Group #36454571 — subgroups

Subgroups — #36454571

Order 22784.

438 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 163 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	#26028615	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂×C₄⋊#10623494	—	2	2	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151496	—	2	2	no	—
9	#19151496	C₄	1	4	yes	—
10	#19151468	C₂²	1	2	yes	—
11	#19151464	—	2	1	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×#97×C₈₉	—	2	1	no	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	#336×C₈₉	—	2	1	no	—
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	4	4	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
21	C₂×#13083553	—	2	4	no	—
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	4	4	no	—
24	#15667919	—	2	4	no	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	2	yes	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	2	2	no	—
32	C₄⋊C₈×C₈₉	Q₈	1	4	yes	—
33	#97×C₈₉	—	4	2	no	—
34	C₂²×Dic₈₉	—	2	4	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	4	no	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	4	no	—
41	#13083553	—	4	8	no	—
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	4	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	—	4	2	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	—	2	2	no	—
53	C₄²×C₈₉	—	4	2	no	—
54	C₄²×C₈₉	SD₁₆	1	2	yes	—
55	C₄²×C₈₉	Q₁₆	1	2	yes	—
56	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
57	C₄²×C₈₉	#41	1	2	yes	—
58	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	#103	1	4	yes	—
66	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₈₉	#137	1	2	yes	—
75	C₂×Dic₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	4	4	no	—
77	C₂×Dic₈₉	—	4	4	no	—
78	C₈×C₈₉	—	4	4	no	—
79	#10623494	—	8	2	no	—
80	C₂²×C₈₉	#391	1	1	yes	characteristic
81	C₂²×C₈₉	#413	1	2	yes	—
82	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
83	C₂²×C₈₉	#417	1	2	yes	—
84	C₂²×C₈₉	#418	1	1	yes	characteristic
85	C₄×C₈₉	—	2	4	no	—
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	#476	1	2	yes	—
89	C₄×C₈₉	#477	1	2	yes	—
90	C₄×C₈₉	#490	1	1	yes	characteristic
91	C₄×C₈₉	#491	1	1	yes	characteristic
92	C₄×C₈₉	#501	1	2	yes	—
93	Dic₈₉	—	8	4	no	—
94	#9902	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	#1550	1	2	yes	—
96	C₂×C₈₉	#1557	1	2	yes	—
97	C₂×C₈₉	#1987	1	1	yes	characteristic
98	C₂×C₈₉	#1992	1	1	yes	characteristic
99	C₂×C₈₉	#1998	1	1	yes	characteristic
100	#2859	—	89	1	no	—
101	#1842	—	89	1	no	—
102	#1767	D₈₉	1	1	yes	characteristic
103	#1296	—	89	4	no	—
104	C₈₉	#9902	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	178	1	no	—
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₂×C₄⋊C₈	—	178	2	no	—
108	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
109	C₂×#97	—	2	1	no	—
110	#389	—	89	4	no	—
111	#389	—	178	2	no	—
112	#384	—	89	2	no	—
113	#382	—	178	1	no	—
114	C₄³	C₂×D₈₉	1	1	yes	characteristic
115	#336	—	2	1	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	178	2	no	—
118	#125	—	356	4	no	—
119	C₂×C₄⋊C₄	—	89	4	no	—
120	C₂×C₄⋊C₄	—	178	4	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄⋊C₄	—	178	2	no	—
123	C₂×C₄²	C₄×D₈₉	1	2	yes	—
124	C₂×C₄²	D₃₅₆	1	2	yes	—
125	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	356	4	no	—
130	C₄⋊C₈	—	178	4	no	—
131	C₄⋊C₈	Dic₁₇₈	1	4	yes	—
132	#97	—	4	2	no	—
133	#95	—	178	4	no	—
134	C₂²×C₄	—	178	4	no	—
135	C₂²×C₄	#13083551	1	2	yes	—
136	C₂²×C₄	#13083551	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	356	4	no	—
145	C₄⋊C₄	—	356	4	no	—
146	C₄⋊C₄	—	356	8	no	—
147	C₄⋊C₄	—	356	4	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#13083562	1	2	yes	—
155	C₄²	#13083564	1	2	yes	—
156	C₄²	C₂×Dic₁₇₈	1	2	yes	—
157	C₄²	#13083574	1	2	yes	—
158	C₂³	C₂×#13083551	1	1	yes	characteristic
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	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	356	4	no	—
166	C₂×C₄	—	356	4	no	—
167	C₂×C₄	—	356	4	no	—
168	C₂×C₄	#15667935	1	4	yes	—
169	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
170	C₂×C₄	#15667991	1	2	yes	—
171	C₂×C₄	#15668000	1	1	yes	characteristic
172	C₂×C₄	#15668012	1	1	yes	characteristic
173	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
174	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
175	C₂×C₄	#15668037	1	2	yes	—
176	C₂×C₄	#15668040	1	2	yes	—
177	C₂×C₄	#15668045	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₈	—	712	2	no	—
180	C₂²	#19151504	1	1	yes	characteristic
181	C₂²	#19151580	1	2	yes	—
182	C₂²	C₂×#15667935	1	1	yes	characteristic
183	C₂²	#19151600	1	2	yes	—
184	C₂²	#19151602	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	712	4	no	—
189	C₄	#19151888	1	2	yes	—
190	C₄	#19151893	1	2	yes	—
191	C₄	#19151963	1	1	yes	characteristic
192	C₄	#19151967	1	1	yes	characteristic
193	C₄	#19152007	1	2	yes	—
194	C₂	#26018943	1	2	yes	—
195	C₂	#26018975	1	2	yes	—
196	C₂	#26021200	1	1	yes	characteristic
197	C₂	#26021221	1	1	yes	characteristic
198	C₂	#26021245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028615

