Group #36448811 — subgroups

Subgroups — #36448811

Order 22784.

396 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 111 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	#26023438	C₂	1	1	yes	maximal, characteristic
2	#26020387	C₂	1	2	yes	maximal
3	C₂×#19151340	C₂	1	1	yes	maximal, characteristic
4	#26018134	C₂	1	2	yes	maximal
5	C₈₉×#1992	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667923	C₂²	1	2	yes	—
7	#19151497	—	2	2	no	—
8	#19151484	—	2	2	no	—
9	#19151484	—	2	2	no	—
10	#19151464	C₂²	1	2	yes	—
11	#19151360	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	#19151340	C₄	1	4	yes	complemented
14	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
16	#384×C₈₉	C₂²	1	1	yes	characteristic
17	#340×C₈₉	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
22	C₂×C₄×Dic₈₉	—	2	2	no	—
23	#15667923	—	2	2	no	—
24	#15667923	C₂×C₄	1	4	yes	—
25	#15667919	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	#15667919	—	2	2	no	—
28	#15667919	—	2	2	no	—
29	C₂²×C₈×C₈₉	—	2	1	no	—
30	#125×C₈₉	—	2	2	no	—
31	#125×C₈₉	C₂×C₄	1	2	yes	—
32	#125×C₈₉	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	#97×C₈₉	—	2	2	no	—
39	C₂²×Dic₈₉	—	2	2	no	—
40	C₂²×Dic₈₉	—	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	—	8	2	no	—
47	C₄×Dic₈₉	—	8	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	4	1	no	—
54	C₂×C₈×C₈₉	—	4	1	no	—
55	C₄⋊C₄×C₈₉	—	2	4	no	—
56	C₄⋊C₄×C₈₉	—	2	4	no	—
57	C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₄⋊C₄×C₈₉	—	4	2	no	—
59	C₄⋊C₄×C₈₉	—	4	4	no	—
60	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
61	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₈₉	#41	1	2	yes	—
64	C₂³×C₈₉	#120	1	1	yes	characteristic
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	—	8	1	no	—
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
78	C₂×Dic₈₉	—	4	2	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	4	2	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	2	no	—
84	C₈×C₈₉	—	4	2	no	—
85	#10623494	—	4	4	no	—
86	C₂²×C₈₉	#385	1	1	yes	characteristic
87	C₂²×C₈₉	#454	1	1	yes	characteristic
88	C₂²×C₈₉	#454	1	2	yes	—
89	C₂²×C₈₉	#455	1	2	yes	—
90	C₂²×C₈₉	#456	1	1	yes	characteristic
91	C₄×C₈₉	—	2	1	no	—
92	C₄×C₈₉	—	2	1	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	—	4	4	no	—
96	C₄×C₈₉	—	16	1	no	—
97	Dic₈₉	—	8	2	no	—
98	Dic₈₉	—	8	2	no	—
99	#8915	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	#1658	1	1	yes	characteristic
101	C₂×C₈₉	#1662	1	1	yes	characteristic
102	C₂×C₈₉	#1664	1	2	yes	—
103	C₂×C₈₉	#1857	1	1	yes	characteristic
104	C₂×C₈₉	#1860	1	2	yes	—
105	#2382	—	89	1	no	—
106	#1992	D₈₉	1	1	yes	characteristic
107	#1836	—	89	2	no	—
108	C₂×#330	—	89	1	no	—
109	#1295	—	89	2	no	—
110	C₈₉	#8915	1	1	yes	characteristic, complemented, Sylow
111	C₂×#125	C₂×D₈₉	1	1	yes	characteristic
112	C₂×#97	—	89	2	no	—
113	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
114	#389	—	178	2	no	—
115	#387	—	178	2	no	—
116	#387	—	178	2	no	—
117	#384	C₂×D₈₉	1	1	yes	characteristic
118	#382	—	89	2	no	—
119	#340	—	2	2	no	—
120	#340	—	178	2	no	—
121	#340	—	178	2	no	—
122	#330	—	89	4	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	178	2	no	—
125	#125	—	2	2	no	—
126	#125	Dic₁₇₈	1	2	yes	—
127	#125	C₄×D₈₉	1	2	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	89	2	no	—
132	C₂×C₄⋊C₄	—	89	2	no	—
133	C₂×C₄⋊C₄	—	178	2	no	—
134	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
135	C₂×C₄²	—	178	2	no	—
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	#97	—	2	2	no	—
138	#97	—	89	4	no	—
139	#97	—	178	2	no	—
140	#95	—	178	2	no	—
141	#95	—	178	2	no	—
142	#95	—	178	2	no	—
143	#95	—	178	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	178	2	no	—
147	C₂²×C₄	—	178	2	no	—
148	C₂²×C₄	D₄×D₈₉	1	2	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	356	2	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	—	178	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	356	2	no	—
162	C₄⋊C₄	—	356	2	no	—
163	C₄⋊C₄	—	712	2	no	—
164	C₄⋊C₄	#13083553	1	4	yes	—
165	C₄²	—	712	2	no	—
166	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
167	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
168	C₄²	#13083573	1	2	yes	—
169	C₂³	#15667980	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	356	2	no	—
181	C₂×C₄	—	356	2	no	—
182	C₂×C₄	—	356	2	no	—
183	C₂×C₄	—	356	2	no	—
184	C₂×C₄	—	356	2	no	—
185	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
186	C₂×C₄	#15667972	1	2	yes	—
187	C₂×C₄	#15667989	1	2	yes	—
188	C₂×C₄	#15667993	1	1	yes	characteristic
189	C₈	—	4	2	no	—
190	C₈	—	356	4	no	—
191	C₂²	#19151470	1	1	yes	characteristic
192	C₂²	#19151769	1	2	yes	—
193	C₂²	#19151770	1	1	yes	characteristic
194	C₂²	#19151774	1	2	yes	—
195	C₂²	#19151780	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	16	1	no	—
202	C₄	—	712	2	no	—
203	C₄	—	712	2	no	—
204	C₂	#26019553	1	1	yes	characteristic
205	C₂	#26019573	1	1	yes	characteristic
206	C₂	#26019584	1	2	yes	—
207	C₂	#26020466	1	1	yes	characteristic
208	C₂	#26020483	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26023438

