Group #26030022 — subgroups

Subgroups — C₂×#19151892 #26030022

Order 11392.

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×#15668011	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667977	C₂	1	1	yes	maximal, characteristic
3	#19151892	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15667945	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667937	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667935	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	—	2	1	no	—
10	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	#15668011	—	2	2	no	—
13	#15668011	C₂²	1	4	yes	—
14	#15667977	—	2	2	no	—
15	#15667977	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083553	—	2	1	no	—
18	C₂×#13083553	—	2	1	no	—
19	C₂×C₄×Dic₈₉	—	2	1	no	—
20	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
21	#15667945	C₂²	1	4	yes	—
22	#15667941	C₂²	1	4	yes	—
23	#15667937	C₂²	1	4	yes	—
24	#15667935	C₂²	1	4	yes	—
25	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
28	#107×C₈₉	C₂²	1	4	yes	—
29	C₂×Dic₁₇₈	—	2	1	no	—
30	C₂×Dic₁₇₈	—	2	2	no	—
31	C₂×Dic₁₇₈	—	2	4	no	—
32	C₂×Dic₁₇₈	—	4	4	no	—
33	C₂×Dic₁₇₈	Q₈	1	4	yes	complemented
34	C₂×Dic₁₇₈	C₂³	1	2	yes	—
35	C₂²×Dic₈₉	—	2	1	no	—
36	C₂²×Dic₈₉	—	2	1	no	—
37	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	C₂³	1	2	yes	—
41	C₄⋊Dic₈₉	—	2	1	no	—
42	C₄⋊Dic₈₉	C₂³	1	2	yes	—
43	#13083553	—	2	4	no	—
44	#13083553	—	2	4	no	—
45	C₄×Dic₈₉	—	2	4	no	—
46	C₄×Dic₈₉	D₄	1	2	yes	—
47	C₄×Dic₈₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₈₉	—	2	1	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
56	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
57	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₈₉	#41	1	2	yes	—
65	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₈₉	—	2	4	no	—
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	4	2	no	—
71	C₂×Dic₈₉	Q₁₆	1	4	yes	—
72	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
73	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
74	Dic₁₇₈	—	2	4	no	—
75	Dic₁₇₈	—	2	2	no	—
76	Dic₁₇₈	—	4	8	no	—
77	Dic₁₇₈	C₂×Q₈	1	4	yes	complemented
78	C₈×C₈₉	—	2	2	no	—
79	#10623494	—	2	2	no	—
80	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
82	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₈₉	—	4	2	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
89	Dic₈₉	—	2	4	no	—
90	Dic₈₉	—	2	2	no	—
91	Dic₈₉	—	4	2	no	—
92	Dic₈₉	C₂×Q₁₆	1	4	yes	—
93	C₂×C₈₉	#477	1	4	yes	complemented
94	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
96	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×#477	—	89	1	no	maximal, Sylow
98	C₈₉	C₂×#477	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄⋊Q₈	—	89	1	no	—
100	C₂×C₄×Q₈	—	89	1	no	—
101	#477	—	89	8	no	—
102	C₂×#107	—	89	1	no	—
103	C₂×#107	D₈₉	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	89	1	no	—
105	C₂×#103	—	89	1	no	—
106	C₂×#103	—	89	1	no	—
107	C₂²×Q₈	—	89	1	no	—
108	C₂²×Q₈	—	178	1	no	—
109	C₂²×C₈	—	89	1	no	—
110	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
111	C₄⋊Q₈	—	89	4	no	—
112	C₄⋊Q₈	—	178	2	no	—
113	C₄×Q₈	—	89	4	no	—
114	C₄×Q₈	—	178	2	no	—
115	C₂×C₄⋊C₄	—	89	1	no	—
116	C₂×C₄⋊C₄	—	178	1	no	—
117	C₂×C₄⋊C₄	—	178	1	no	—
118	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
120	C₂×C₄²	—	89	1	no	—
121	C₂×C₄²	—	178	1	no	—
122	#107	—	89	4	no	—
123	#107	C₂×D₈₉	1	4	yes	—
124	C₄⋊C₈	—	89	4	no	—
125	#103	—	89	4	no	—
126	#103	—	89	4	no	—
127	C₂×Q₈	—	89	4	no	—
128	C₂×Q₈	—	89	2	no	—
129	C₂×Q₈	—	356	4	no	—
130	C₂×Q₈	—	178	4	no	—
131	C₂×Q₈	—	178	1	no	—
132	C₂×Q₈	—	178	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	89	1	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	—	178	1	no	—
138	C₂²×C₄	—	178	1	no	—
139	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	89	2	no	—
142	C₂×C₈	—	178	2	no	—
143	C₂×C₈	C₂²×D₈₉	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	89	2	no	—
147	C₄⋊C₄	—	178	4	no	—
148	C₄⋊C₄	—	178	4	no	—
149	C₄⋊C₄	—	178	1	no	—
150	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
151	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
152	C₄²	—	89	2	no	—
153	C₄²	—	89	2	no	—
154	C₄²	—	178	4	no	—
155	C₂³	D₄×D₈₉	1	1	yes	characteristic
156	Q₈	—	89	4	no	—
157	Q₈	—	356	8	no	—
158	Q₈	—	178	4	no	—
159	Q₈	—	178	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	—	89	4	no	—
165	C₂×C₄	—	89	2	no	—
166	C₂×C₄	—	89	2	no	—
167	C₂×C₄	—	356	2	no	—
168	C₂×C₄	—	178	2	no	—
169	C₂×C₄	—	178	4	no	—
170	C₂×C₄	—	178	2	no	—
171	C₂×C₄	—	178	2	no	—
172	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₈₉	1	2	yes	—
174	C₂×C₄	#13083574	1	2	yes	—
175	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	178	2	no	—
178	C₂²	#15667990	1	2	yes	—
179	C₂²	Q₁₆×D₈₉	1	2	yes	—
180	C₂²	#15668036	1	2	yes	—
181	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	89	4	no	—
185	C₄	—	356	2	no	—
186	C₄	—	178	4	no	—
187	C₄	—	178	2	no	—
188	C₄	#15667990	1	2	yes	—
189	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
190	C₄	C₂×#13083574	1	1	yes	characteristic
191	C₂	#19151892	1	4	yes	direct product factor
192	C₂	C₂×#15667990	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
194	C₂	C₂×#15668036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668011

