Group #22569973 — subgroups

Subgroups — C₂×#16863680 #22569973

Order 8576.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14524042	C₂	1	1	yes	maximal, characteristic
2	C₂×#14524008	C₂	1	1	yes	maximal, characteristic
3	#16863680	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14523974	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523968	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523966	C₂	1	1	yes	maximal, characteristic
8	C₂×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₃₄	—	2	1	no	—
10	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#14524042	—	2	2	no	—
13	#14524042	C₂²	1	4	yes	—
14	#14524008	—	2	2	no	—
15	#14524008	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₇⋊C₄	—	2	1	no	—
18	C₂×Dic₆₇⋊C₄	—	2	1	no	—
19	C₂×C₄×Dic₆₇	—	2	1	no	—
20	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
21	#14523974	C₂²	1	4	yes	—
22	Dic₆₇⋊C₈	C₂²	1	4	yes	—
23	#14523968	C₂²	1	4	yes	—
24	#14523966	C₂²	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	#106×C₆₇	C₂²	1	4	yes	—
29	C₂×Dic₁₃₄	—	2	1	no	—
30	C₂×Dic₁₃₄	—	2	2	no	—
31	C₂×Dic₁₃₄	—	2	4	no	—
32	C₂×Dic₁₃₄	—	4	4	no	—
33	C₂×Dic₁₃₄	Q₈	1	4	yes	complemented
34	C₂×Dic₁₃₄	C₂³	1	2	yes	—
35	C₂²×Dic₆₇	—	2	1	no	—
36	C₂²×Dic₆₇	—	2	1	no	—
37	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
38	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₆₇	—	2	1	no	—
42	C₄⋊Dic₆₇	C₂³	1	2	yes	—
43	Dic₆₇⋊C₄	—	2	4	no	—
44	Dic₆₇⋊C₄	—	2	4	no	—
45	C₄×Dic₆₇	—	2	4	no	—
46	C₄×Dic₆₇	D₄	1	2	yes	—
47	C₄×Dic₆₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	C₂³	1	2	yes	—
53	C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	#41	1	2	yes	—
65	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	4	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	4	2	no	—
71	C₂×Dic₆₇	SD₁₆	1	4	yes	—
72	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
73	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
74	Dic₁₃₄	—	2	4	no	—
75	Dic₁₃₄	—	2	2	no	—
76	Dic₁₃₄	—	4	8	no	—
77	Dic₁₃₄	C₂×Q₈	1	4	yes	complemented
78	C₈×C₆₇	—	2	2	no	—
79	C₆₇⋊C₈	—	2	2	no	—
80	C₂²×C₆₇	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
82	C₂²×C₆₇	#137	1	2	yes	—
83	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₆₇	—	4	2	no	—
85	C₄×C₆₇	—	4	2	no	—
86	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
89	Dic₆₇	—	2	4	no	—
90	Dic₆₇	—	2	2	no	—
91	Dic₆₇	—	4	2	no	—
92	Dic₆₇	C₂×SD₁₆	1	4	yes	—
93	C₂×C₆₇	#475	1	4	yes	complemented
94	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
97	C₂×#475	—	67	1	no	maximal, Sylow
98	C₆₇	C₂×#475	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄⋊Q₈	—	67	1	no	—
100	C₂×C₄×Q₈	—	67	1	no	—
101	#475	—	67	8	no	—
102	C₂×#106	—	67	1	no	—
103	C₂×#106	D₆₇	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	67	1	no	—
105	C₂×#103	—	67	1	no	—
106	C₂×#103	—	67	1	no	—
107	C₂²×Q₈	—	67	1	no	—
108	C₂²×Q₈	—	134	1	no	—
109	C₂²×C₈	—	67	1	no	—
110	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	67	4	no	—
112	C₄⋊Q₈	—	134	2	no	—
113	C₄×Q₈	—	67	4	no	—
114	C₄×Q₈	—	134	2	no	—
115	C₂×C₄⋊C₄	—	67	1	no	—
116	C₂×C₄⋊C₄	—	134	1	no	—
117	C₂×C₄⋊C₄	—	134	1	no	—
118	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
120	C₂×C₄²	—	67	1	no	—
121	C₂×C₄²	—	134	1	no	—
122	#106	—	67	4	no	—
123	#106	C₂×D₆₇	1	4	yes	—
124	C₄⋊C₈	—	67	4	no	—
125	#103	—	67	4	no	—
126	#103	—	67	4	no	—
127	C₂×Q₈	—	268	4	no	—
128	C₂×Q₈	—	67	4	no	—
129	C₂×Q₈	—	67	2	no	—
130	C₂×Q₈	—	134	4	no	—
131	C₂×Q₈	—	134	1	no	—
132	C₂×Q₈	—	134	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	67	1	no	—
136	C₂²×C₄	—	67	1	no	—
137	C₂²×C₄	—	134	1	no	—
138	C₂²×C₄	—	134	1	no	—
139	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	67	2	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	C₂²×D₆₇	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	67	2	no	—
147	C₄⋊C₄	—	134	4	no	—
148	C₄⋊C₄	—	134	4	no	—
149	C₄⋊C₄	—	134	1	no	—
150	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
151	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
152	C₄²	—	67	2	no	—
153	C₄²	—	67	2	no	—
154	C₄²	—	134	4	no	—
155	C₂³	D₄×D₆₇	1	1	yes	characteristic
156	Q₈	—	268	8	no	—
157	Q₈	—	67	4	no	—
158	Q₈	—	134	4	no	—
159	Q₈	—	134	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	268	2	no	—
165	C₂×C₄	—	67	4	no	—
166	C₂×C₄	—	67	2	no	—
167	C₂×C₄	—	67	2	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	—	134	4	no	—
170	C₂×C₄	—	134	2	no	—
171	C₂×C₄	—	134	2	no	—
172	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₆₇	1	2	yes	—
174	C₂×C₄	#11761628	1	2	yes	—
175	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	134	2	no	—
178	C₂²	#14524021	1	2	yes	—
179	C₂²	SD₁₆×D₆₇	1	2	yes	—
180	C₂²	#14524072	1	2	yes	—
181	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	268	2	no	—
185	C₄	—	67	4	no	—
186	C₄	—	134	4	no	—
187	C₄	—	134	2	no	—
188	C₄	#14524021	1	2	yes	—
189	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
190	C₄	C₂×#11761628	1	1	yes	characteristic
191	C₂	#16863680	1	4	yes	direct product factor
192	C₂	C₂×#14524021	1	1	yes	characteristic
193	C₂	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
194	C₂	C₂×#14524072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524042

