Group #26025462 — subgroups

Subgroups — #26025462

Order 11392.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151519	C₂	1	4	yes	maximal
4	#19151505	C₂	1	1	yes	maximal, characteristic
5	#19151495	C₂	1	4	yes	maximal
6	#19151468	C₂	1	1	yes	maximal, characteristic
7	#19151467	C₂	1	1	yes	maximal, characteristic
8	C₄×#13083553	C₂	1	1	yes	maximal, characteristic
9	#391×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
11	#15668011	—	2	4	no	—
12	Q₈×Dic₈₉	—	2	4	no	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
17	C₂×#13083553	C₂²	1	2	yes	—
18	C₂×#13083553	C₂²	1	4	yes	—
19	C₂×#13083553	C₂²	1	2	yes	—
20	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
23	#15667919	C₂²	1	2	yes	—
24	#15667919	C₂²	1	4	yes	—
25	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	#95×C₈₉	C₂²	1	4	yes	—
29	C₂×Dic₁₇₈	—	2	4	no	—
30	C₂×Dic₁₇₈	—	4	4	no	—
31	C₂²×Dic₈₉	C₂³	1	2	yes	—
32	C₂²×Dic₈₉	C₂³	1	2	yes	—
33	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₈₉	C₂³	1	2	yes	—
35	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₈₉	—	2	2	no	—
37	C₄⋊Dic₈₉	—	2	2	no	—
38	C₄⋊Dic₈₉	—	2	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	#13083553	—	2	4	no	—
41	#13083553	—	4	4	no	—
42	#13083553	Q₈	1	8	yes	complemented
43	C₄×Dic₈₉	—	2	2	no	—
44	C₄×Dic₈₉	—	2	4	no	—
45	C₄×Dic₈₉	D₄	1	4	yes	—
46	C₂×Q₈×C₈₉	—	2	2	no	—
47	C₂×Q₈×C₈₉	—	2	2	no	—
48	C₂×Q₈×C₈₉	Q₈	1	4	yes	complemented
49	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₄²×C₈₉	—	2	2	no	—
55	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
56	Q₈×C₈₉	—	2	8	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	4	no	—
63	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	#41	1	2	yes	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	C₂×Dic₈₉	—	2	2	no	—
73	C₂×Dic₈₉	—	2	4	no	—
74	C₂×Dic₈₉	—	2	4	no	—
75	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
76	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
77	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
78	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
79	Dic₁₇₈	—	4	8	no	—
80	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
83	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₈₉	—	2	4	no	—
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
89	Dic₈₉	—	2	2	no	—
90	Dic₈₉	—	2	4	no	—
91	Dic₈₉	—	4	2	no	—
92	Dic₈₉	—	4	2	no	—
93	Dic₈₉	C₄⋊Q₈	1	4	yes	—
94	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₈₉	#535	1	1	yes	characteristic
97	C₂×C₈₉	#554	1	2	yes	—
98	C₂×C₈₉	Q₈²	1	2	yes	—
99	#2567	—	89	1	no	maximal, Sylow
100	C₈₉	#2567	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	89	1	no	—
102	C₂×C₄×Q₈	—	89	1	no	—
103	#395	—	89	4	no	—
104	#391	—	89	1	no	—
105	#391	D₈₉	1	1	yes	characteristic
106	#389	—	89	4	no	—
107	#384	—	89	1	no	—
108	#384	—	89	1	no	—
109	C₄×C₄⋊C₄	—	89	1	no	—
110	C₂²×Q₈	—	89	2	no	—
111	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
112	C₄⋊Q₈	—	178	4	no	—
113	C₄×Q₈	—	178	4	no	—
114	C₂×C₄⋊C₄	—	89	2	no	—
115	C₂×C₄⋊C₄	—	89	4	no	—
116	C₂×C₄⋊C₄	—	89	4	no	—
117	C₂×C₄⋊C₄	—	89	1	no	—
118	C₂×C₄⋊C₄	—	89	2	no	—
119	C₂×C₄⋊C₄	—	89	1	no	—
120	C₂×C₄⋊C₄	—	89	2	no	—
121	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
122	C₂×C₄²	—	89	1	no	—
123	C₂×C₄²	—	89	2	no	—
124	C₂×C₄²	—	89	1	no	—
125	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
126	#95	—	89	2	no	—
127	#95	—	89	4	no	—
128	#95	C₂×D₈₉	1	4	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	356	4	no	—
132	C₂×Q₈	—	178	4	no	—
133	C₂×Q₈	Dic₁₇₈	1	4	yes	complemented
134	C₂²×C₄	—	89	1	no	—
135	C₂²×C₄	—	89	2	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	—	89	2	no	—
138	C₂²×C₄	—	89	2	no	—
139	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
140	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
141	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
142	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	89	8	no	—
145	C₄⋊C₄	—	356	4	no	—
146	C₄⋊C₄	—	356	4	no	—
147	C₄⋊C₄	—	178	4	no	—
148	C₄⋊C₄	—	178	4	no	—
149	C₄⋊C₄	—	178	2	no	—
150	C₄⋊C₄	—	178	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	89	4	no	—
153	C₄²	—	178	4	no	—
154	C₄²	—	178	2	no	—
155	C₂³	C₂³×D₈₉	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	Q₈	—	356	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	89	2	no	—
165	C₂×C₄	—	89	4	no	—
166	C₂×C₄	—	89	2	no	—
167	C₂×C₄	—	89	4	no	—
168	C₂×C₄	—	178	2	no	—
169	C₂×C₄	—	178	2	no	—
170	C₂×C₄	—	178	4	no	—
171	C₂×C₄	—	178	4	no	—
172	C₂×C₄	—	178	2	no	—
173	C₂×C₄	—	178	2	no	—
174	C₂×C₄	—	178	2	no	—
175	C₂×C₄	D₄×D₈₉	1	2	yes	—
176	C₂×C₄	Q₈×D₈₉	1	4	yes	—
177	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
178	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
179	C₂×C₄	#13083573	1	2	yes	—
180	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₈₉	1	2	yes	—
182	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
183	C₂²	C₂×#13083573	1	1	yes	characteristic
184	C₂²	#15668058	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	89	4	no	—
189	C₄	—	356	2	no	—
190	C₄	—	356	2	no	—
191	C₄	—	178	2	no	—
192	C₄	—	178	4	no	—
193	C₄	#15667989	1	4	yes	—
194	C₂	C₂×#15667989	1	1	yes	characteristic
195	C₂	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
196	C₂	#19152166	1	1	yes	characteristic
197	C₂	#19152306	1	2	yes	—
198	C₂	Q₈×Dic₁₇₈	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668011

