
115	C₂×Q₁₆	—	67	1	no	—
116	C₂×SD₁₆	—	67	2	no	—
117	C₂×SD₁₆	—	67	2	no	—
118	C₂×D₈	—	67	1	no	—
119	C₄⋊Q₈	—	67	2	no	—
120	#126	—	134	2	no	—
121	#126	—	134	2	no	—
122	#124	—	67	2	no	—
123	#124	—	67	2	no	—
124	C₂²⋊Q₈	—	134	2	no	—
125	C₄×Q₈	—	67	2	no	—
126	C₄×Q₈	—	67	2	no	—
127	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
128	C₄×D₄	—	67	2	no	—
129	#117	—	134	2	no	—
130	#107	—	67	1	no	—
131	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
132	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
133	#103	—	67	2	no	—
134	#103	—	67	2	no	—
135	#102	—	67	2	no	—
136	#97	C₂×D₆₇	1	2	yes	—
137	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×Q₈	—	67	2	no	—
139	C₂×Q₈	—	67	2	no	—
140	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
141	C₂×D₄	—	67	2	no	—
142	C₂²×C₄	—	134	2	no	—
143	Q₁₆	—	268	1	no	—
144	Q₁₆	—	268	2	no	—
145	Q₁₆	—	134	2	no	—
146	SD₁₆	—	268	2	no	—
147	SD₁₆	—	134	4	no	—
148	D₈	—	268	1	no	—
149	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₆₇	1	2	yes	—
151	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
152	C₄⋊C₄	—	67	2	no	—
153	C₄⋊C₄	—	134	2	no	—
154	C₄⋊C₄	—	134	2	no	—
155	C₄⋊C₄	—	134	2	no	—
156	C₄⋊C₄	—	134	2	no	—
157	C₄⋊C₄	D₂₆₈	1	2	yes	—
158	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
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	—	134	2	no	—
166	C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₄²	C₂²×D₆₇	1	2	yes	—
168	C₂³	—	134	2	no	—
169	Q₈	—	4	1	no	—
170	Q₈	—	134	2	no	—
171	Q₈	—	134	2	no	—
172	Q₈	—	134	2	no	—
173	Q₈	—	134	2	no	—
174	D₄	—	134	2	no	—
175	D₄	—	134	2	no	—
176	C₂×C₄	—	268	4	no	—
177	C₂×C₄	—	134	2	no	—
178	C₂×C₄	—	134	2	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
182	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
183	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
184	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#11761629	1	2	yes	—
188	C₂²	—	268	2	no	—
189	C₂²	—	268	2	no	—
190	C₂²	—	268	2	no	—
191	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	268	2	no	—
197	C₄	—	268	2	no	—
198	C₄	—	268	2	no	—
199	C₄	C₂×#11761629	1	1	yes	characteristic
200	C₄	#14524089	1	1	yes	characteristic
201	C₂	—	536	2	no	—
202	C₂	#16864068	1	1	yes	characteristic
203	C₂	#16864196	1	1	yes	characteristic
204	C₂	#16864213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864119

