Group #26029735 — subgroups

Subgroups — #26029735

Order 11392.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15668043	C₂	1	2	yes	maximal
2	C₄×#13083574	C₂	1	1	yes	maximal, characteristic
3	#19151805	C₂	1	4	yes	maximal, complemented
4	#19151800	C₂	1	4	yes	maximal, complemented
5	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151607	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	1	yes	characteristic
9	#15668043	—	2	8	no	—
10	C₂×#13083564	C₂²	1	4	yes	—
11	C₂×#13083561	C₂²	1	4	yes	—
12	C₂×#13083558	C₂²	1	2	yes	—
13	C₄×Dic₁₇₈	C₂²	1	2	yes	—
14	C₄×D₃₅₆	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₈₉	—	2	2	no	—
16	#15667971	C₂²	1	1	yes	characteristic
17	#15667969	—	2	1	no	—
18	C₄²×D₈₉	—	2	1	no	—
19	C₄⋊#10623494	C₂²	1	4	yes	—
20	#15667936	C₂²	1	8	yes	—
21	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₈₉	—	2	1	no	—
23	C₄⋊Q₈×C₈₉	—	2	1	no	—
24	C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
27	#13083574	—	2	2	no	—
28	#13083574	D₄	1	4	yes	—
29	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
30	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₈₉	—	2	1	no	—
32	C₂×C₄×D₈₉	—	2	1	no	—
33	C₂×C₄×D₈₉	—	2	2	no	—
34	#13083564	—	2	8	no	—
35	#13083561	—	2	8	no	—
36	#13083558	—	2	4	no	—
37	C₂×#10623494	C₂³	1	4	yes	—
38	C₄⋊Dic₈₉	C₂³	1	2	yes	—
39	#13083553	—	2	2	no	—
40	C₂²⋊Dic₈₉	—	2	1	no	—
41	#13083551	—	2	2	no	—
42	C₄×Dic₈₉	—	2	1	no	—
43	C₄×Dic₈₉	—	2	2	no	—
44	C₂×Q₈×C₈₉	—	2	2	no	—
45	C₂×Q₈×C₈₉	—	2	4	no	—
46	C₂×Q₈×C₈₉	—	2	4	no	—
47	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	D₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₈₉	—	2	4	no	—
52	C₄⋊C₄×C₈₉	—	2	4	no	—
53	C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂³	1	2	yes	—
56	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₈₉	—	2	1	no	—
58	Q₈×C₈₉	—	2	4	no	—
59	Q₈×C₈₉	—	2	4	no	—
60	Q₈×C₈₉	—	4	4	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	2	4	no	—
63	C₂×C₄×C₈₉	—	4	4	no	—
64	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₈₉	#41	1	2	yes	—
69	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₈₉	—	2	4	no	—
71	C₂×Dic₈₉	—	2	1	no	—
72	C₂×Dic₈₉	—	2	2	no	—
73	D₃₅₆	—	2	1	no	—
74	D₃₅₆	C₂×D₄	1	2	yes	—
75	C₄×D₈₉	—	2	4	no	—
76	C₄×D₈₉	—	4	4	no	—
77	Dic₁₇₈	—	2	1	no	—
78	Dic₁₇₈	C₂×D₄	1	2	yes	—
79	#10623494	—	2	4	no	—
80	C₂²×C₈₉	—	2	2	no	—
81	C₂²×C₈₉	#121	1	2	yes	—
82	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₈₉	—	2	2	no	—
84	C₂×D₈₉	—	4	2	no	—
85	C₄×C₈₉	—	2	2	no	—
86	C₄×C₈₉	—	4	4	no	—
87	C₄×C₈₉	#121	1	2	yes	—
88	C₄×C₈₉	#137	1	4	yes	complemented
89	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
91	Dic₈₉	—	2	2	no	—
92	Dic₈₉	—	4	2	no	—
93	C₂×C₈₉	—	2	2	no	—
94	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
95	C₂×C₈₉	C₂×#137	1	2	yes	—
96	D₈₉	—	4	2	no	—
97	#3039	—	89	1	no	maximal, Sylow
98	C₈₉	#3039	1	1	yes	characteristic, complemented, Sylow
99	C₂×#137	—	89	2	no	—
100	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
101	C₄×#41	—	89	1	no	—
102	#462	—	89	4	no	—
103	#461	—	89	4	no	—
104	#423	—	89	1	no	—
105	#419	—	89	2	no	—
106	C₂×#41	—	89	1	no	—
107	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
108	#137	—	178	8	no	—
109	C₂×Q₁₆	—	89	4	no	—
110	C₂×SD₁₆	—	89	4	no	—
111	C₂×M₁₆	—	89	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₈₉	1	4	yes	—
115	C₄×Q₈	—	89	2	no	—
116	C₄×D₄	—	89	2	no	—
117	C₄×D₄	—	178	2	no	—
118	#117	—	89	1	no	—
119	#117	—	178	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	178	1	no	—
122	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
123	C₄⋊C₈	—	89	4	no	—
124	#103	—	89	8	no	—
125	#41	—	89	4	no	—
126	#41	—	178	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	89	1	no	—
131	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
132	C₂×D₄	—	89	1	no	—
133	C₂×D₄	—	178	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	178	2	no	—
136	C₂²×C₄	—	178	1	no	—
137	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
138	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
139	Q₁₆	—	178	8	no	—
140	SD₁₆	—	178	8	no	—
141	M₁₆	—	178	4	no	—
142	C₂×C₈	—	89	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	89	2	no	—
146	C₄⋊C₄	—	178	2	no	—
147	C₂²⋊C₄	—	178	1	no	—
148	C₂²⋊C₄	—	178	2	no	—
149	C₄²	—	178	1	no	—
150	C₄²	—	178	2	no	—
151	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
152	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
153	C₄²	C₂²×D₈₉	1	2	yes	—
154	C₂³	—	178	1	no	—
155	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	89	2	no	—
160	Q₈	—	178	1	no	—
161	D₄	—	89	2	no	—
162	D₄	—	178	4	no	—
163	D₄	—	178	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	356	4	no	—
168	C₂×C₄	—	178	2	no	—
169	C₂×C₄	—	178	1	no	—
170	C₂×C₄	—	178	4	no	—
171	C₂×C₄	D₄×D₈₉	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
175	C₂×C₄	#13083573	1	2	yes	—
176	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
177	C₈	—	178	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	356	2	no	—
180	C₂²	—	178	2	no	—
181	C₂²	#15667983	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	356	2	no	—
186	C₄	—	178	2	no	—
187	C₄	#15667983	1	2	yes	—
188	C₄	#15668043	1	4	yes	complemented
189	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
190	C₄	C₂×#13083573	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	356	2	no	—
193	C₂	C₂×#15667983	1	1	yes	characteristic
194	C₂	C₂×#15668043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668043

