Group #36459532 — subgroups

Subgroups — #36459532

Order 22784.

526 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 153 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	#26021058	C₂	1	1	yes	maximal, characteristic
2	#26021057	C₂	1	2	yes	maximal
3	#26020408	C₂	1	2	yes	maximal, complemented
4	#26018079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668009	C₂²	1	1	yes	characteristic
7	C₂×#15667943	—	2	4	no	—
8	C₂×#15667934	—	2	2	no	—
9	C₂×#15667932	—	2	4	no	—
10	C₂×C₄×#10623494	C₂²	1	2	yes	—
11	#19151468	C₂²	1	1	yes	characteristic
12	C₂×#106×C₈₉	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#384×C₈₉	C₂²	1	2	yes	—
15	C₂²×D₃₅₆	—	2	2	no	—
16	C₂²×D₃₅₆	—	2	1	no	—
17	C₂²×#10623494	D₄	1	4	yes	—
18	#15668009	—	2	2	no	—
19	#15668009	—	2	1	no	—
20	#15668009	Q₈	1	2	yes	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	1	no	—
23	#15667943	—	4	8	no	—
24	#15667934	—	4	4	no	—
25	#15667932	—	4	8	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₂²×C₈×C₈₉	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	#106×C₈₉	—	4	4	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
35	C₂³×D₈₉	—	4	1	no	—
36	C₂×D₃₅₆	—	2	2	no	—
37	C₂×D₃₅₆	—	2	4	no	—
38	C₂×D₃₅₆	—	4	4	no	—
39	C₂×D₃₅₆	—	4	2	no	—
40	C₂×D₃₅₆	—	4	2	no	—
41	C₂×D₃₅₆	—	4	4	no	—
42	C₂×D₃₅₆	—	4	1	no	—
43	C₂×D₃₅₆	—	4	2	no	—
44	C₂²×Dic₈₉	—	4	1	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	2	4	no	—
47	C₂×#10623494	SD₁₆	1	8	yes	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₄⋊Dic₈₉	—	4	4	no	—
50	C₂²×C₄×C₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
56	C₄⋊C₄×C₈₉	—	4	4	no	—
57	C₄⋊C₄×C₈₉	—	4	4	no	—
58	C₄⋊C₄×C₈₉	—	4	4	no	—
59	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	#41	1	1	yes	characteristic
62	C₂³×C₈₉	#127	1	1	yes	characteristic
63	C₂²×D₈₉	—	4	2	no	—
64	C₂²×D₈₉	—	8	2	no	—
65	C₂²×D₈₉	—	8	1	no	—
66	C₂²×D₈₉	—	8	2	no	—
67	C₂²×D₈₉	—	8	1	no	—
68	C₂×C₄×C₈₉	—	8	2	no	—
69	C₂×C₄×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	—	8	2	no	—
71	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#123	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
79	C₂×Dic₈₉	—	8	1	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	D₃₅₆	—	4	4	no	—
82	D₃₅₆	—	4	8	no	—
83	D₃₅₆	—	8	2	no	—
84	D₃₅₆	—	8	4	no	—
85	C₈×C₈₉	—	2	4	no	—
86	#10623494	—	2	8	no	—
87	C₂²×C₈₉	#399	1	1	yes	characteristic
88	C₂²×C₈₉	#492	1	2	yes	—
89	C₂²×C₈₉	#492	1	1	yes	characteristic
90	C₂²×C₈₉	#496	1	2	yes	—
91	C₂²×C₈₉	#496	1	1	yes	characteristic
92	C₂×D₈₉	—	8	4	no	—
93	C₂×D₈₉	—	8	2	no	—
94	C₂×D₈₉	—	16	1	no	—
95	C₂×D₈₉	—	16	2	no	—
96	C₂×D₈₉	—	16	1	no	—
97	C₄×C₈₉	—	16	2	no	—
98	C₄×C₈₉	#476	1	2	yes	—
99	C₄×C₈₉	#476	1	4	yes	—
100	C₄×C₈₉	#481	1	2	yes	—
101	C₄×C₈₉	#481	1	4	yes	—
102	Dic₈₉	—	16	1	no	—
103	#10683	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	#1705	1	1	yes	characteristic
105	C₂×C₈₉	#1715	1	2	yes	—
106	C₂×C₈₉	#1715	1	1	yes	characteristic
107	C₂×C₈₉	#2094	1	2	yes	—
108	C₂×C₈₉	#2094	1	1	yes	characteristic