C₂

yes

maximal, characteristic

C₂×#15667977

C₂

yes

maximal, characteristic

#19151892

C₂

yes

maximal, direct product factor

C₂×#15667945

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal, characteristic

C₂×#15667935

C₂

yes

maximal, characteristic

C₂×#107×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15668011

—

#15668011

C₂²

yes

—

#15667977

—

#15667977

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×#13083553

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667945

C₂²

yes

—

#15667941

C₂²

yes

—

#15667937

C₂²

yes

—

#15667935

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

complemented

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

characteristic

C₂²×Dic₈₉

D₄

yes

characteristic

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

Q₁₆

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₂×Q₈

yes

complemented

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×Q₁₆

yes

—

C₂×C₈₉

#477

yes

complemented

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

100

C₂×C₄×Q₈

—

101

#477

—

102

C₂×#107

—

103

C₂×#107

D₈₉

yes

characteristic

104

C₂×C₄⋊C₈

—

105

C₂×#103

—

106

C₂×#103

—

107

C₂²×Q₈

—

108

C₂²×Q₈

—

178

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₈₉

yes

characteristic

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

178

—

113

C₄×Q₈

—

114

C₄×Q₈

—

178

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

178

—

117

C₂×C₄⋊C₄

—

178

—

118

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

178

—

122

#107

—

123

#107

C₂×D₈₉

yes

—

124

C₄⋊C₈

—

125

#103

—

126

#103

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

356

—

130

C₂×Q₈

—

178

—

131

C₂×Q₈

—

178

—

132

C₂×Q₈

—

178

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

178

—

138

C₂²×C₄

—

178

—

139

C₂²×C₄

C₂²×D₈₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

178

—

143

C₂×C₈

C₂²×D₈₉

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

178

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄⋊C₄

C₂²×D₈₉

yes

—

151

C₄⋊C₄

C₂²×D₈₉

yes

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

178

—

155

C₂³

D₄×D₈₉

yes

characteristic

156

Q₈

—

157

Q₈

—

356

—

158

Q₈

—

178

—

159

Q₈

—

178

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

356

—

168

C₂×C₄

—

178

—

169

C₂×C₄

—

178

—

170

C₂×C₄

—

178

—

171

C₂×C₄

—

178

—

172

C₂×C₄

D₄×D₈₉

yes

characteristic

173

C₂×C₄

Q₈×D₈₉

yes

—

174

C₂×C₄

#13083574

yes

—

175

C₂×C₄

C₂³×D₈₉

yes

characteristic

176

C₈

—

177

C₈

—

178

—

178

C₂²

#15667990

yes

—

179

C₂²

Q₁₆×D₈₉

yes

—

180

C₂²

#15668036

yes

—

181

C₂²

C₂×D₄×D₈₉

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

356

—

186

C₄

—

178

—

187

C₄

—

178

—

188

C₄

#15667990

yes

—

189

C₄

C₂×Q₈×D₈₉

yes

characteristic

190

C₄

C₂×#13083574

yes

characteristic

191

C₂

#19151892

yes

direct product factor

192

C₂

C₂×#15667990

yes

characteristic

193

C₂

C₂×Q₁₆×D₈₉

yes

characteristic

194

C₂

C₂×#15668036

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×#107	D₈₉
hypocenter	C₈₉	C₂×#477
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#107×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #19151892	C₂	direct (N × H)
33. C₂×Dic₁₇₈	Q₈	semidirect
77. Dic₁₇₈	C₂×Q₈	semidirect
93. C₂×C₈₉	#477	semidirect
98. C₈₉	C₂×#477	semidirect
191. C₂	#19151892	direct (N × H)

H ≅ G/N

type

3. #19151892

C₂

direct (N × H)

33. C₂×Dic₁₇₈

Q₈

semidirect

77. Dic₁₇₈

C₂×Q₈

semidirect

93. C₂×C₈₉

#477

semidirect

98. C₈₉

C₂×#477

semidirect

191. C₂

#19151892

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15668011	2
C₂×#15667977	2
#19151892	2
C₂×#15667945	2
C₂×#15667941	2
C₂×#15667937	2
C₂×#15667935	2
C₂×#107×C₈₉	2
C₂×#477	89

subgroup

index

C₂×#107×C₈₉

Subgroups — C₂×#19151892 #26030022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151892 #26030022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151892 #26030022