C₂

yes

maximal

C₄×#13083574

C₂

yes

maximal, characteristic

#19151805

C₂

yes

maximal, complemented

#19151800

C₂

yes

maximal, complemented

#19151622

C₂

yes

maximal, characteristic, complemented

#19151607

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

#15668043

—

C₂×#13083564

C₂²

yes

—

C₂×#13083561

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×D₃₅₆

C₂²

yes

—

C₄×C₂²⋊D₈₉

—

#15667971

C₂²

yes

characteristic

#15667969

—

C₄²×D₈₉

—

C₄⋊#10623494

C₂²

yes

—

#15667936

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

D₄

yes

—

C₂×Dic₁₇₈

C₂³

yes

characteristic

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

C₂×D₄

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

C₂×D₄

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#137

yes

complemented

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#121

yes

characteristic

C₂×C₈₉

C₂×#137

yes

—

D₈₉

—

#3039

—

maximal, Sylow

C₈₉

#3039

yes

characteristic, complemented, Sylow

C₂×#137

—

100

C₂×C₄⋊Q₈

D₈₉

yes

characteristic, complemented

101

C₄×#41

—

102

#462

—

103

#461

—

104

#423

—

105

#419

—

106

C₂×#41

—

107

C₂²×Q₈

C₂×D₈₉

yes

—

108

#137

—

178

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₈₉

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

178

—

118

#117

—

119

#117

—

178

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

178

—

122

C₂×C₄²

C₂×D₈₉

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

—

178

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

C₂²×D₈₉

yes

—

132

C₂×D₄

—

133

C₂×D₄

—

178

—

134

C₂²×C₄

—

135

C₂²×C₄

—

178

—

136

C₂²×C₄

—

178

—

137

C₂²×C₄

C₂²⋊D₈₉

yes

—

138

C₂²×C₄

C₂²×D₈₉

yes

characteristic

139

Q₁₆

—

178

—

140

SD₁₆

—

178

—

141

M₁₆

—

178

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

178

—

147

C₂²⋊C₄

—

178

—

148

C₂²⋊C₄

—

178

—

149

C₄²

—

178

—

150

C₄²

—

178

—

151

C₄²

C₂²⋊D₈₉

yes

characteristic

152

C₄²

C₂²⋊D₈₉

yes

characteristic

153

C₄²

C₂²×D₈₉

yes

—

154

C₂³

—

178

—

155

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

178

—

161

D₄

—

162

D₄

—

178

—

163

D₄

—

178

—

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

D₄×D₈₉

yes

—

172

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

173

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

175

C₂×C₄

#13083573

yes

—

176

C₂×C₄

C₂³×D₈₉

yes

characteristic

177

C₈

—

178

—

178

C₂²

—

179

C₂²

—

356

—

180

C₂²

—

178

—

181

C₂²

#15667983

yes

—

182

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

356

—

186

C₄

—

178

—

187

C₄

#15667983

yes

—

188

C₄

#15668043

yes

complemented

189

C₄

C₂×D₄×D₈₉

yes

characteristic

190

C₄

C₂×#13083573

yes

characteristic

191

C₂

—

192

C₂

—

356

—

193

C₂

C₂×#15667983

yes

characteristic

194

C₂

C₂×#15668043

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#3039
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×#13083574

C₂²

hypercenter

C₂×C₄⋊Q₈

D₈₉

hypocenter

C₈₉

#3039

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #19151805	C₂	semidirect
4. #19151800	C₂	semidirect
5. #19151622	C₂	semidirect
6. #19151607	C₂	semidirect
7. C₂×C₄⋊Q₈×C₈₉	C₂	semidirect
88. C₄×C₈₉	#137	semidirect
98. C₈₉	#3039	semidirect
100. C₂×C₄⋊Q₈	D₈₉	semidirect
188. C₄	#15668043	semidirect

H ≅ G/N

type

3. #19151805

C₂

semidirect

4. #19151800

C₂

semidirect

5. #19151622

C₂

semidirect

6. #19151607

C₂

semidirect

7. C₂×C₄⋊Q₈×C₈₉

C₂

semidirect

88. C₄×C₈₉

#137

semidirect

98. C₈₉

#3039

semidirect

100. C₂×C₄⋊Q₈

D₈₉

semidirect

188. C₄

#15668043

semidirect

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

subgroup	index
C₂×#15668043	2
C₄×#13083574	2
#19151805	2
#19151800	2
#19151622	2
#19151607	2
C₂×C₄⋊Q₈×C₈₉	2
#3039	89

subgroup

index

C₂×C₄⋊Q₈×C₈₉

Subgroups — #26029735

Conjugacy classes of subgroups

Complements

Maximal subgroups