Group #36448933 — subgroups

Subgroups — #36448933

Order 22784.

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

normal subgroups only

This group has 412 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	#26029319	C₂	1	1	yes	maximal, characteristic
2	#26021081	C₂	1	1	yes	maximal, characteristic
3	#26019041	C₂	1	2	yes	maximal, complemented
4	#26018084	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1543	C₂	1	1	yes	maximal, characteristic
6	#19151676	—	2	2	no	—
7	D₄×#10623494	—	2	2	no	—
8	#19151651	C₂²	1	2	yes	—
9	#19151622	C₂²	1	1	yes	characteristic
10	C₂×#15667939	—	2	2	no	—
11	#19151565	—	2	2	no	—
12	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
13	#19151351	—	2	2	no	—
14	#19151349	—	2	2	no	—
15	#532×C₈₉	C₂²	1	1	yes	characteristic
16	#431×C₈₉	C₂²	1	2	yes	—
17	#333×C₈₉	C₄	1	2	yes	complemented
18	#331×C₈₉	C₄	1	1	yes	characteristic, complemented
19	#331×C₈₉	C₄	1	1	yes	characteristic, complemented
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂²×#10623494	—	2	2	no	—
23	C₂×C₄×Dic₈₉	—	2	1	no	—
24	C₂×C₄×Dic₈₉	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	D₄	1	2	yes	—
27	C₄⋊#10623494	D₄	1	2	yes	—
28	#15667939	—	4	4	no	—
29	C₂²⋊#10623494	—	2	4	no	—
30	C₂²⋊#10623494	—	2	2	no	—
31	C₄×#10623494	—	2	2	no	—
32	C₂×#41×C₈₉	D₄	1	2	yes	—
33	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
35	#124×C₈₉	Q₈	1	2	yes	—
36	#121×C₈₉	—	2	2	no	—
37	C₄×D₄×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈×C₈₉	—	2	2	no	—
42	C₄×C₈×C₈₉	—	2	2	no	—
43	C₂²×Dic₈₉	—	4	1	no	—
44	#13083558	—	4	2	no	—
45	#13083558	—	8	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	4	1	no	—
49	C₂×#10623494	—	8	2	no	—
50	C₄×Dic₈₉	—	2	2	no	—
51	C₄×Dic₈₉	—	2	4	no	—
52	C₄×Dic₈₉	—	4	2	no	—
53	C₄×Dic₈₉	—	4	1	no	—
54	#41×C₈₉	—	4	2	no	—
55	#41×C₈₉	—	4	2	no	—
56	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂×D₄×C₈₉	—	2	2	no	—
58	C₂×D₄×C₈₉	—	2	2	no	—
59	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₈₉	—	2	2	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	#41	1	2	yes	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₄⋊C₄×C₈₉	—	2	2	no	—
66	C₂²⋊C₄×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₈₉	—	2	2	no	—
71	C₂³×C₈₉	#123	1	1	yes	characteristic
72	Q₈×C₈₉	—	4	2	no	—
73	D₄×C₈₉	—	4	2	no	—
74	D₄×C₈₉	—	4	2	no	—
75	D₄×C₈₉	—	4	2	no	—
76	D₄×C₈₉	—	8	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
88	C₂×Dic₈₉	—	4	2	no	—
89	C₂×Dic₈₉	—	4	4	no	—
90	C₈×C₈₉	—	4	2	no	—
91	C₈×C₈₉	—	4	2	no	—
92	#10623494	—	4	2	no	—
93	#10623494	—	4	2	no	—
94	#10623494	—	8	1	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	2	no	—
97	C₂²×C₈₉	—	4	2	no	—
98	C₂²×C₈₉	—	4	4	no	—
99	C₂²×C₈₉	#385	1	1	yes	characteristic
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	—	4	2	no	—
103	C₄×C₈₉	#443	1	2	yes	—
104	C₄×C₈₉	#444	1	2	yes	—
105	C₄×C₈₉	#453	1	1	yes	characteristic
106	C₄×C₈₉	#455	1	1	yes	characteristic
107	Dic₈₉	—	4	4	no	—
108	#8934	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	—	4	1	no	—
