Group #22571604 — subgroups

Subgroups — #22571604

Order 8576.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16863927	C₂	1	2	yes	maximal, complemented
2	#16863706	C₂	1	2	yes	maximal, complemented
3	#16863698	C₂	1	2	yes	maximal, complemented
4	#16863654	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863640	C₂	1	2	yes	maximal, complemented
6	#16863617	C₂	1	2	yes	maximal, complemented
7	#16863609	C₂	1	2	yes	maximal, complemented
8	#16863419	C₂	1	1	yes	maximal, characteristic, complemented
9	#469×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
11	C₂×#11761616	C₂²	1	2	yes	—
12	C₂×#11761612	C₂²	1	1	yes	characteristic
13	#14524036	—	2	2	no	—
14	#14524034	C₂²	1	2	yes	—
15	#14524030	C₂²	1	2	yes	—
16	#14524024	—	2	2	no	—
17	#14524022	C₂²	1	2	yes	—
18	#14524008	C₂²	1	2	yes	—
19	D₄×Dic₆₇	—	2	2	no	—
20	D₄×Dic₆₇	C₂²	1	2	yes	—
21	#14523999	—	2	2	no	—
22	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
23	#14523976	C₂²	1	2	yes	—
24	#14523975	C₂²	1	1	yes	characteristic
25	Dic₆₇⋊C₈	C₂²	1	2	yes	—
26	Dic₆₇⋊C₈	C₂²	1	2	yes	—
27	#14523968	C₂²	1	2	yes	—
28	#14523966	C₂²	1	2	yes	—
29	#14523964	C₂²	1	2	yes	—
30	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	#121×C₆₇	C₂²	1	2	yes	—
33	#106×C₆₇	C₂²	1	1	yes	characteristic
34	#102×C₆₇	C₂²	1	2	yes	—
35	C₂×Dic₁₃₄	C₂³	1	2	yes	—
36	C₂²×Dic₆₇	—	2	2	no	—
37	C₂²×Dic₆₇	D₄	1	2	yes	—
38	Dic₂₆₈	—	4	1	no	—
39	#11761616	—	4	2	no	—
40	#11761612	—	4	1	no	—
41	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₄⋊Dic₆₇	—	2	2	no	—
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	C₂²⋊Dic₆₇	—	2	2	no	—
48	C₂²⋊Dic₆₇	—	2	2	no	—
49	C₂²⋊Dic₆₇	—	2	2	no	—
50	C₂²⋊Dic₆₇	—	2	2	no	—
51	C₄×Dic₆₇	—	2	2	no	—
52	C₄×Dic₆₇	D₄	1	1	yes	characteristic
53	C₄×Dic₆₇	D₄	1	1	yes	characteristic
54	C₄×Dic₆₇	C₂³	1	2	yes	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	D₈×C₆₇	—	4	1	no	—
59	M₁₆×C₆₇	—	4	1	no	—
60	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂³×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
66	D₄×C₆₇	—	2	2	no	—
67	D₄×C₆₇	—	2	2	no	—
68	D₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	2	2	no	—
76	C₂×Dic₆₇	—	4	2	no	—
77	C₂×Dic₆₇	—	4	4	no	—
78	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
79	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
80	Dic₁₃₄	—	2	2	no	—
81	Dic₁₃₄	—	2	2	no	—
82	C₈×C₆₇	—	2	1	no	—
83	C₈×C₆₇	—	2	1	no	—
84	C₆₇⋊C₈	—	2	2	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	—	4	2	no	—
88	C₂²×C₆₇	—	4	2	no	—
89	C₂²×C₆₇	—	4	2	no	—
90	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
95	Dic₆₇	—	2	2	no	—
96	Dic₆₇	—	2	2	no	—
97	Dic₆₇	—	4	2	no	—
98	Dic₆₇	—	4	2	no	—
99	C₂×C₆₇	—	4	1	no	—
100	C₂×C₆₇	—	8	2	no	—
101	C₂×C₆₇	#537	1	1	yes	characteristic
102	C₂×C₆₇	#575	1	2	yes	—
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	D₆₇	1	1	yes	characteristic
110	#468	—	67	2	no	—
111	#464	—	67	2	no	—
112	#463	—	67	2	no	—
113	#423	—	67	1	no	—
114	C₂×Q₁₆	—	67	1	no	—
115	C₂×SD₁₆	—	67	2	no	—
116	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
117	C₂×M₁₆	—	67	1	no	—
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	C₂×D₆₇	1	2	yes	—
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	—	67	1	no	—
130	#107	—	67	2	no	—
131	#106	—	67	1	no	—
132	#106	C₂×D₆₇	1	1	yes	characteristic
133	C₄⋊C₈	—	67	2	no	—
134	C₄⋊C₈	—	67	2	no	—
135	#103	—	67	2	no	—
136	#103	—	67	2	no	—
137	#102	—	67	2	no	—
138	#102	C₂×D₆₇	1	2	yes	—
139	C₂×Q₈	—	67	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₂²×D₆₇	1	2	yes	—
142	C₂²×C₄	—	67	2	no	—
143	C₂²×C₄	—	134	2	no	—
144	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
145	Q₁₆	—	268	1	no	—
146	SD₁₆	—	268	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	268	1	no	—
150	C₂×C₈	—	67	2	no	—
151	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	C₄⋊C₄	—	67	2	no	—
154	C₄⋊C₄	—	134	2	no	—
155	C₄⋊C₄	—	134	2	no	—
156	C₄⋊C₄	—	134	2	no	—
157	C₄⋊C₄	—	134	2	no	—
158	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
159	C₂²⋊C₄	—	2	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	—	67	1	no	—
165	C₄²	—	67	1	no	—
166	C₄²	—	67	2	no	—
167	C₄²	—	134	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	D₄×D₆₇	1	1	yes	characteristic
170	Q₈	—	134	2	no	—
171	Q₈	—	134	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	268	4	no	—
179	C₂×C₄	—	268	2	no	—
180	C₂×C₄	—	67	2	no	—
181	C₂×C₄	—	67	2	no	—
182	C₂×C₄	—	134	2	no	—
183	C₂×C₄	—	134	2	no	—
184	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	134	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	268	2	no	—
198	C₄	—	268	2	no	—
199	C₄	—	134	2	no	—
200	C₄	—	134	2	no	—
201	C₄	#14524088	1	1	yes	characteristic
202	C₄	#14524090	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	#16863973	1	1	yes	characteristic
206	C₂	#16864192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863927

