Group #22570667 — subgroups

Subgroups — #22570667

Order 8576.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14524075	C₂	1	4	yes	maximal, complemented
2	#16863947	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863789	C₂	1	4	yes	maximal, complemented
5	#16863775	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863770	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863769	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
9	#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11761629	C₂²	1	2	yes	—
11	C₂×Q₈×D₆₇	C₂²	1	2	yes	—
12	#14524075	—	2	16	no	—
13	C₂×#11761618	C₂²	1	4	yes	—
14	C₂×#11761617	C₂²	1	4	yes	—
15	C₂×#11761615	C₂²	1	4	yes	—
16	C₂×#11761611	C₂²	1	2	yes	—
17	#14524043	C₂²	1	1	yes	characteristic
18	#14524042	—	2	2	no	—
19	#14524041	C₂²	1	2	yes	—
20	#14524040	C₂²	1	1	yes	characteristic
21	#14524032	C₂²	1	2	yes	—
22	#14524018	—	2	2	no	—
23	#14524005	—	2	2	no	—
24	C₄⋊C₄×D₆₇	—	2	2	no	—
25	C₄²×D₆₇	C₂²	1	1	yes	characteristic
26	#14523953	C₂²	1	2	yes	—
27	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₆₇	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#11761629	—	2	8	no	—
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂×Dic₁₃₄	C₂³	1	2	yes	—
35	Q₈×D₆₇	—	2	8	no	—
36	C₂×D₂₆₈	—	2	2	no	—
37	C₂×D₂₆₈	—	2	2	no	—
38	C₂×D₂₆₈	C₂³	1	2	yes	—
39	C₂×C₄×D₆₇	—	2	2	no	—
40	C₂×C₄×D₆₇	—	2	2	no	—
41	C₂×C₄×D₆₇	D₄	1	2	yes	—
42	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
43	#11761618	—	2	8	no	—
44	#11761617	—	2	8	no	—
45	#11761615	—	2	8	no	—
46	#11761611	D₄	1	8	yes	complemented
47	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	Dic₆₇⋊C₄	—	2	4	no	—
50	(C₂×D₆₇)⋊C₄	—	2	4	no	—
51	C₄×Dic₆₇	D₄	1	2	yes	—
52	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
54	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
55	Q₁₆×C₆₇	—	2	8	no	—
56	C₂×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×D₆₇	—	2	2	no	—
61	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	4	no	—
63	Q₈×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
70	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
71	D₂₆₈	—	2	4	no	—
72	D₂₆₈	—	4	4	no	—
73	D₂₆₈	—	4	4	no	—
74	C₄×D₆₇	—	2	4	no	—
75	C₄×D₆₇	—	4	4	no	—
76	C₄×D₆₇	—	4	4	no	—
77	C₄×D₆₇	C₂×D₄	1	4	yes	—
78	Dic₁₃₄	—	2	4	no	—
79	Dic₁₃₄	—	4	4	no	—
80	C₈×C₆₇	C₂×D₄	1	4	yes	—
81	C₆₇⋊C₈	C₂×D₄	1	4	yes	—
82	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₆₇	—	2	2	no	—
84	C₂×D₆₇	—	4	2	no	—
85	C₂×D₆₇	—	4	4	no	—
86	C₂×D₆₇	#127	1	2	yes	—
87	C₄×C₆₇	—	4	2	no	—
88	C₄×C₆₇	—	4	2	no	—
89	C₄×C₆₇	#137	1	4	yes	complemented
90	C₄×C₆₇	C₂²×D₄	1	2	yes	—
91	Dic₆₇	—	2	2	no	—
92	Dic₆₇	—	4	2	no	—
93	Dic₆₇	#127	1	2	yes	—
