Group #22561121 — subgroups

Subgroups — #22561121

Order 8576.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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
2	C₂×#14523978	C₂	1	1	yes	maximal, characteristic
3	#16863406	C₂	1	1	yes	maximal, characteristic
4	#16863405	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523961	C₂	1	1	yes	maximal, characteristic
6	#16863354	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂×#11761618	—	2	1	no	—
10	C₂×#11761618	—	2	1	no	—
11	C₂×#11761618	—	2	2	no	—
12	C₂×#11761618	C₄	1	4	yes	complemented
13	C₂×#11761612	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#14524002	C₂²	1	1	yes	characteristic
16	#14523978	—	2	2	no	—
17	#14523968	—	2	2	no	—
18	#14523967	—	2	2	no	—
19	#14523961	—	2	2	no	—
20	#14523953	—	2	1	no	—
21	C₈×Dic₆₇	—	2	1	no	—
22	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#101×C₆₇	—	2	2	no	—
26	C₂×Dic₁₃₄	—	2	1	no	—
27	C₂×Dic₁₃₄	—	2	1	no	—
28	C₂×Dic₁₃₄	—	2	1	no	—
29	C₂×Dic₁₃₄	—	2	1	no	—
30	C₂×Dic₁₃₄	—	2	1	no	—
31	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
32	C₂²×Dic₆₇	—	2	1	no	—
33	#11761618	—	2	4	no	—
34	#11761618	—	2	4	no	—
35	#11761612	—	2	1	no	—
36	#11761612	—	2	1	no	—
37	C₂×C₆₇⋊C₈	—	2	1	no	—
38	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
42	C₄⋊Dic₆₇	D₄	1	2	yes	—
43	C₂²⋊Dic₆₇	—	2	1	no	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₂×Q₈×C₆₇	—	2	1	no	—
46	C₂×Q₈×C₆₇	—	2	1	no	—
47	C₂×Q₈×C₆₇	—	2	1	no	—
48	C₂×Q₈×C₆₇	—	2	1	no	—
49	C₂×Q₈×C₆₇	—	2	1	no	—
50	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	1	no	—
54	M₁₆×C₆₇	—	2	1	no	—
55	M₁₆×C₆₇	—	2	1	no	—
56	M₁₆×C₆₇	D₄	1	2	yes	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂³×C₆₇	#41	1	1	yes	characteristic
60	Q₈×C₆₇	—	2	2	no	—
61	Q₈×C₆₇	—	2	2	no	—
62	Q₈×C₆₇	—	4	1	no	—
63	Q₈×C₆₇	—	4	1	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	4	1	no	—
66	C₂×C₄×C₆₇	—	4	1	no	—
67	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	4	1	no	—
76	C₂×Dic₆₇	—	4	1	no	—
77	Dic₁₃₄	—	2	2	no	—
78	Dic₁₃₄	—	2	2	no	—
79	Dic₁₃₄	—	4	1	no	—
80	Dic₁₃₄	—	4	1	no	—
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	—	4	1	no	—
83	C₆₇⋊C₈	—	4	1	no	—
84	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
85	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₆₇	#121	1	1	yes	characteristic
89	C₂²×C₆₇	#124	1	1	yes	characteristic
90	C₄×C₆₇	—	4	2	no	—
91	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₄×C₆₇	#121	1	1	yes	characteristic
94	C₄×C₆₇	#127	1	1	yes	characteristic
95	Dic₆₇	—	4	2	no	—
96	Dic₆₇	—	4	2	no	—
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	#390	1	1	yes	characteristic
100	C₂×C₆₇	#473	1	2	yes	—
101	#1979	—	67	1	no	maximal, Sylow
102	C₆₇	#1979	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₁₆	—	67	1	no	—
104	C₂×#108	—	67	1	no	—
105	#419	—	67	1	no	—
106	#419	—	67	1	no	—
107	C₂×#101	—	67	1	no	—
108	C₂×#101	D₆₇	1	1	yes	characteristic
109	#405	—	67	1	no	—
110	C₂²×Q₈	—	67	1	no	—
111	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
112	C₂×Q₁₆	—	67	4	no	—
113	C₂×Q₁₆	—	134	1	no	—
114	C₂×Q₁₆	—	134	1	no	—
115	C₂×Q₁₆	—	134	2	no	—
116	C₂×M₁₆	—	67	1	no	—
117	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
118	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	—	67	1	no	—
120	#117	—	67	1	no	—
121	#108	—	134	2	no	—
122	#103	—	134	2	no	—
123	#103	—	134	2	no	—
124	#101	—	2	2	no	—
125	#101	—	134	2	no	—
126	#97	—	134	1	no	—
127	C₄×C₈	—	134	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	67	2	no	—
134	C₂×Q₈	—	134	1	no	—
135	C₂×Q₈	—	134	1	no	—
136	C₂×Q₈	—	134	1	no	—
137	C₂×Q₈	—	134	1	no	—
138	C₂×Q₈	—	134	1	no	—
139	C₂×Q₈	C₄×D₆₇	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	134	1	no	—
142	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
143	Q₁₆	—	134	4	no	—
144	Q₁₆	—	134	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	2	1	no	—
148	M₁₆	—	134	1	no	—
149	M₁₆	—	134	1	no	—
150	M₁₆	C₂²⋊D₆₇	1	2	yes	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	67	1	no	—
155	C₂×C₈	—	67	1	no	—
156	C₂×C₈	—	67	2	no	—
157	C₂×C₈	—	134	1	no	—
158	C₄⋊C₄	—	67	2	no	—
159	C₂²⋊C₄	—	134	1	no	—
160	C₄²	—	134	1	no	—
161	C₂³	#11761628	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	268	1	no	—
167	Q₈	—	268	1	no	—
168	Q₈	—	134	2	no	—
169	Q₈	—	134	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	268	1	no	—
174	C₂×C₄	—	268	1	no	—
175	C₂×C₄	—	134	2	no	—
176	C₂×C₄	—	134	2	no	—
177	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
180	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×C₄	#11761627	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	268	1	no	—
186	C₈	—	67	2	no	—
187	C₈	—	67	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²	#14524016	1	1	yes	characteristic
191	C₂²	#14524029	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	268	2	no	—
194	C₄	—	268	2	no	—
195	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
196	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
197	C₄	#14524015	1	1	yes	characteristic
198	C₄	#14524039	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#16863296	1	1	yes	characteristic
202	C₂	#16863670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761618