C₂

yes

maximal, complemented

#16863706

C₂

yes

maximal, complemented

#16863698

C₂

yes

maximal, complemented

#16863654

C₂

yes

maximal, characteristic, complemented

#16863640

C₂

yes

maximal, complemented

#16863617

C₂

yes

maximal, complemented

#16863609

C₂

yes

maximal, complemented

#16863419

C₂

yes

maximal, characteristic, complemented

#469×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761616

C₂²

yes

—

C₂×#11761612

C₂²

yes

characteristic

#14524036

—

#14524034

C₂²

yes

—

#14524030

C₂²

yes

—

#14524024

—

#14524022

C₂²

yes

—

#14524008

C₂²

yes

—

D₄×Dic₆₇

—

D₄×Dic₆₇

C₂²

yes

—

#14523999

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523976

C₂²

yes

—

#14523975

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523966

C₂²

yes

—

#14523964

C₂²

yes

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#121×C₆₇

C₂²

yes

—

#106×C₆₇

C₂²

yes

characteristic

#102×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

Dic₂₆₈

—

#11761616

—

#11761612

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₄×Dic₆₇

D₄

yes

characteristic

C₄×Dic₆₇

C₂³

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

D₈×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

—

101

C₂×C₆₇

#537

yes

characteristic

102

C₂×C₆₇

#575

yes

—

103

#3256

—

maximal, Sylow

104

C₆₇

#3256

yes

characteristic, complemented, Sylow

105

#529

—

106

#479

—

107

#478

—

108

#470

—

109

#469

D₆₇

yes

characteristic

110

#468

—

111

#464

—

112

#463

—

113

#423

—

114

C₂×Q₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

C₂×D₆₇

yes

characteristic

117

C₂×M₁₆

—

118

C₂×M₁₆

C₂×D₆₇

yes

characteristic

119

#126

—

134

—

120

#125

—

121

#124

—

122

#123

—

134

—

123

C₂²⋊Q₈

—

124

#121

C₂×D₆₇

yes

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

C₄×D₄

—

134

—

128

#117

—

134

—

129

C₂×C₄²

—

130

#107

—

131

#106

—

132

#106

C₂×D₆₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

—

137

#102

—

138

#102

C₂×D₆₇

yes

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

C₂²×D₆₇

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

134

—

144

C₂²×C₄

C₂²×D₆₇

yes

characteristic

145

Q₁₆

—

268

—

146

SD₁₆

—

268

—

147

D₈

—

148

M₁₆

—

149

M₁₆

—

268

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×D₆₇

yes

characteristic

152

C₂×C₈

C₂²×D₆₇

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

134

—

155

C₄⋊C₄

—

134

—

156

C₄⋊C₄

—

134

—

157

C₄⋊C₄

—

134

—

158

C₄⋊C₄

C₂²×D₆₇

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

134

—

161

C₂²⋊C₄

—

134

—

162

C₂²⋊C₄

—

134

—

163

C₂²⋊C₄

—

134

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

134

—

168

C₂³

—

169

C₂³

D₄×D₆₇

yes

characteristic

170

Q₈

—

134

—

171

Q₈

—

134

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

268

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

134

—

183

C₂×C₄

—

134

—

184

C₂×C₄

D₄×D₆₇

yes

characteristic

185

C₂×C₄

C₂³×D₆₇

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

134

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

C₂×D₄×D₆₇

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

268

—

198

C₄

—

268

—

199

C₄

—

134

—

200

C₄

—

134

—

201

C₄

#14524088

yes

characteristic

202

C₄

#14524090

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#16863973

yes

characteristic

206

C₂

#16864192

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#469×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#469	D₆₇
hypocenter	C₆₇	#3256
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#469×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

N	H ≅ G/N	type
1. #16863927	C₂	semidirect
2. #16863706	C₂	semidirect
3. #16863698	C₂	semidirect
4. #16863654	C₂	semidirect
5. #16863640	C₂	semidirect
6. #16863617	C₂	semidirect
7. #16863609	C₂	semidirect
8. #16863419	C₂	semidirect
104. C₆₇	#3256	semidirect

H ≅ G/N

type

1. #16863927

C₂

semidirect

2. #16863706

C₂

semidirect

3. #16863698

C₂

semidirect

4. #16863654

C₂

semidirect

5. #16863640

C₂

semidirect

6. #16863617

C₂

semidirect

7. #16863609

C₂

semidirect

8. #16863419

C₂

semidirect

104. C₆₇

#3256

semidirect

Subgroups — #22571604

Conjugacy classes of subgroups

Complements