C₂

yes

maximal, characteristic

C₂×#14524008

C₂

yes

maximal, characteristic

#16863680

C₂

yes

maximal, direct product factor

C₂×#14523974

C₂

yes

maximal, characteristic

C₂×Dic₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#14523968

C₂

yes

maximal, characteristic

C₂×#14523966

C₂

yes

maximal, characteristic

C₂×#106×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524042

—

#14524042

C₂²

yes

—

#14524008

—

#14524008

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523974

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523966

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₈

yes

complemented

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

SD₁₆

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂×Q₈

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

C₂×SD₁₆

yes

—

C₂×C₆₇

#475

yes

complemented

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

100

C₂×C₄×Q₈

—

101

#475

—

102

C₂×#106

—

103

C₂×#106

D₆₇

yes

characteristic

104

C₂×C₄⋊C₈

—

105

C₂×#103

—

106

C₂×#103

—

107

C₂²×Q₈

—

108

C₂²×Q₈

—

134

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₆₇

yes

characteristic

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

134

—

113

C₄×Q₈

—

114

C₄×Q₈

—

134

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

134

—

117

C₂×C₄⋊C₄

—

134

—

118

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

134

—

122

#106

—

123

#106

C₂×D₆₇

yes

—

124

C₄⋊C₈

—

125

#103

—

126

#103

—

127

C₂×Q₈

—

268

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

134

—

131

C₂×Q₈

—

134

—

132

C₂×Q₈

—

134

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

134

—

138

C₂²×C₄

—

134

—

139

C₂²×C₄

C₂²×D₆₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

134

—

143

C₂×C₈

C₂²×D₆₇

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

134

—

148

C₄⋊C₄

—

134

—

149

C₄⋊C₄

—

134

—

150

C₄⋊C₄

C₂²×D₆₇

yes

—

151

C₄⋊C₄

C₂²×D₆₇

yes

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

134

—

155

C₂³

D₄×D₆₇

yes

characteristic

156

Q₈

—

268

—

157

Q₈

—

158

Q₈

—

134

—

159

Q₈

—

134

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

268

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

134

—

169

C₂×C₄

—

134

—

170

C₂×C₄

—

134

—

171

C₂×C₄

—

134

—

172

C₂×C₄

D₄×D₆₇

yes

characteristic

173

C₂×C₄

Q₈×D₆₇

yes

—

174

C₂×C₄

#11761628

yes

—

175

C₂×C₄

C₂³×D₆₇

yes

characteristic

176

C₈

—

177

C₈

—

134

—

178

C₂²

#14524021

yes

—

179

C₂²

SD₁₆×D₆₇

yes

—

180

C₂²

#14524072

yes

—

181

C₂²

C₂×D₄×D₆₇

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

268

—

185

C₄

—

186

C₄

—

134

—

187

C₄

—

134

—

188

C₄

#14524021

yes

—

189

C₄

C₂×Q₈×D₆₇

yes

characteristic

190

C₄

C₂×#11761628

yes

characteristic

191

C₂

#16863680

yes

direct product factor

192

C₂

C₂×#14524021

yes

characteristic

193

C₂

C₂×SD₁₆×D₆₇

yes

characteristic

194

C₂

C₂×#14524072

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#106×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#106	D₆₇
hypocenter	C₆₇	C₂×#475
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#106×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #16863680	C₂	direct (N × H)
33. C₂×Dic₁₃₄	Q₈	semidirect
77. Dic₁₃₄	C₂×Q₈	semidirect
93. C₂×C₆₇	#475	semidirect
98. C₆₇	C₂×#475	semidirect
191. C₂	#16863680	direct (N × H)

H ≅ G/N

type

3. #16863680

C₂

direct (N × H)

33. C₂×Dic₁₃₄

Q₈

semidirect

77. Dic₁₃₄

C₂×Q₈

semidirect

93. C₂×C₆₇

#475

semidirect

98. C₆₇

C₂×#475

semidirect

191. C₂

#16863680

direct (N × H)

Subgroups — C₂×#16863680 #22569973

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863680 #22569973

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863680 #22569973