C₂

yes

maximal, characteristic

#26020404

C₂

yes

maximal, characteristic

#26018136

C₂

yes

maximal

C₈₉×#1767

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

#19151496

—

#19151496

C₄

yes

—

#19151468

C₂²

yes

—

#19151464

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×#97×C₈₉

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

Q₈

yes

—

#97×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#391

yes

characteristic

C₂²×C₈₉

#413

yes

—

C₂²×C₈₉

C₂×#103

yes

characteristic

C₂²×C₈₉

#417

yes

—

C₂²×C₈₉

#418

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

#477

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#9902

—

maximal, Sylow

C₂×C₈₉

#1550

yes

—

C₂×C₈₉

#1557

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

100

—

101

#1842

—

102

#1767

D₈₉

yes

characteristic

103

#1296

—

104

C₈₉

#9902

yes

characteristic, complemented, Sylow

105

C₂×#125

—

178

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

178

—

108

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

109

C₂×#97

—

110

#389

—

111

#389

—

178

—

112

#384

—

113

#382

—

178

—

114

C₄³

C₂×D₈₉

yes

characteristic

115

#336

—

116

C₂²×C₈

—

117

C₂²×C₈

—

178

—

118

#125

—

356

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

178

—

121

C₂×C₄⋊C₄

—

178

—

122

C₂×C₄⋊C₄

—

178

—

123

C₂×C₄²

C₄×D₈₉

yes

—

124

C₂×C₄²

D₃₅₆

yes

—

125

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

126

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

127

C₂×C₄²

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

356

—

130

C₄⋊C₈

—

178

—

131

C₄⋊C₈

Dic₁₇₈

yes

—

132

#97

—

133

#95

—

178

—

134

C₂²×C₄

—

178

—

135

C₂²×C₄

#13083551

yes

—

136

C₂²×C₄

#13083551

yes

—

137

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

356

—

144

C₂×C₈

—

356

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

356

—

149

C₄⋊C₄

—

356

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#13083562

yes

—

155

C₄²

#13083564

yes

—

156

C₄²

C₂×Dic₁₇₈

yes

—

157

C₄²

#13083574

yes

—

158

C₂³

C₂×#13083551

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

356

—

166

C₂×C₄

—

356

—

167

C₂×C₄

—

356

—

168

C₂×C₄

#15667935

yes

—

169

C₂×C₄

C₄×Dic₁₇₈

yes

—

170

C₂×C₄

#15667991

yes

—

171

C₂×C₄

#15668000

yes

characteristic

172

C₂×C₄

#15668012

yes

characteristic

173

C₂×C₄

C₂×#13083562

yes

characteristic

174

C₂×C₄

C₂×#13083564

yes

characteristic

175

C₂×C₄

#15668037

yes

—

176

C₂×C₄

#15668040

yes

—

177

C₂×C₄

#15668045

yes

—

178

C₈

—

179

C₈

—

712

—

180

C₂²

#19151504

yes

characteristic

181

C₂²

#19151580

yes

—

182

C₂²

C₂×#15667935

yes

characteristic

183

C₂²

#19151600

yes

—

184

C₂²

#19151602

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

712

—

189

C₄

#19151888

yes

—

190

C₄

#19151893

yes

—

191

C₄

#19151963

yes

characteristic

192

C₄

#19151967

yes

characteristic

193

C₄

#19152007

yes

—

194

C₂

#26018943

yes

—

195

C₂

#26018975

yes

—

196

C₂

#26021200

yes

characteristic

197

C₂

#26021221

yes

characteristic

198

C₂

#26021245

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#9902
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1767

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
104. C₈₉	#9902	semidirect

H ≅ G/N

type

104. C₈₉

#9902

semidirect

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

subgroup	index
#26028615	2
#26020404	2
#26018136	2
C₈₉×#1767	2
#9902	89

subgroup

index

C₈₉×#1767

Subgroups — #36454571

Conjugacy classes of subgroups

Complements

Maximal subgroups