C₂

yes

maximal, characteristic

#26020387

C₂

yes

maximal

C₂×#19151340

C₂

yes

maximal, characteristic

#26018134

C₂

yes

maximal

C₈₉×#1992

C₂

yes

maximal, characteristic

C₂×#15667923

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151360

—

#19151360

—

#19151340

C₄

yes

complemented

C₂×#125×C₈₉

C₂²

yes

characteristic

C₂×#97×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

#340×C₈₉

—

C₂²×#10623494

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄×Dic₈₉

—

#15667923

—

#15667923

C₂×C₄

yes

—

—

—

—

—

C₂²×C₈×C₈₉

—

#125×C₈₉

—

#125×C₈₉

C₂×C₄

yes

—

#125×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#97×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#120

yes

characteristic

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⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#123

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#455

yes

—

C₂²×C₈₉

#456

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

#8915

—

maximal, Sylow

100

C₂×C₈₉

#1658

yes

characteristic

101

C₂×C₈₉

#1662

yes

characteristic

102

C₂×C₈₉

#1664

yes

—

103

C₂×C₈₉

#1857

yes

characteristic

104

C₂×C₈₉

#1860

yes

—

105

#2382

—

106

#1992

D₈₉

yes

characteristic

107

#1836

—

108

C₂×#330

—

109

#1295

—

110

C₈₉

#8915

yes

characteristic, complemented, Sylow

111

C₂×#125

C₂×D₈₉

yes

characteristic

112

C₂×#97

—

113

C₂×#97

C₂×D₈₉

yes

characteristic

114

#389

—

178

—

115

#387

—

178

—

116

#387

—

178

—

117

#384

C₂×D₈₉

yes

characteristic

118

#382

—

119

#340

—

120

#340

—

178

—

121

#340

—

178

—

122

#330

—

123

C₂²×C₈

—

124

C₂²×C₈

—

178

—

125

#125

—

126

#125

Dic₁₇₈

yes

—

127

#125

C₄×D₈₉

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

178

—

134

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

135

C₂×C₄²

—

178

—

136

C₂×C₄²

C₂²×D₈₉

yes

characteristic

137

#97

—

138

#97

—

139

#97

—

178

—

140

#95

—

178

—

141

#95

—

178

—

142

#95

—

178

—

143

#95

—

178

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

178

—

147

C₂²×C₄

—

178

—

148

C₂²×C₄

D₄×D₈₉

yes

—

149

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

356

—

154

C₂×C₈

—

356

—

155

C₂×C₈

—

178

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

712

—

164

C₄⋊C₄

#13083553

yes

—

165

C₄²

—

712

—

166

C₄²

C₂×C₄×D₈₉

yes

characteristic

167

C₄²

C₂×Dic₁₇₈

yes

characteristic

168

C₄²

#13083573

yes

—

169

C₂³

#15667980

yes

characteristic

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

356

—

180

C₂×C₄

—

356

—

181

C₂×C₄

—

356

—

182

C₂×C₄

—

356

—

183

C₂×C₄

—

356

—

184

C₂×C₄

—

356

—

185

C₂×C₄

C₂×#13083553

yes

characteristic

186

C₂×C₄

#15667972

yes

—

187

C₂×C₄

#15667989

yes

—

188

C₂×C₄

#15667993

yes

characteristic

189

C₈

—

190

C₈

—

356

—

191

C₂²

#19151470

yes

characteristic

192

C₂²

#19151769

yes

—

193

C₂²

#19151770

yes

characteristic

194

C₂²

#19151774

yes

—

195

C₂²

#19151780

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

712

—

203

C₄

—

712

—

204

C₂

#26019553

yes

characteristic

205

C₂

#26019573

yes

characteristic

206

C₂

#26019584

yes

—

207

C₂

#26020466

yes

characteristic

208

C₂

#26020483

yes

—

	subgroup	G/H
center	C₂³	#15667980
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1992	C₂
socle, prime radical	C₂³×C₈₉	#120
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1992	D₈₉
hypocenter	C₈₉	#8915
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1992

subgroup

G/H

center

C₂³

#15667980

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1992

C₂

socle, prime radical

C₂³×C₈₉

#120

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1992

N	H ≅ G/N	type
13. #19151340	C₄	semidirect
110. C₈₉	#8915	semidirect

H ≅ G/N

type

13. #19151340

C₄

semidirect

110. C₈₉

#8915

semidirect

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

subgroup	index
#26023438	2
#26020387	2
C₂×#19151340	2
#26018134	2
C₈₉×#1992	2
#8915	89

subgroup

index

C₈₉×#1992

Subgroups — #36448811

Conjugacy classes of subgroups

Complements

Maximal subgroups