Group #22569741 — subgroups

Subgroups — #22569741

Order 8576.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×#11761618	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863967	C₂	1	1	yes	maximal, characteristic
3	#16863874	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863616	C₂	1	4	yes	maximal, complemented
5	#16863604	C₂	1	4	yes	maximal, complemented
6	#16863424	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863406	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863392	C₂	1	1	yes	maximal, characteristic
9	C₂×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₂×#11761618	—	2	4	no	—
12	C₂×#11761618	C₂²	1	4	yes	—
13	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#14524041	C₂²	1	2	yes	—
15	#14524030	C₂²	1	2	yes	—
16	#14524024	—	2	2	no	—
17	#14524022	—	2	2	no	—
18	Q₈×Dic₆₇	—	2	2	no	—
19	Q₈×Dic₆₇	C₂²	1	4	yes	—
20	#14524002	—	2	2	no	—
21	#14524002	C₂²	1	1	yes	characteristic
22	#14524002	C₂²	1	1	yes	characteristic
23	Dic₆₇⋊C₈	C₂²	1	4	yes	—
24	#14523968	C₂²	1	4	yes	—
25	#14523967	C₂²	1	2	yes	—
26	#14523964	C₂²	1	2	yes	—
27	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	#137×C₆₇	—	2	4	no	—
30	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂×SD₁₆×C₆₇	C₂²	1	2	yes	—
32	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂×Dic₁₃₄	—	2	2	no	—
35	C₂×Dic₁₃₄	—	2	1	no	—
36	C₂×Dic₁₃₄	C₂³	1	2	yes	—
37	C₂²×Dic₆₇	—	2	1	no	—
38	#11761618	—	2	8	no	—
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₆₇	—	2	4	no	—
42	C₄⋊Dic₆₇	D₄	1	2	yes	—
43	C₄⋊Dic₆₇	C₂³	1	2	yes	—
44	Dic₆₇⋊C₄	—	2	4	no	—
45	C₂²⋊Dic₆₇	—	2	1	no	—
46	C₂²⋊Dic₆₇	—	2	4	no	—
47	C₂²⋊Dic₆₇	D₄	1	2	yes	—
48	C₄×Dic₆₇	—	2	1	no	—
49	C₄×Dic₆₇	—	2	4	no	—
50	C₄×Dic₆₇	C₂³	1	2	yes	—
51	#41×C₆₇	—	2	2	no	—
52	#41×C₆₇	—	2	2	no	—
53	C₂×Q₈×C₆₇	—	2	1	no	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	D₄	1	4	yes	—
56	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₂×D₄×C₆₇	—	2	1	no	—
59	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	Q₁₆×C₆₇	—	2	4	no	—
64	SD₁₆×C₆₇	—	2	4	no	—
65	M₁₆×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₆₇	—	2	1	no	—
68	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₆₇	—	2	4	no	—
70	Q₈×C₆₇	—	2	2	no	—
71	Q₈×C₆₇	—	2	1	no	—
72	Q₈×C₆₇	—	2	1	no	—
73	Q₈×C₆₇	C₂×D₄	1	4	yes	—
74	D₄×C₆₇	—	2	1	no	—
75	D₄×C₆₇	—	2	1	no	—
76	D₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	#41	1	2	yes	—
84	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	4	2	no	—
88	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
89	Dic₁₃₄	—	2	2	no	—
90	Dic₁₃₄	—	2	2	no	—
91	Dic₁₃₄	—	4	2	no	—
92	C₈×C₆₇	—	2	2	no	—
93	C₆₇⋊C₈	—	2	2	no	—
94	C₂²×C₆₇	—	2	2	no	—
95	C₂²×C₆₇	—	4	1	no	—
96	C₂²×C₆₇	—	4	2	no	—
97	C₂²×C₆₇	#121	1	2	yes	—
98	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₆₇	—	2	4	no	—
100	C₄×C₆₇	—	4	1	no	—
101	C₄×C₆₇	#121	1	2	yes	—
102	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	Dic₆₇	—	2	4	no	—
105	Dic₆₇	—	4	2	no	—
106	Dic₆₇	—	4	2	no	—
