Group #36458879 — subgroups

Subgroups — #36458879

Order 22784.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#26021015	C₂	1	2	yes	maximal
2	#26020411	C₂	1	1	yes	maximal, characteristic
3	#26018135	C₂	1	1	yes	maximal, characteristic
4	#26018093	C₂	1	2	yes	maximal
5	C₈₉×#2571	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667945	—	2	2	no	—
7	C₂×#15667943	—	2	2	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151468	C₂²	1	1	yes	characteristic
11	#19151360	—	2	2	no	—
12	#400×C₈₉	—	2	2	no	—
13	#389×C₈₉	—	2	2	no	—
14	#384×C₈₉	C₂²	1	1	yes	characteristic
15	#382×C₈₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
17	C₂²×#10623494	—	2	1	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	1	no	—
21	#15667945	—	4	4	no	—
22	#15667943	—	4	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	Q₈	1	4	yes	—
25	C₄×#10623494	—	2	1	no	—
26	C₄×#10623494	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	#95×C₈₉	—	2	2	no	—
35	#95×C₈₉	—	2	4	no	—
36	#95×C₈₉	—	2	2	no	—
37	C₂²×Dic₈₉	—	4	1	no	—
38	C₂×#10623494	—	2	2	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₄⋊C₄×C₈₉	—	2	2	no	—
55	C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₄⋊C₄×C₈₉	—	4	2	no	—
59	C₄⋊C₄×C₈₉	—	8	2	no	—
60	C₄²×C₈₉	—	4	2	no	—
61	C₄²×C₈₉	—	4	2	no	—
62	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
64	C₄²×C₈₉	#41	1	1	yes	characteristic
65	C₂³×C₈₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	4	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₈₉	#135	1	2	yes	—
81	C₂×C₄×C₈₉	#135	1	2	yes	—
82	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₈₉	#137	1	2	yes	—
84	C₂×Dic₈₉	—	8	2	no	—
85	C₂×Dic₈₉	—	8	1	no	—
86	#10623494	—	2	4	no	—
87	#10623494	—	4	4	no	—
88	C₂²×C₈₉	#395	1	1	yes	characteristic
89	C₂²×C₈₉	#449	1	2	yes	—
90	C₂²×C₈₉	#452	1	2	yes	—
91	C₂²×C₈₉	#453	1	1	yes	characteristic
92	C₂²×C₈₉	#456	1	1	yes	characteristic
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	8	2	no	—
96	C₄×C₈₉	—	8	2	no	—
97	C₄×C₈₉	—	8	2	no	—
98	C₄×C₈₉	#478	1	2	yes	—
99	C₄×C₈₉	#479	1	2	yes	—
100	C₄×C₈₉	#499	1	2	yes	—
101	C₄×C₈₉	#501	1	2	yes	—
102	Dic₈₉	—	16	1	no	—
103	#10576	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	#1620	1	1	yes	characteristic
105	C₂×C₈₉	#1634	1	1	yes	characteristic
106	C₂×C₈₉	#1650	1	2	yes	—
107	C₂×C₈₉	#2059	1	2	yes	—
108	C₂×C₈₉	#2060	1	1	yes	characteristic
109	#2571	D₈₉	1	1	yes	characteristic
110	#1948	—	89	2	no	—
111	#1845	—	89	1	no	—
112	#1296	—	89	1	no	—
113	#1279	—	89	2	no	—
114	C₈₉	#10576	1	1	yes	characteristic, complemented, Sylow
115	C₂×#107	—	178	2	no	—
116	C₂×#106	—	178	2	no	—
117	C₂×C₄⋊C₈	—	89	2	no	—
118	C₂×C₄×C₈	—	89	1	no	—
119	#400	—	2	2	no	—
120	#389	—	2	2	no	—
121	#384	—	89	1	no	—
122	#384	C₂×D₈₉	1	1	yes	characteristic
123	#382	—	2	2	no	—
124	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
125	#340	—	178	2	no	—
126	C₂²×C₈	—	178	2	no	—
127	C₂²×C₈	—	178	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	178	2	no	—
131	C₂×C₄⋊C₄	—	178	1	no	—
132	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	#107	—	356	4	no	—
138	#106	—	356	4	no	—
139	C₄⋊C₈	—	89	4	no	—
140	C₄⋊C₈	—	178	2	no	—
141	C₄×C₈	—	89	2	no	—
142	C₄×C₈	—	178	1	no	—
143	#95	—	2	2	no	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	356	1	no	—
150	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
152	C₂×C₈	—	356	2	no	—
153	C₂×C₈	—	356	2	no	—
154	C₂×C₈	—	178	4	no	—
155	C₂×C₈	—	178	2	no	—
156	C₂×C₈	—	178	2	no	—
157	C₂×C₈	—	178	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	356	2	no	—
167	C₄⋊C₄	—	356	2	no	—
168	C₄⋊C₄	—	356	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	Q₈×D₈₉	1	1	yes	characteristic
172	C₄²	Q₈×D₈₉	1	2	yes	—
173	C₄²	#13083575	1	1	yes	characteristic
174	C₂³	#15667981	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	712	2	no	—
186	C₂×C₄	—	712	1	no	—
187	C₂×C₄	#15667988	1	1	yes	characteristic
188	C₂×C₄	#15667988	1	2	yes	—
189	C₂×C₄	#15668010	1	1	yes	characteristic
190	C₂×C₄	#15668010	1	2	yes	—
191	C₂×C₄	#15668032	1	2	yes	—
192	C₂×C₄	#15668032	1	2	yes	—
193	C₂×C₄	#15668037	1	2	yes	—
194	C₂×C₄	#15668042	1	2	yes	—
195	C₈	—	356	4	no	—
196	C₈	—	178	4	no	—
197	C₂²	#19151517	1	1	yes	characteristic
198	C₂²	#19151743	1	2	yes	—
199	C₂²	#19151760	1	2	yes	—
200	C₂²	#19151766	1	1	yes	characteristic
201	C₂²	#19151781	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	8	2	no	—
207	C₄	—	1424	1	no	—
208	C₄	#19151895	1	2	yes	—
209	C₄	#19151902	1	2	yes	—
210	C₄	#19151997	1	2	yes	—
211	C₄	#19152008	1	2	yes	—
212	C₂	#26019311	1	1	yes	characteristic
213	C₂	#26019397	1	1	yes	characteristic
214	C₂	#26019508	1	2	yes	—
215	C₂	#26021587	1	2	yes	—
216	C₂	#26021594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021015