C₂

yes

maximal, characteristic

C₂×#14523978

C₂

yes

maximal, characteristic

#16863406

C₂

yes

maximal, characteristic

#16863405

C₂

yes

maximal, characteristic

C₂×#14523961

C₂

yes

maximal, characteristic

#16863354

C₂

yes

maximal, characteristic

C₂×#101×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

—

—

—

—

C₈×Dic₆₇

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#101×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#390

yes

characteristic

100

C₂×C₆₇

#473

yes

—

101

#1979

—

maximal, Sylow

102

C₆₇

#1979

yes

characteristic, complemented, Sylow

103

C₂²×Q₁₆

—

104

C₂×#108

—

105

#419

—

106

#419

—

107

C₂×#101

—

108

C₂×#101

D₆₇

yes

characteristic

109

#405

—

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₆₇

yes

characteristic

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

134

—

114

C₂×Q₁₆

—

134

—

115

C₂×Q₁₆

—

134

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×D₆₇

yes

characteristic

118

C₂×M₁₆

C₂×D₆₇

yes

characteristic

119

C₂²×C₈

—

120

#117

—

121

#108

—

134

—

122

#103

—

134

—

123

#103

—

134

—

124

#101

—

125

#101

—

134

—

126

#97

—

134

—

127

C₄×C₈

—

134

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

134

—

135

C₂×Q₈

—

134

—

136

C₂×Q₈

—

134

—

137

C₂×Q₈

—

134

—

138

C₂×Q₈

—

134

—

139

C₂×Q₈

C₄×D₆₇

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

134

—

142

C₂²×C₄

C₂²×D₆₇

yes

characteristic

143

Q₁₆

—

134

—

144

Q₁₆

—

134

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

134

—

149

M₁₆

—

134

—

150

M₁₆

C₂²⋊D₆₇

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

134

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

134

—

160

C₄²

—

134

—

161

C₂³

#11761628

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

268

—

167

Q₈

—

268

—

168

Q₈

—

134

—

169

Q₈

—

134

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

268

—

174

C₂×C₄

—

268

—

175

C₂×C₄

—

134

—

176

C₂×C₄

—

134

—

177

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

178

C₂×C₄

D₄×D₆₇

yes

characteristic

179

C₂×C₄

D₄×D₆₇

yes

characteristic

180

C₂×C₄

D₄×D₆₇

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

182

C₂×C₄

#11761627

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

268

—

186

C₈

—

187

C₈

—

188

C₂²

—

189

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

190

C₂²

#14524016

yes

characteristic

191

C₂²

#14524029

yes

characteristic

192

C₄

—

193

C₄

—

268

—

194

C₄

—

268

—

195

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

196

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

197

C₄

#14524015

yes

characteristic

198

C₄

#14524039

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#16863296

yes

characteristic

202

C₂

#16863670

yes

—

	subgroup	G/H
center	C₂²	#14524016
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#101×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₂×#101	D₆₇
hypocenter	C₆₇	#1979
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#14524016

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#101×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
12. C₂×#11761618	C₄	semidirect
102. C₆₇	#1979	semidirect

H ≅ G/N

type

12. C₂×#11761618

C₄

semidirect

102. C₆₇

#1979

semidirect

Subgroups — #22561121

Conjugacy classes of subgroups

Complements