109	D₈₉	—	16	2	no	—
110	#1957	—	89	1	no	—
111	#1957	—	89	2	no	—
112	#1844	—	89	2	no	—
113	#1844	D₈₉	1	1	yes	characteristic, complemented
114	#1273	—	89	1	no	—
115	C₈₉	#10683	1	1	yes	characteristic, complemented, Sylow
116	C₂×#127	—	89	1	no	—
117	C₂×#106	—	2	2	no	—
118	C₂×#106	—	178	4	no	—
119	C₂×#102	—	178	4	no	—
120	C₂×#102	—	178	2	no	—
121	C₂×C₄×C₈	—	89	2	no	—
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	#384	—	89	1	no	—
124	#384	C₂×D₈₉	1	2	yes	—
125	C₂²×D₄	—	178	2	no	—
126	C₂²×D₄	—	178	1	no	—
127	C₂²×C₈	—	89	4	no	—
128	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
129	#127	—	89	2	no	—
130	#127	—	178	2	no	—
131	#127	—	178	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	178	2	no	—
136	C₂×C₄⋊C₄	—	178	1	no	—
137	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
138	#106	—	4	4	no	—
139	#106	—	356	8	no	—
140	#102	—	356	8	no	—
141	#102	—	356	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	178	2	no	—
144	C₄×C₈	—	178	2	no	—
145	C₂⁴	—	356	1	no	—
146	C₂×D₄	—	356	1	no	—
147	C₂×D₄	—	356	2	no	—
148	C₂×D₄	—	356	4	no	—
149	C₂×D₄	—	356	2	no	—
150	C₂×D₄	—	356	2	no	—
151	C₂×D₄	—	356	4	no	—
152	C₂×D₄	—	178	2	no	—
153	C₂×D₄	—	178	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	356	1	no	—
156	C₂²×C₄	D₄×D₈₉	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	89	8	no	—
161	C₂×C₈	—	178	4	no	—
162	C₂×C₈	—	178	4	no	—
163	C₂×C₈	#13083561	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	356	2	no	—
168	C₄⋊C₄	—	356	4	no	—
169	C₄²	Q₈×D₈₉	1	1	yes	characteristic
170	C₄²	#13083574	1	2	yes	—
171	C₄²	#13083575	1	1	yes	characteristic
172	C₂³	—	356	2	no	—
173	C₂³	—	712	2	no	—
174	C₂³	—	712	1	no	—
175	C₂³	—	712	2	no	—
176	C₂³	—	712	1	no	—
177	C₂³	#15668008	1	1	yes	characteristic
178	D₄	—	356	4	no	—
179	D₄	—	356	8	no	—
180	D₄	—	712	2	no	—
181	D₄	—	712	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	712	1	no	—
186	C₂×C₄	—	712	2	no	—
187	C₂×C₄	#15667988	1	1	yes	characteristic
188	C₂×C₄	#15667990	1	2	yes	—
189	C₂×C₄	#15667995	1	1	yes	characteristic
190	C₂×C₄	#15667997	1	2	yes	—
191	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
192	C₂×C₄	C₂×#13083561	1	2	yes	—
193	C₂×C₄	#15668038	1	2	yes	—
194	C₂×C₄	#15668039	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	178	8	no	—
197	C₂²	—	1424	1	no	—
198	C₂²	—	1424	2	no	—
199	C₂²	—	1424	1	no	—
200	C₂²	—	712	4	no	—
201	C₂²	—	712	2	no	—
202	C₂²	#19151533	1	1	yes	characteristic
203	C₂²	#19151969	1	2	yes	—
204	C₂²	#19151972	1	1	yes	characteristic
205	C₂²	#19151984	1	2	yes	—
206	C₂²	#19151985	1	1	yes	characteristic
207	C₄	—	16	2	no	—
208	C₄	—	1424	1	no	—
209	C₄	#19151885	1	2	yes	—
210	C₄	#19151887	1	4	yes	—
211	C₄	#19151916	1	2	yes	—
212	C₄	#19151918	1	4	yes	—
213	C₂	—	1424	2	no	—
214	C₂	#26019840	1	1	yes	characteristic
215	C₂	#26019879	1	2	yes	—
216	C₂	#26019881	1	1	yes	characteristic
217	C₂	#26021782	1	2	yes	—
218	C₂	#26021786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021058