94	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
95	C₂×C₆₇	C₂×#137	1	2	yes	—
96	D₆₇	—	2	2	no	—
97	D₆₇	—	8	2	no	—
98	#3152	—	67	1	no	maximal, Sylow
99	C₆₇	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	67	4	no	—
101	#532	—	67	1	no	—
102	C₂×C₄⋊Q₈	—	67	1	no	—
103	#497	—	67	4	no	—
104	#494	—	67	1	no	—
105	#494	D₆₇	1	1	yes	characteristic, complemented
106	#492	—	67	1	no	—
107	#492	—	67	1	no	—
108	C₄×M₁₆	—	67	1	no	—
109	C₂×#41	—	67	2	no	—
110	C₂²×Q₈	—	67	2	no	—
111	#137	—	134	16	no	—
112	C₂×Q₁₆	—	67	4	no	—
113	C₂×Q₁₆	C₂×D₆₇	1	4	yes	—
114	C₂×SD₁₆	—	67	4	no	—
115	C₂×SD₁₆	—	67	4	no	—
116	C₂×M₁₆	—	67	2	no	—
117	C₄⋊Q₈	—	67	2	no	—
118	C₄⋊Q₈	—	67	1	no	—
119	C₄⋊Q₈	—	134	2	no	—
120	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
122	#127	—	67	1	no	—
123	#124	—	67	2	no	—
124	#121	—	134	2	no	—
125	C₄×D₄	—	134	2	no	—
126	C₂×C₄⋊C₄	—	134	2	no	—
127	C₂×C₄²	—	67	1	no	—
128	#97	—	67	2	no	—
129	C₄×C₈	—	67	1	no	—
130	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	#41	—	134	8	no	—
132	C₂×Q₈	—	67	2	no	—
133	C₂×Q₈	—	134	2	no	—
134	C₂×Q₈	—	134	8	no	—
135	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
136	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
137	C₂×D₄	—	67	2	no	—
138	C₂×D₄	—	134	2	no	—
139	C₂×D₄	—	134	2	no	—
140	C₂²×C₄	—	67	1	no	—
141	C₂²×C₄	—	67	2	no	—
142	C₂²×C₄	—	134	2	no	—
143	C₂²×C₄	—	134	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	134	8	no	—
146	SD₁₆	—	134	8	no	—
147	SD₁₆	—	134	8	no	—
148	M₁₆	—	67	8	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	C₂²×D₆₇	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	134	4	no	—
154	C₄⋊C₄	—	134	2	no	—
155	C₂²⋊C₄	—	134	4	no	—
156	C₄²	—	67	2	no	—
157	C₄²	—	67	1	no	—
158	C₄²	C₂²×D₆₇	1	1	yes	characteristic
159	C₂³	—	67	1	no	—
160	C₂³	—	134	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	268	4	no	—
164	Q₈	—	134	4	no	—
165	D₄	—	268	4	no	—
166	D₄	—	268	4	no	—
167	D₄	—	134	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	268	4	no	—
171	C₂×C₄	—	268	4	no	—
172	C₂×C₄	—	67	1	no	—
173	C₂×C₄	—	67	4	no	—
174	C₂×C₄	—	67	2	no	—
175	C₂×C₄	—	134	2	no	—
176	C₂×C₄	—	134	4	no	—
177	C₂×C₄	D₄×D₆₇	1	2	yes	—
178	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
179	C₈	—	67	4	no	—
180	C₈	D₄×D₆₇	1	4	yes	—
181	C₂²	—	268	2	no	—
182	C₂²	—	268	4	no	—
183	C₂²	—	67	2	no	—
184	C₂²	—	134	2	no	—
185	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	268	2	no	—
189	C₄	—	67	2	no	—
190	C₄	—	134	2	no	—
191	C₄	#14524075	1	4	yes	complemented
192	C₄	C₂×D₄×D₆₇	1	2	yes	—
193	C₂	—	536	2	no	—
194	C₂	—	134	2	no	—
195	C₂	#127×D₆₇	1	1	yes	characteristic
196	C₂	C₂×#14524075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524075

