Group #36473193 — subgroups

Subgroups — #36473193

Order 22784.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151874	C₂	1	2	yes	maximal
3	C₂×#19151413	C₂	1	2	yes	maximal, complemented
4	#26021865	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668045	—	2	2	no	—
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	C₂×Dic₇₁₂	—	2	2	no	—
10	C₂×Dic₇₁₂	—	2	4	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	#19151874	—	2	4	no	—
13	#19151874	—	2	4	no	—
14	C₂×#15667930	—	2	2	no	—
15	#19151413	—	2	4	no	—
16	#19151377	—	2	2	no	—
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#108×C₈₉	C₂²	1	2	yes	—
19	C₂×#13083574	D₄	1	1	yes	characteristic
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	#15668045	—	2	8	no	—
22	#15668045	—	4	4	no	—
23	C₂×Dic₃₅₆	—	2	2	no	—
24	C₂×Dic₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	—	2	2	no	—
26	C₂×Dic₃₅₆	—	2	2	no	—
27	C₂×Dic₃₅₆	—	2	2	no	—
28	C₂×Dic₃₅₆	—	4	2	no	—
29	C₂×Dic₃₅₆	D₄	1	4	yes	—
30	C₂×#13083563	—	4	2	no	—
31	#15668018	—	2	2	no	—
32	#15668018	—	2	2	no	—
33	C₂×#13083558	D₄	1	1	yes	characteristic
34	C₂×#13083557	—	2	1	no	—
35	C₂×C₈×D₈₉	—	2	1	no	—
36	Dic₇₁₂	—	2	8	no	—
37	#15667952	—	2	2	no	—
38	C₂×#13083549	—	2	1	no	—
39	#15667930	—	2	8	no	—
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₈₉	—	2	2	no	—
43	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
44	#108×C₈₉	—	2	2	no	—
45	#108×C₈₉	D₄	1	4	yes	—
46	#13083574	—	2	2	no	—
47	#13083574	—	2	2	no	—
48	C₂×Dic₁₇₈	—	2	4	no	—
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂×Dic₁₇₈	—	4	2	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
55	C₂×D₃₅₆	D₈	1	1	yes	characteristic
56	C₂×C₂²⋊D₈₉	—	2	1	no	—
57	C₂²×Dic₈₉	—	4	2	no	—
58	C₂×C₄×D₈₉	—	2	1	no	—
59	Dic₃₅₆	—	2	4	no	—
60	Dic₃₅₆	—	2	4	no	—
61	Dic₃₅₆	—	4	8	no	—
62	Dic₃₅₆	—	4	2	no	—
63	Dic₃₅₆	—	4	2	no	—
64	#13083563	—	4	8	no	—
65	#13083558	—	2	2	no	—
66	#13083557	—	4	2	no	—
67	C₂×#10623494	—	2	1	no	—
68	C₈×D₈₉	—	4	1	no	—
69	C₈×D₈₉	—	4	1	no	—
70	#13083549	—	4	1	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₈₉	—	2	4	no	—
73	M₁₆×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
79	C₁₆×C₈₉	—	2	2	no	—
80	C₂³×C₈₉	#135	1	1	yes	characteristic
81	C₂²×D₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	#120	1	2	yes	—
83	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
86	C₂²⋊D₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	2	1	no	—
88	C₂×Dic₈₉	—	4	4	no	—
89	C₂×Dic₈₉	—	8	2	no	—
90	C₂×Dic₈₉	—	8	2	no	—
91	D₃₅₆	—	2	2	no	—
92	C₄×D₈₉	—	4	1	no	—
93	C₄×D₈₉	—	4	1	no	—
94	Dic₁₇₈	—	2	2	no	—
95	Dic₁₇₈	—	4	4	no	—