C₂

yes

maximal, characteristic

#26021057

C₂

yes

maximal

#26020408

C₂

yes

maximal, complemented

#26018079

C₂

yes

maximal, characteristic, complemented

C₈₉×#1844

C₂

yes

maximal, characteristic, complemented

C₂×#15668009

C₂²

yes

—

—

—

C₂²

yes

—

#19151468

C₂²

yes

characteristic

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

—

C₂²×D₃₅₆

—

C₂²×#10623494

D₄

yes

—

#15668009

—

#15668009

—

#15668009

Q₈

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#106×C₈₉

—

C₄×C₈×C₈₉

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

SD₁₆

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

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

#123

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#399

yes

characteristic

C₂²×C₈₉

#492

yes

—

C₂²×C₈₉

#492

yes

characteristic

C₂²×C₈₉

#496

yes

—

C₂²×C₈₉

#496

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

#476

yes

—

100

C₄×C₈₉

#481

yes

—

101

C₄×C₈₉

#481

yes

—

102

Dic₈₉

—

103

#10683

—

maximal, Sylow

104

C₂×C₈₉

#1705

yes

characteristic

105

C₂×C₈₉

#1715

yes

—

106

C₂×C₈₉

#1715

yes

characteristic

107

C₂×C₈₉

#2094

yes

—

108

C₂×C₈₉

#2094

yes

characteristic

109

D₈₉

—

110

#1957

—

111

#1957

—

112

#1844

—

113

#1844

D₈₉

yes

characteristic, complemented

114

#1273

—

115

C₈₉

#10683

yes

characteristic, complemented, Sylow

116

C₂×#127

—

117

C₂×#106

—

118

C₂×#106

—

178

—

119

C₂×#102

—

178

—

120

C₂×#102

—

178

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

123

#384

—

124

#384

C₂×D₈₉

yes

—

125

C₂²×D₄

—

178

—

126

C₂²×D₄

—

178

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₈₉

yes

—

129

#127

—

130

#127

—

178

—

131

#127

—

178

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

178

—

136

C₂×C₄⋊C₄

—

178

—

137

C₂×C₄²

C₂²×D₈₉

yes

characteristic

138

#106

—

139

#106

—

356

—

140

#102

—

356

—

141

#102

—

356

—

142

C₄×C₈

—

143

C₄×C₈

—

178

—

144

C₄×C₈

—

178

—

145

C₂⁴

—

356

—

146

C₂×D₄

—

356

—

147

C₂×D₄

—

356

—

148

C₂×D₄

—

356

—

149

C₂×D₄

—

356

—

150

C₂×D₄

—

356

—

151

C₂×D₄

—

356

—

152

C₂×D₄

—

178

—

153

C₂×D₄

—

178

—

154

C₂²×C₄

—

155

C₂²×C₄

—

356

—

156

C₂²×C₄

D₄×D₈₉

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

178

—

162

C₂×C₈

—

178

—

163

C₂×C₈

#13083561

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

356

—

168

C₄⋊C₄

—

356

—

169

C₄²

Q₈×D₈₉

yes

characteristic

170

C₄²

#13083574

yes

—

171

C₄²

#13083575

yes

characteristic

172

C₂³

—

356

—

173

C₂³

—

712

—

174

C₂³

—

712

—

175

C₂³

—

712

—

176

C₂³

—

712

—

177

C₂³

#15668008

yes

characteristic

178

D₄

—

356

—

179

D₄

—

356

—

180

D₄

—

712

—

181

D₄

—

712

—

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

#15667990

yes

—

189

C₂×C₄

#15667995

yes

characteristic

190

C₂×C₄

#15667997

yes

—

191

C₂×C₄

SD₁₆×D₈₉

yes

—

192

C₂×C₄

C₂×#13083561

yes

—

193

C₂×C₄

#15668038

yes

—

194

C₂×C₄

#15668039

yes

—

195

C₈

—

196

C₈

—

178

—

197

C₂²

—

1424

—

198

C₂²

—

1424

—

199

C₂²

—

1424

—

200

C₂²

—

712

—

201

C₂²

—

712

—

202

C₂²

#19151533

yes

characteristic

203

C₂²

#19151969

yes

—

204

C₂²

#19151972

yes

characteristic

205

C₂²

#19151984

yes

—

206

C₂²

#19151985

yes

characteristic

207

C₄

—

208

C₄

—

1424

—

209

C₄

#19151885

yes

—

210

C₄

#19151887

yes

—

211

C₄

#19151916

yes

—

212

C₄

#19151918

yes

—

213

C₂

—

1424

—

214

C₂

#26019840

yes

characteristic

215

C₂

#26019879

yes

—

216

C₂

#26019881

yes

characteristic

217

C₂

#26021782

yes

—

218

C₂

#26021786

yes

characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#10683
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#15668008

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1844

C₂

C₂³×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
3. #26020408	C₂	semidirect
4. #26018079	C₂	semidirect
5. C₈₉×#1844	C₂	semidirect
113. #1844	D₈₉	semidirect
115. C₈₉	#10683	semidirect

H ≅ G/N

type

3. #26020408

C₂

semidirect

4. #26018079

C₂

semidirect

5. C₈₉×#1844

C₂

semidirect

113. #1844

D₈₉

semidirect

115. C₈₉

#10683

semidirect

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

subgroup	index
#26021058	2
#26021057	2
#26020408	2
#26018079	2
C₈₉×#1844	2
#10683	89

subgroup

index

C₈₉×#1844

Subgroups — #36459532

Conjugacy classes of subgroups

Complements

Maximal subgroups