C₂

yes

maximal, complemented

#16863947

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₆₇

C₂

yes

maximal, characteristic, complemented

#16863789

C₂

yes

maximal, complemented

#16863775

C₂

yes

maximal, characteristic, complemented

#16863770

C₂

yes

maximal, characteristic, complemented

#16863769

C₂

yes

maximal, characteristic, complemented

C₄×#11761611

C₂

yes

maximal, characteristic, complemented

#494×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761629

C₂²

yes

—

C₂×Q₈×D₆₇

C₂²

yes

—

#14524075

—

C₂×#11761618

C₂²

yes

—

C₂×#11761617

C₂²

yes

—

C₂×#11761615

C₂²

yes

—

C₂×#11761611

C₂²

yes

—

#14524043

C₂²

yes

characteristic

#14524042

—

#14524041

C₂²

yes

—

#14524040

C₂²

yes

characteristic

#14524032

C₂²

yes

—

#14524018

—

#14524005

—

C₄⋊C₄×D₆₇

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761629

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

Q₈×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

—

—

D₄

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

—

C₆₇⋊C₈

C₂×D₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

#127

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#137

yes

complemented

C₄×C₆₇

C₂²×D₄

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#127

yes

—

C₂×C₆₇

C₂×#127

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

D₆₇

—

D₆₇

—

#3152

—

maximal, Sylow

C₆₇

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

101

#532

—

102

C₂×C₄⋊Q₈

—

103

#497

—

104

#494

—

105

#494

D₆₇

yes

characteristic, complemented

106

#492

—

107

#492

—

108

C₄×M₁₆

—

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

134

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₆₇

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

134

—

120

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

121

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

134

—

125

C₄×D₄

—

134

—

126

C₂×C₄⋊C₄

—

134

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₆₇

yes

characteristic

131

#41

—

134

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

—

134

C₂×Q₈

—

134

—

135

C₂×Q₈

C₂²×D₆₇

yes

—

136

C₂×Q₈

C₂²×D₆₇

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

134

—

139

C₂×D₄

—

134

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

134

—

143

C₂²×C₄

—

134

—

144

Q₁₆

—

145

Q₁₆

—

134

—

146

SD₁₆

—

134

—

147

SD₁₆

—

134

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₆₇

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

134

—

154

C₄⋊C₄

—

134

—

155

C₂²⋊C₄

—

134

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₆₇

yes

characteristic

159

C₂³

—

160

C₂³

—

134

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

268

—

164

Q₈

—

134

—

165

D₄

—

268

—

166

D₄

—

268

—

167

D₄

—

134

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

268

—

171

C₂×C₄

—

268

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

134

—

176

C₂×C₄

—

134

—

177

C₂×C₄

D₄×D₆₇

yes

—

178

C₂×C₄

C₂³×D₆₇

yes

characteristic

179

C₈

—

180

C₈

D₄×D₆₇

yes

—

181

C₂²

—

268

—

182

C₂²

—

268

—

183

C₂²

—

184

C₂²

—

134

—

185

C₂²

C₂×D₄×D₆₇

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

268

—

189

C₄

—

190

C₄

—

134

—

191

C₄

#14524075

yes

complemented

192

C₄

C₂×D₄×D₆₇

yes

—

193

C₂

—

536

—

194

C₂

—

134

—

195

C₂

#127×D₆₇

yes

characteristic

196

C₂

C₂×#14524075

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#494	D₆₇
hypocenter	C₆₇	#3152
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#494×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14524075	C₂	semidirect
3. C₄⋊Q₈×D₆₇	C₂	semidirect
4. #16863789	C₂	semidirect
5. #16863775	C₂	semidirect
6. #16863770	C₂	semidirect
7. #16863769	C₂	semidirect
8. C₄×#11761611	C₂	semidirect
9. #494×C₆₇	C₂	semidirect
46. #11761611	D₄	semidirect
89. C₄×C₆₇	#137	semidirect
99. C₆₇	#3152	semidirect
105. #494	D₆₇	semidirect
191. C₄	#14524075	semidirect

H ≅ G/N

type

1. C₂×#14524075

C₂

semidirect

3. C₄⋊Q₈×D₆₇

C₂

semidirect

4. #16863789

C₂

semidirect

5. #16863775

C₂

semidirect

6. #16863770

C₂

semidirect

7. #16863769

C₂

semidirect

8. C₄×#11761611

C₂

semidirect

9. #494×C₆₇

C₂

semidirect

46. #11761611

D₄

semidirect

89. C₄×C₆₇

#137

semidirect

99. C₆₇

#3152

semidirect

105. #494

D₆₇

semidirect

191. C₄

#14524075

semidirect

Subgroups — #22570667

Conjugacy classes of subgroups

Complements