Group #22569986 — subgroups

Subgroups — C₂×#16863703 #22569986

Order 8576.

384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14524035	C₂	1	1	yes	maximal, characteristic
2	#16863703	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523976	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523974	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523967	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523967	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#14524035	—	2	2	no	—
12	#14524035	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₇⋊C₄	—	2	1	no	—
16	C₂×Dic₆₇⋊C₄	—	2	1	no	—
17	#14523976	C₂²	1	4	yes	—
18	#14523974	C₂²	1	4	yes	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	#14523967	C₂²	1	4	yes	—
21	#14523967	C₂²	1	4	yes	—
22	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₆₇	—	2	2	no	—
24	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	—	2	1	no	—
28	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₆₇	—	2	1	no	—
30	C₂²×Dic₆₇	—	2	1	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₄⋊Dic₆₇	—	2	1	no	—
36	C₄⋊Dic₆₇	—	2	1	no	—
37	C₄⋊Dic₆₇	C₂³	1	2	yes	—
38	C₄⋊Dic₆₇	C₂³	1	2	yes	—
39	Dic₆₇⋊C₄	—	2	4	no	—
40	Dic₆₇⋊C₄	—	2	4	no	—
41	C₂×Q₈×C₆₇	—	2	1	no	—
42	C₂×Q₈×C₆₇	—	2	2	no	—
43	C₂×Q₈×C₆₇	Q₈	1	4	yes	—
44	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₆₇	—	2	4	no	—
51	C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₄²×C₆₇	—	2	4	no	—
54	C₄²×C₆₇	D₄	1	2	yes	—
55	C₄²×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₆₇	—	2	4	no	—
58	Q₈×C₆₇	—	2	2	no	—
59	Q₈×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₆₇	#41	1	2	yes	—
71	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₆₇	—	2	2	no	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	—	4	2	no	—
75	C₂×Dic₆₇	—	4	2	no	—
76	C₆₇⋊C₈	—	2	2	no	—
77	C₆₇⋊C₈	—	2	2	no	—
78	C₂²×C₆₇	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₆₇	#135	1	2	yes	—
80	C₂²×C₆₇	#137	1	2	yes	—
81	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₆₇	—	2	4	no	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	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	—	4	2	no	—
90	Dic₆₇	—	4	2	no	—
91	C₂×C₆₇	#479	1	4	yes	complemented
92	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
94	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
95	C₂×#479	—	67	1	no	maximal, Sylow
96	C₆₇	C₂×#479	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	67	1	no	—
98	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
99	#479	—	67	8	no	—
100	C₂×#107	—	67	1	no	—
101	C₂×#106	—	67	1	no	—
102	C₂×C₄⋊C₈	—	67	1	no	—
103	C₂×#103	—	67	1	no	—
104	C₂×#103	—	67	1	no	—
105	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
106	C₂²×C₈	—	67	1	no	—
107	C₂²×C₈	—	67	1	no	—
108	#125	—	67	4	no	—
109	#125	—	134	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₄×Q₈	C₂×D₆₇	1	4	yes	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	67	1	no	—
114	C₂×C₄⋊C₄	—	67	1	no	—
115	C₂×C₄⋊C₄	—	134	1	no	—
116	C₂×C₄⋊C₄	—	134	1	no	—
117	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
120	#107	—	67	4	no	—
121	#106	—	67	4	no	—
122	C₄⋊C₈	—	67	4	no	—
123	#103	—	67	4	no	—
124	#103	—	67	4	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	Dic₁₃₄	1	4	yes	—
128	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	134	1	no	—
132	C₂²×C₄	—	134	1	no	—
133	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
136	C₂×C₈	—	67	2	no	—
137	C₂×C₈	—	67	2	no	—
138	C₂×C₈	—	134	2	no	—
139	C₂×C₈	—	134	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	67	2	no	—
143	C₄⋊C₄	—	67	2	no	—
144	C₄⋊C₄	—	134	1	no	—
145	C₄⋊C₄	—	134	1	no	—
146	C₄⋊C₄	—	134	4	no	—
147	C₄⋊C₄	—	134	4	no	—
148	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	C₂²⋊D₆₇	1	2	yes	—
151	C₄²	C₂²×D₆₇	1	2	yes	—
152	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	Q₈	—	2	4	no	—
154	Q₈	—	2	2	no	—
155	Q₈	C₂×Dic₁₃₄	1	4	yes	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	268	2	no	—
163	C₂×C₄	—	268	2	no	—
164	C₂×C₄	—	134	2	no	—
165	C₂×C₄	—	134	2	no	—
166	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
170	C₂×C₄	#11761628	1	2	yes	—
171	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
172	C₈	—	134	2	no	—
173	C₈	—	134	2	no	—
174	C₂²	#14524022	1	2	yes	—
175	C₂²	#14524063	1	2	yes	—
176	C₂²	#14524074	1	2	yes	—
177	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	268	2	no	—
183	C₄	—	268	2	no	—
184	C₄	#14524022	1	2	yes	—
185	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
186	C₄	C₂×#11761628	1	1	yes	characteristic
187	C₂	#16863703	1	4	yes	direct product factor
188	C₂	C₂×#14524022	1	1	yes	characteristic
189	C₂	C₂×#14524063	1	1	yes	characteristic
190	C₂	C₂×#14524074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