C₂

yes

maximal

#26020411

C₂

yes

maximal, characteristic

#26018135

C₂

yes

maximal, characteristic

#26018093

C₂

yes

maximal

C₈₉×#2571

C₂

yes

maximal, characteristic

C₂×#15667945

—

C₂×#15667943

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

characteristic

#19151468

C₂²

yes

characteristic

#19151360

—

#400×C₈₉

—

#389×C₈₉

—

#384×C₈₉

C₂²

yes

characteristic

#382×C₈₉

—

C₄×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂²×#10623494

—

C₂²×#10623494

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

Q₈

yes

—

C₄×#10623494

—

C₄×#10623494

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#95×C₈₉

—

#95×C₈₉

—

#95×C₈₉

—

C₂²×Dic₈₉

—

—

—

—

—

—

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

#395

yes

characteristic

C₂²×C₈₉

#449

yes

—

C₂²×C₈₉

#452

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#478

yes

—

C₄×C₈₉

#479

yes

—

100

C₄×C₈₉

#499

yes

—

101

C₄×C₈₉

#501

yes

—

102

Dic₈₉

—

103

#10576

—

maximal, Sylow

104

C₂×C₈₉

#1620

yes

characteristic

105

C₂×C₈₉

#1634

yes

characteristic

106

C₂×C₈₉

#1650

yes

—

107

C₂×C₈₉

#2059

yes

—

108

C₂×C₈₉

#2060

yes

characteristic

109

#2571

D₈₉

yes

characteristic

110

#1948

—

111

#1845

—

112

#1296

—

113

#1279

—

114

C₈₉

#10576

yes

characteristic, complemented, Sylow

115

C₂×#107

—

178

—

116

C₂×#106

—

178

—

117

C₂×C₄⋊C₈

—

118

C₂×C₄×C₈

—

119

#400

—

120

#389

—

121

#384

—

122

#384

C₂×D₈₉

yes

characteristic

123

#382

—

124

C₄×C₄⋊C₄

C₂×D₈₉

yes

—

125

#340

—

178

—

126

C₂²×C₈

—

178

—

127

C₂²×C₈

—

178

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

178

—

131

C₂×C₄⋊C₄

—

178

—

132

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

133

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

134

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₈₉

yes

characteristic

137

#107

—

356

—

138

#106

—

356

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

178

—

141

C₄×C₈

—

142

C₄×C₈

—

178

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

356

—

150

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

151

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

152

C₂×C₈

—

356

—

153

C₂×C₈

—

356

—

154

C₂×C₈

—

178

—

155

C₂×C₈

—

178

—

156

C₂×C₈

—

178

—

157

C₂×C₈

—

178

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

356

—

167

C₄⋊C₄

—

356

—

168

C₄⋊C₄

—

356

—

169

C₄²

—

170

C₄²

—

171

C₄²

Q₈×D₈₉

yes

characteristic

172

C₄²

Q₈×D₈₉

yes

—

173

C₄²

#13083575

yes

characteristic

174

C₂³

#15667981

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

712

—

186

C₂×C₄

—

712

—

187

C₂×C₄

#15667988

yes

characteristic

188

C₂×C₄

#15667988

yes

—

189

C₂×C₄

#15668010

yes

characteristic

190

C₂×C₄

#15668010

yes

—

191

C₂×C₄

#15668032

yes

—

192

C₂×C₄

#15668032

yes

—

193

C₂×C₄

#15668037

yes

—

194

C₂×C₄

#15668042

yes

—

195

C₈

—

356

—

196

C₈

—

178

—

197

C₂²

#19151517

yes

characteristic

198

C₂²

#19151743

yes

—

199

C₂²

#19151760

yes

—

200

C₂²

#19151766

yes

characteristic

201

C₂²

#19151781

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

1424

—

208

C₄

#19151895

yes

—

209

C₄

#19151902

yes

—

210

C₄

#19151997

yes

—

211

C₄

#19152008

yes

—

212

C₂

#26019311

yes

characteristic

213

C₂

#26019397

yes

characteristic

214

C₂

#26019508

yes

—

215

C₂

#26021587

yes

—

216

C₂

#26021594

yes

characteristic

	subgroup	G/H
center	C₂³	#15667981
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2571	C₂
socle, prime radical	C₂³×C₈₉	#120
hypercenter	#2571	D₈₉
hypocenter	C₈₉	#10576
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

subgroup

G/H

center

C₂³

#15667981

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#2571

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

N	H ≅ G/N	type
114. C₈₉	#10576	semidirect

H ≅ G/N

type

114. C₈₉

#10576

semidirect

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

subgroup	index
#26021015	2
#26020411	2
#26018135	2
#26018093	2
C₈₉×#2571	2
#10576	89

subgroup

index

C₈₉×#2571

Subgroups — #36458879

Conjugacy classes of subgroups

Complements

Maximal subgroups