107	C₂×C₆₇	—	2	2	no	—
108	C₂×C₆₇	—	8	1	no	—
109	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
110	C₂×C₆₇	#578	1	2	yes	—
111	#3044	—	67	1	no	maximal, Sylow
112	C₆₇	#3044	1	1	yes	characteristic, complemented, Sylow
113	C₂×#137	D₆₇	1	1	yes	characteristic
114	C₂²×Q₁₆	—	67	1	no	—
115	#536	—	67	1	no	—
116	#519	—	67	1	no	—
117	#464	—	67	4	no	—
118	#462	—	67	4	no	—
119	#424	—	67	1	no	—
120	#419	—	67	1	no	—
121	#416	—	67	1	no	—
122	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
123	C₂²×Q₈	—	67	1	no	—
124	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
125	#137	—	2	4	no	—
126	C₂×Q₁₆	—	67	4	no	—
127	C₂×Q₁₆	—	134	4	no	—
128	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
129	C₂×SD₁₆	C₂×D₆₇	1	2	yes	—
130	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
131	C₂²×C₈	—	67	1	no	—
132	C₄⋊Q₈	—	67	2	no	—
133	#124	—	67	2	no	—
134	#123	—	134	2	no	—
135	C₂²⋊Q₈	—	134	2	no	—
136	C₄×Q₈	—	67	4	no	—
137	C₄×Q₈	—	134	2	no	—
138	#117	—	67	1	no	—
139	#117	—	67	1	no	—
140	#117	—	134	2	no	—
141	C₄⋊C₈	—	67	4	no	—
142	#103	—	67	2	no	—
143	#103	—	67	4	no	—
144	#102	—	67	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	67	2	no	—
150	C₂×Q₈	—	134	2	no	—
151	C₂×Q₈	—	134	2	no	—
152	C₂×Q₈	—	134	1	no	—
153	C₂×Q₈	C₂²⋊D₆₇	1	4	yes	—
154	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
155	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	134	1	no	—
161	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
162	Q₁₆	—	2	4	no	—
163	Q₁₆	—	134	8	no	—
164	SD₁₆	—	2	4	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	67	2	no	—
167	C₂×C₈	—	134	2	no	—
168	C₂×C₈	C₂²×D₆₇	1	2	yes	—
169	C₄⋊C₄	—	67	2	no	—
170	C₄⋊C₄	—	67	2	no	—
171	C₄⋊C₄	—	134	4	no	—
172	C₄⋊C₄	—	134	4	no	—
173	C₂²⋊C₄	—	67	2	no	—
174	C₂²⋊C₄	—	134	4	no	—
175	C₂²⋊C₄	—	134	1	no	—
176	C₄²	—	67	2	no	—
177	C₄²	—	134	4	no	—
178	C₄²	—	134	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	D₄×D₆₇	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	2	1	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	268	2	no	—
186	Q₈	—	134	2	no	—
187	Q₈	—	134	2	no	—
188	Q₈	C₂×C₂²⋊D₆₇	1	4	yes	—
189	D₄	—	2	1	no	—
190	D₄	—	2	1	no	—
191	D₄	—	4	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	268	2	no	—
197	C₂×C₄	—	67	4	no	—
198	C₂×C₄	—	134	2	no	—
199	C₂×C₄	—	134	2	no	—
200	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
202	C₂×C₄	#11761627	1	2	yes	—
203	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	134	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	#14524015	1	2	yes	—
210	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	4	1	no	—
213	C₄	—	268	2	no	—
214	C₄	—	268	2	no	—
215	C₄	—	134	4	no	—
216	C₄	#14524015	1	2	yes	—
217	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₄	C₂×#11761627	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	8	1	no	—
221	C₂	C₂×#14524015	1	1	yes	characteristic
222	C₂	#16864212	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761618