#16863703

C₂

yes

maximal, direct product factor

C₂×#14523976

C₂

yes

maximal, characteristic

C₂×#14523974

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524035

—

#14524035

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×Dic₆₇⋊C₄

—

#14523976

C₂²

yes

—

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523967

C₂²

yes

—

#14523967

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

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

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

#135

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#479

yes

complemented

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×Q₈

D₆₇

yes

characteristic

#479

—

100

C₂×#107

—

101

C₂×#106

—

102

C₂×C₄⋊C₈

—

103

C₂×#103

—

104

C₂×#103

—

105

C₂²×Q₈

C₂×D₆₇

yes

characteristic

106

C₂²×C₈

—

107

C₂²×C₈

—

108

#125

—

109

#125

—

134

—

110

C₄×Q₈

—

111

C₄×Q₈

C₂×D₆₇

yes

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

134

—

116

C₂×C₄⋊C₄

—

134

—

117

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

118

C₂×C₄²

—

119

C₂×C₄²

C₂×D₆₇

yes

characteristic

120

#107

—

121

#106

—

122

C₄⋊C₈

—

123

#103

—

124

#103

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

Dic₁₃₄

yes

—

128

C₂×Q₈

C₂²×D₆₇

yes

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

134

—

132

C₂²×C₄

—

134

—

133

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

134

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

135

C₂²×C₄

C₂²×D₆₇

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

134

—

139

C₂×C₈

—

134

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

134

—

145

C₄⋊C₄

—

134

—

146

C₄⋊C₄

—

134

—

147

C₄⋊C₄

—

134

—

148

C₄⋊C₄

C₂²×D₆₇

yes

—

149

C₄²

—

150

C₄²

C₂²⋊D₆₇

yes

—

151

C₄²

C₂²×D₆₇

yes

—

152

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

153

Q₈

—

154

Q₈

—

155

Q₈

C₂×Dic₁₃₄

yes

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

268

—

163

C₂×C₄

—

268

—

164

C₂×C₄

—

134

—

165

C₂×C₄

—

134

—

166

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

167

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

168

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

169

C₂×C₄

C₂×Dic₁₃₄

yes

—

170

C₂×C₄

#11761628

yes

—

171

C₂×C₄

C₂³×D₆₇

yes

characteristic

172

C₈

—

134

—

173

C₈

—

134

—

174

C₂²

#14524022

yes

—

175

C₂²

#14524063

yes

—

176

C₂²

#14524074

yes

—

177

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

268

—

183

C₄

—

268

—

184

C₄

#14524022

yes

—

185

C₄

C₂²×Dic₁₃₄

yes

characteristic

186

C₄

C₂×#11761628

yes

characteristic

187

C₂

#16863703

yes

direct product factor

188

C₂

C₂×#14524022

yes

characteristic

189

C₂

C₂×#14524063

yes

characteristic

190

C₂

C₂×#14524074

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#479
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₆₇

hypocenter

C₆₇

C₂×#479

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #16863703	C₂	direct (N × H)
91. C₂×C₆₇	#479	semidirect
96. C₆₇	C₂×#479	semidirect
187. C₂	#16863703	direct (N × H)

H ≅ G/N

type

2. #16863703

C₂

direct (N × H)

91. C₂×C₆₇

#479

semidirect

96. C₆₇

C₂×#479

semidirect

187. C₂

#16863703

direct (N × H)

Subgroups — C₂×#16863703 #22569986

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863703 #22569986

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863703 #22569986