110	C₂×C₈₉	—	8	2	no	—
111	C₂×C₈₉	#1856	1	1	yes	characteristic
112	C₂×C₈₉	#1866	1	2	yes	—
113	#2979	—	89	1	no	—
114	#1962	—	89	1	no	—
115	#1574	—	89	2	no	—
116	#1543	D₈₉	1	1	yes	characteristic
117	#1276	—	89	2	no	—
118	C₈₉	#8934	1	1	yes	characteristic, complemented, Sylow
119	#532	C₂×D₈₉	1	1	yes	characteristic
120	#436	—	178	2	no	—
121	C₈×D₄	—	178	2	no	—
122	#431	—	89	2	no	—
123	#431	C₂×D₈₉	1	2	yes	—
124	#423	—	89	1	no	—
125	C₂×#104	—	178	2	no	—
126	#408	—	178	2	no	—
127	C₄³	—	89	1	no	—
128	#335	—	178	2	no	—
129	#334	—	178	2	no	—
130	#333	Dic₈₉	1	2	yes	complemented
131	#331	Dic₈₉	1	1	yes	characteristic, complemented
132	#331	Dic₈₉	1	1	yes	characteristic, complemented
133	C₂×#41	D₃₅₆	1	2	yes	—
134	C₂×M₁₆	—	178	1	no	—
135	C₂×M₁₆	—	178	2	no	—
136	C₂²×C₈	—	178	2	no	—
137	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
138	#127	C₂×Dic₈₉	1	1	yes	characteristic
139	#124	Dic₁₇₈	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	178	1	no	—
143	C₂×C₄²	—	178	2	no	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊C₈	—	89	2	no	—
146	C₄⋊C₈	—	89	2	no	—
147	C₄⋊C₈	—	178	2	no	—
148	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
149	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
150	#104	—	356	4	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	178	4	no	—
153	C₂²⋊C₈	—	178	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	178	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	C₄⋊Dic₈₉	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₄⋊Dic₈₉	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	356	1	no	—
164	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
165	C₂²×C₄	#13083573	1	2	yes	—
166	M₁₆	—	356	2	no	—
167	M₁₆	—	712	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	356	1	no	—
171	C₂×C₈	—	178	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	—	712	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	356	2	no	—
177	C₄²	—	356	1	no	—
178	C₄²	—	178	2	no	—
179	C₄²	—	178	4	no	—
180	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
181	C₄²	D₄×D₈₉	1	2	yes	—
182	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#15667996	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	356	2	no	—
198	C₂×C₄	—	356	4	no	—
199	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
200	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
201	C₂×C₄	#15667982	1	1	yes	characteristic
202	C₂×C₄	#15667989	1	2	yes	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	356	2	no	—
206	C₈	—	356	2	no	—
207	C₈	—	712	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#19151473	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	356	4	no	—
217	C₄	#19151715	1	2	yes	—
218	C₄	#19151720	1	2	yes	—
219	C₄	#19151768	1	1	yes	characteristic
220	C₄	#19151778	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#26020464	1	1	yes	characteristic
224	C₂	#26020529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029319