C₂

yes

maximal, characteristic, complemented

#16863967

C₂

yes

maximal, characteristic

#16863874

C₂

yes

maximal, characteristic, complemented

#16863616

C₂

yes

maximal, complemented

#16863604

C₂

yes

maximal, complemented

#16863424

C₂

yes

maximal, characteristic, complemented

#16863406

C₂

yes

maximal, characteristic, complemented

#16863392

C₂

yes

maximal, characteristic

C₂×#137×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂×#11761618

—

C₂×#11761618

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524041

C₂²

yes

—

#14524030

C₂²

yes

—

#14524024

—

#14524022

—

Q₈×Dic₆₇

—

Q₈×Dic₆₇

C₂²

yes

—

#14524002

—

#14524002

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523967

C₂²

yes

—

#14523964

C₂²

yes

—

C₂×#41×C₆₇

C₂²

yes

characteristic

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

#137×C₆₇

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₂×SD₁₆×C₆₇

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

D₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×Q₈×C₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂³

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

C₂×D₄

yes

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×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

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

100

C₄×C₆₇

—

101

C₄×C₆₇

#121

yes

—

102

C₄×C₆₇

C₂²×D₄

yes

characteristic

103

C₄×C₆₇

C₂×#41

yes

characteristic

104

Dic₆₇

—

105

Dic₆₇

—

106

Dic₆₇

—

107

C₂×C₆₇

—

108

C₂×C₆₇

—

109

C₂×C₆₇

C₂×#121

yes

characteristic

110

C₂×C₆₇

#578

yes

—

111

#3044

—

maximal, Sylow

112

C₆₇

#3044

yes

characteristic, complemented, Sylow

113

C₂×#137

D₆₇

yes

characteristic

114

C₂²×Q₁₆

—

115

#536

—

116

#519

—

117

#464

—

118

#462

—

119

#424

—

120

#419

—

121

#416

—

122

C₂×#41

C₂×D₆₇

yes

characteristic

123

C₂²×Q₈

—

124

C₂²×Q₈

C₂×D₆₇

yes

characteristic

125

#137

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

134

—

128

C₂×Q₁₆

C₂×D₆₇

yes

—

129

C₂×SD₁₆

C₂×D₆₇

yes

—

130

C₂×M₁₆

C₂×D₆₇

yes

characteristic

131

C₂²×C₈

—

132

C₄⋊Q₈

—

133

#124

—

134

#123

—

134

—

135

C₂²⋊Q₈

—

134

—

136

C₄×Q₈

—

137

C₄×Q₈

—

134

—

138

#117

—

139

#117

—

140

#117

—

134

—

141

C₄⋊C₈

—

142

#103

—

143

#103

—

144

#102

—

145

#41

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

134

—

151

C₂×Q₈

—

134

—

152

C₂×Q₈

—

134

—

153

C₂×Q₈

C₂²⋊D₆₇

yes

—

154

C₂×Q₈

C₂²×D₆₇

yes

characteristic

155

C₂×Q₈

C₂²×D₆₇

yes

—

156

C₂×D₄

—

157

C₂×D₄

C₂²×D₆₇

yes

characteristic

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

134

—

161

C₂²×C₄

C₂²×D₆₇

yes

characteristic

162

Q₁₆

—

163

Q₁₆

—

134

—

164

SD₁₆

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

134

—

168

C₂×C₈

C₂²×D₆₇

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

134

—

172

C₄⋊C₄

—

134

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

134

—

175

C₂²⋊C₄

—

134

—

176

C₄²

—

177

C₄²

—

134

—

178

C₄²

—

134

—

179

C₂³

—

180

C₂³

D₄×D₆₇

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

Q₈

—

268

—

186

Q₈

—

134

—

187

Q₈

—

134

—

188

Q₈

C₂×C₂²⋊D₆₇

yes

—

189

D₄

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

268

—

197

C₂×C₄

—

198

C₂×C₄

—

134

—

199

C₂×C₄

—

134

—

200

C₂×C₄

D₄×D₆₇

yes

characteristic

201

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

202

C₂×C₄

#11761627

yes

—

203

C₂×C₄

C₂³×D₆₇

yes

characteristic

204

C₈

—

205

C₈

—

134

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#14524015

yes

—

210

C₂²

C₂×D₄×D₆₇

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

268

—

214

C₄

—

268

—

215

C₄

—

134

—

216

C₄

#14524015

yes

—

217

C₄

C₂²×C₂²⋊D₆₇

yes

characteristic

218

C₄

C₂×#11761627

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

C₂×#14524015

yes

characteristic

222

C₂

#16864212

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#137×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×#137	D₆₇
hypocenter	C₆₇	#3044
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#137×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

N	H ≅ G/N	type
1. C₂²×#11761618	C₂	semidirect
3. #16863874	C₂	semidirect
4. #16863616	C₂	semidirect
5. #16863604	C₂	semidirect
6. #16863424	C₂	semidirect
7. #16863406	C₂	semidirect
112. C₆₇	#3044	semidirect

H ≅ G/N

type

1. C₂²×#11761618

C₂

semidirect

3. #16863874

C₂

semidirect

4. #16863616

C₂

semidirect

5. #16863604

C₂

semidirect

6. #16863424

C₂

semidirect

7. #16863406

C₂

semidirect

112. C₆₇

#3044

semidirect

Subgroups — #22569741

Conjugacy classes of subgroups

Complements