C₂

yes

maximal, complemented

#16863790

C₂

yes

maximal, complemented

#16863780

C₂

yes

maximal, characteristic

#16863619

C₂

yes

maximal, complemented

#16863612

C₂

yes

maximal

#16863607

C₂

yes

maximal, complemented

#16863498

C₂

yes

maximal, complemented

C₄×Dic₂₆₈

C₂

yes

maximal, characteristic, complemented

#446×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761617

C₂²

yes

—

C₂×#11761617

C₂²

yes

—

C₂×D₅₃₆

C₂²

yes

characteristic

#14524043

C₂²

yes

—

#14524038

—

#14524038

—

#14524031

C₂²

yes

—

#14524031

C₂²

yes

—

#14524019

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×D₂₆₈

C₂²

yes

—

#14524001

—

#14523977

C₂²

yes

characteristic

#14523968

C₂²

yes

—

#14523968

C₂²

yes

—

#14523965

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761617

—

#11761617

—

D₅₃₆

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₂×C₆₇

#578

yes

characteristic

101

D₆₇

—

102

#3402

—

maximal, Sylow

103

C₆₇

#3402

yes

characteristic, complemented, Sylow

104

#556

—

105

#497

—

106

#495

—

107

#464

—

108

#463

—

109

#462

—

110

#446

D₆₇

yes

characteristic, complemented

111

#440

—

112

C₄×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

134

—

121

#126

—

134

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

134

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₆₇

yes

characteristic

128

C₄×D₄

—

129

#117

—

134

—

130

#107

—

131

C₄⋊C₈

C₂×D₆₇

yes

—

132

C₄⋊C₈

C₂×D₆₇

yes

characteristic

133

#103

—

134

#103

—

135

#102

—

136

#97

C₂×D₆₇

yes

—

137

C₄×C₈

C₂×D₆₇

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

D₂₆₈

yes

characteristic

141

C₂×D₄

—

142

C₂²×C₄

—

134

—

143

Q₁₆

—

268

—

144

Q₁₆

—

268

—

145

Q₁₆

—

134

—

146

SD₁₆

—

268

—

147

SD₁₆

—

134

—

148

D₈

—

268

—

149

C₂×C₈

C₂²×D₆₇

yes

characteristic

150

C₂×C₈

C₂²×D₆₇

yes

—

151

C₂×C₈

C₂²×D₆₇

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

134

—

154

C₄⋊C₄

—

134

—

155

C₄⋊C₄

—

134

—

156

C₄⋊C₄

—

134

—

157

C₄⋊C₄

D₂₆₈

yes

—

158

C₄⋊C₄

D₂₆₈

yes

characteristic

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

—

134

—

166

C₄²

C₂²×D₆₇

yes

characteristic

167

C₄²

C₂²×D₆₇

yes

—

168

C₂³

—

134

—

169

Q₈

—

170

Q₈

—

134

—

171

Q₈

—

134

—

172

Q₈

—

134

—

173

Q₈

—

134

—

174

D₄

—

134

—

175

D₄

—

134

—

176

C₂×C₄

—

268

—

177

C₂×C₄

—

134

—

178

C₂×C₄

—

134

—

179

C₂×C₄

—

134

—

180

C₂×C₄

C₂×D₂₆₈

yes

characteristic

181

C₂×C₄

C₂×D₂₆₈

yes

—

182

C₂×C₄

C₂×D₂₆₈

yes

characteristic

183

C₂×C₄

C₂×D₂₆₈

yes

—

184

C₂×C₄

C₂³×D₆₇

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#11761629

yes

—

188

C₂²

—

268

—

189

C₂²

—

268

—

190

C₂²

—

268

—

191

C₂²

C₂²×D₂₆₈

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

268

—

197

C₄

—

268

—

198

C₄

—

268

—

199

C₄

C₂×#11761629

yes

characteristic

200

C₄

#14524089

yes

characteristic

201

C₂

—

536

—

202

C₂

#16864068

yes

characteristic

203

C₂

#16864196

yes

characteristic

204

C₂

#16864213

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#446×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#446	D₆₇
hypocenter	C₆₇	#3402
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#446×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #16864119	C₂	semidirect
2. #16863790	C₂	semidirect
4. #16863619	C₂	semidirect
6. #16863607	C₂	semidirect
7. #16863498	C₂	semidirect
8. C₄×Dic₂₆₈	C₂	semidirect
9. #446×C₆₇	C₂	semidirect
103. C₆₇	#3402	semidirect
110. #446	D₆₇	semidirect

H ≅ G/N

type

1. #16864119

C₂

semidirect

2. #16863790

C₂

semidirect

4. #16863619

C₂

semidirect

6. #16863607

C₂

semidirect

7. #16863498

C₂

semidirect

8. C₄×Dic₂₆₈

C₂

semidirect

9. #446×C₆₇

C₂

semidirect

103. C₆₇

#3402

semidirect

110. #446

D₆₇

semidirect

Subgroups — #22573290

Conjugacy classes of subgroups

Complements