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151519

C₂

yes

maximal

#19151505

C₂

yes

maximal, characteristic

#19151495

C₂

yes

maximal

#19151468

C₂

yes

maximal, characteristic

#19151467

C₂

yes

maximal, characteristic

C₄×#13083553

C₂

yes

maximal, characteristic

#391×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668011

—

Q₈×Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

Q₈

yes

complemented

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₄⋊Q₈

yes

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

Q₈²

yes

—

#2567

—

maximal, Sylow

100

C₈₉

#2567

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

102

C₂×C₄×Q₈

—

103

#395

—

104

#391

—

105

#391

D₈₉

yes

characteristic

106

#389

—

107

#384

—

108

#384

—

109

C₄×C₄⋊C₄

—

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₈₉

yes

characteristic

112

C₄⋊Q₈

—

178

—

113

C₄×Q₈

—

178

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×D₈₉

yes

characteristic

126

#95

—

127

#95

—

128

#95

C₂×D₈₉

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

356

—

132

C₂×Q₈

—

178

—

133

C₂×Q₈

Dic₁₇₈

yes

complemented

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₈₉

yes

—

140

C₂²×C₄

C₂²×D₈₉

yes

—

141

C₂²×C₄

C₂²×D₈₉

yes

—

142

C₂²×C₄

C₂²×D₈₉

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

178

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄⋊C₄

—

178

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

178

—

154

C₄²

—

178

—

155

C₂³

C₂³×D₈₉

yes

characteristic

156

Q₈

—

157

Q₈

—

356

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

178

—

169

C₂×C₄

—

178

—

170

C₂×C₄

—

178

—

171

C₂×C₄

—

178

—

172

C₂×C₄

—

178

—

173

C₂×C₄

—

178

—

174

C₂×C₄

—

178

—

175

C₂×C₄

D₄×D₈₉

yes

—

176

C₂×C₄

Q₈×D₈₉

yes

—

177

C₂×C₄

C₂×Dic₁₇₈

yes

—

178

C₂×C₄

C₂×Dic₁₇₈

yes

—

179

C₂×C₄

#13083573

yes

—

180

C₂²

C₂×D₄×D₈₉

yes

characteristic

181

C₂²

C₂×Q₈×D₈₉

yes

—

182

C₂²

C₂²×Dic₁₇₈

yes

characteristic

183

C₂²

C₂×#13083573

yes

characteristic

184

C₂²

#15668058

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

356

—

190

C₄

—

356

—

191

C₄

—

178

—

192

C₄

—

178

—

193

C₄

#15667989

yes

—

194

C₂

C₂×#15667989

yes

characteristic

195

C₂

C₄⋊Q₈×D₈₉

yes

characteristic

196

C₂

#19152166

yes

characteristic

197

C₂

#19152306

yes

—

198

C₂

Q₈×Dic₁₇₈

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#391×C₈₉	C₂
hypercenter	#391	D₈₉
hypocenter	C₈₉	#2567
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#391×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
42. #13083553	Q₈	semidirect
48. C₂×Q₈×C₈₉	Q₈	semidirect
100. C₈₉	#2567	semidirect
133. C₂×Q₈	Dic₁₇₈	semidirect

H ≅ G/N

type

42. #13083553

Q₈

semidirect

48. C₂×Q₈×C₈₉

Q₈

semidirect

100. C₈₉

#2567

semidirect

133. C₂×Q₈

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15668011	2
C₂×Q₈×Dic₈₉	2
#19151519	2
#19151505	2
#19151495	2
#19151468	2
#19151467	2
C₄×#13083553	2
#391×C₈₉	2
#2567	89

subgroup

index

C₂×Q₈×Dic₈₉

#391×C₈₉

Subgroups — #26025462

Conjugacy classes of subgroups

Complements

Maximal subgroups