C₂

yes

maximal, characteristic

#26021081

C₂

yes

maximal, characteristic

#26019041

C₂

yes

maximal, complemented

#26018084

C₂

yes

maximal, complemented

C₈₉×#1543

C₂

yes

maximal, characteristic

#19151676

—

D₄×#10623494

—

#19151651

C₂²

yes

—

#19151622

C₂²

yes

characteristic

C₂×#15667939

—

#19151565

—

C₄²×Dic₈₉

C₂²

yes

characteristic

#19151351

—

#19151349

—

#532×C₈₉

C₂²

yes

characteristic

#431×C₈₉

C₂²

yes

—

#333×C₈₉

C₄

yes

complemented

#331×C₈₉

C₄

yes

characteristic, complemented

#331×C₈₉

C₄

yes

characteristic, complemented

C₂×#13083558

—

C₂×#13083558

—

C₂²×#10623494

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

—

C₄⋊#10623494

—

C₄⋊#10623494

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

C₂×#41×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

characteristic

#127×C₈₉

C₂×C₄

yes

characteristic

#124×C₈₉

Q₈

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

C₂×C₄

yes

—

C₄⋊C₈×C₈₉

C₂×C₄

yes

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

—

—

—

—

—

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₄⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#123

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×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

#120

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#385

yes

characteristic

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

—

103

C₄×C₈₉

#443

yes

—

104

C₄×C₈₉

#444

yes

—

105

C₄×C₈₉

#453

yes

characteristic

106

C₄×C₈₉

#455

yes

characteristic

107

Dic₈₉

—

108

#8934

—

maximal, Sylow

109

C₂×C₈₉

—

110

C₂×C₈₉

—

111

C₂×C₈₉

#1856

yes

characteristic

112

C₂×C₈₉

#1866

yes

—

113

#2979

—

114

#1962

—

115

#1574

—

116

#1543

D₈₉

yes

characteristic

117

#1276

—

118

C₈₉

#8934

yes

characteristic, complemented, Sylow

119

#532

C₂×D₈₉

yes

characteristic

120

#436

—

178

—

121

C₈×D₄

—

178

—

122

#431

—

123

#431

C₂×D₈₉

yes

—

124

#423

—

125

C₂×#104

—

178

—

126

#408

—

178

—

127

C₄³

—

128

#335

—

178

—

129

#334

—

178

—

130

#333

Dic₈₉

yes

complemented

131

#331

Dic₈₉

yes

characteristic, complemented

132

#331

Dic₈₉

yes

characteristic, complemented

133

C₂×#41

D₃₅₆

yes

—

134

C₂×M₁₆

—

178

—

135

C₂×M₁₆

—

178

—

136

C₂²×C₈

—

178

—

137

C₄⋊Q₈

C₂×Dic₈₉

yes

characteristic

138

#127

C₂×Dic₈₉

yes

characteristic

139

#124

Dic₁₇₈

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

178

—

143

C₂×C₄²

—

178

—

144

C₂×C₄²

C₂²×D₈₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

178

—

148

C₄⋊C₈

C₂×Dic₈₉

yes

—

149

C₄⋊C₈

C₂×Dic₈₉

yes

—

150

#104

—

356

—

151

C₂²⋊C₈

—

152

C₂²⋊C₈

—

178

—

153

C₂²⋊C₈

—

178

—

154

C₄×C₈

—

155

C₄×C₈

—

178

—

156

#41

—

157

#41

—

158

C₂×Q₈

C₄⋊Dic₈₉

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

C₄⋊Dic₈₉

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

356

—

164

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

165

C₂²×C₄

#13083573

yes

—

166

M₁₆

—

356

—

167

M₁₆

—

712

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

356

—

171

C₂×C₈

—

178

—

172

C₂×C₈

—

178

—

173

C₂×C₈

—

712

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

356

—

177

C₄²

—

356

—

178

C₄²

—

178

—

179

C₄²

—

178

—

180

C₄²

C₂²×Dic₈₉

yes

characteristic

181

C₄²

D₄×D₈₉

yes

—

182

C₄²

C₂×Dic₁₇₈

yes

characteristic

183

C₂³

—

184

C₂³

#15667996

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

356

—

198

C₂×C₄

—

356

—

199

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

200

C₂×C₄

D₄×Dic₈₉

yes

—

201

C₂×C₄

#15667982

yes

characteristic

202

C₂×C₄

#15667989

yes

—

203

C₈

—

204

C₈

—

205

C₈

—

356

—

206

C₈

—

356

—

207

C₈

—

712

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#19151473

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

356

—

217

C₄

#19151715

yes

—

218

C₄

#19151720

yes

—

219

C₄

#19151768

yes

characteristic

220

C₄

#19151778

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#26020464

yes

characteristic

224

C₂

#26020529

yes

—

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1543	C₂
socle	C₂²×C₈₉	#385
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1543	D₈₉
hypocenter	C₈₉	#8934
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

subgroup

G/H

center

C₂×C₄

#15667982

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1543

C₂

socle

C₂²×C₈₉

#385

prime radical

#532×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

N	H ≅ G/N	type
3. #26019041	C₂	semidirect
4. #26018084	C₂	semidirect
17. #333×C₈₉	C₄	semidirect
18. #331×C₈₉	C₄	semidirect
19. #331×C₈₉	C₄	semidirect
118. C₈₉	#8934	semidirect
130. #333	Dic₈₉	semidirect
131. #331	Dic₈₉	semidirect
132. #331	Dic₈₉	semidirect

H ≅ G/N

type

3. #26019041

C₂

semidirect

4. #26018084

C₂

semidirect

17. #333×C₈₉

C₄

semidirect

18. #331×C₈₉

C₄

semidirect

19. #331×C₈₉

C₄

semidirect

118. C₈₉

#8934

semidirect

130. #333

Dic₈₉

semidirect

131. #331

Dic₈₉

semidirect

132. #331

Dic₈₉

semidirect

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

subgroup	index
#26029319	2
#26021081	2
#26019041	2
#26018084	2
C₈₉×#1543	2
#8934	89

subgroup

index

C₈₉×#1543

Subgroups — #36448933

Conjugacy classes of subgroups

Complements

Maximal subgroups