96	Dic₁₇₈	—	4	4	no	—
97	Dic₁₇₈	—	8	2	no	—
98	Dic₁₇₈	—	8	2	no	—
99	C₈×C₈₉	—	4	4	no	—
100	C₈×C₈₉	#120	1	1	yes	characteristic
101	C₈×C₈₉	#121	1	2	yes	—
102	C₈×C₈₉	#127	1	1	yes	characteristic
103	#10623494	—	4	1	no	—
104	C₂²×C₈₉	—	2	2	no	—
105	C₂²×C₈₉	#449	1	2	yes	—
106	C₂²×C₈₉	#466	1	1	yes	characteristic
107	C₂×D₈₉	—	4	2	no	—
108	C₂×D₈₉	—	4	1	no	—
109	C₄×C₈₉	#392	1	1	yes	characteristic
110	C₄×C₈₉	#447	1	2	yes	—
111	C₄×C₈₉	#469	1	1	yes	characteristic
112	Dic₈₉	—	4	1	no	—
113	Dic₈₉	—	8	4	no	—
114	#13315	—	89	1	no	maximal, Sylow
115	C₂×C₈₉	—	2	2	no	—
116	C₂×C₈₉	#1999	1	1	yes	characteristic
117	C₂×C₈₉	#2222	1	2	yes	—
118	D₈₉	—	8	1	no	—
119	C₂²×Q₃₂	—	89	1	no	—
120	C₂×#473	—	89	2	no	—
121	C₂×#362	—	89	2	no	—
122	#2113	—	89	1	no	—
123	#1381	D₈₉	1	1	yes	characteristic, complemented
124	C₈₉	#13315	1	1	yes	characteristic, complemented, Sylow
125	C₂×#137	—	178	2	no	—
126	C₂²×Q₁₆	—	89	2	no	—
127	C₂×#131	—	89	1	no	—
128	C₂×Q₃₂	—	178	4	no	—
129	C₂×Q₃₂	—	178	2	no	—
130	C₂×M₃₂	—	89	1	no	—
131	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
132	#473	—	178	4	no	—
133	#473	—	178	4	no	—
134	C₂×#108	C₂×D₈₉	1	2	yes	—
135	C₂×#101	—	178	2	no	—
136	#362	—	178	4	no	—
137	C₂²⋊C₁₆	—	178	2	no	—
138	C₂×#41	—	89	1	no	—
139	C₂²×Q₈	—	178	2	no	—
140	#137	—	356	4	no	—
141	#137	—	178	8	no	—
142	C₂×Q₁₆	—	89	4	no	—
143	C₂×Q₁₆	—	356	2	no	—
144	C₂×Q₁₆	—	178	2	no	—
145	C₂×Q₁₆	—	178	2	no	—
146	C₂×Q₁₆	—	178	2	no	—
147	C₂×Q₁₆	—	178	2	no	—
148	C₂×Q₁₆	—	178	2	no	—
149	C₂×SD₁₆	—	356	2	no	—
150	#131	—	178	2	no	—
151	#131	—	178	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	89	1	no	—
154	C₂×M₁₆	—	178	1	no	—
155	C₂²×C₈	—	178	1	no	—
156	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
157	Q₃₂	—	178	8	no	—
158	M₃₂	—	178	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	178	1	no	—
161	C₂×C₁₆	D₃₅₆	1	2	yes	—
162	#108	—	2	2	no	—
163	#108	D₃₅₆	1	4	yes	—
164	#101	—	178	8	no	—
165	#41	—	178	2	no	—
166	#41	—	178	2	no	—
167	C₂×Q₈	—	89	1	no	—
168	C₂×Q₈	—	356	2	no	—
169	C₂×Q₈	—	356	2	no	—
170	C₂×Q₈	—	356	2	no	—
171	C₂×Q₈	—	356	2	no	—
172	C₂×Q₈	—	356	2	no	—
173	C₂×Q₈	—	178	4	no	—
174	C₂×D₄	—	89	1	no	—
175	C₂×D₄	—	178	1	no	—
176	C₂²×C₄	—	356	2	no	—
177	C₂²×C₄	—	178	1	no	—
178	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
179	Q₁₆	—	356	8	no	—
180	Q₁₆	—	356	2	no	—
181	Q₁₆	—	356	2	no	—
182	Q₁₆	—	178	4	no	—
183	Q₁₆	—	178	4	no	—
184	SD₁₆	—	356	8	no	—
185	M₁₆	—	2	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	356	2	no	—
188	M₁₆	—	178	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	356	1	no	—
191	C₂×C₈	—	356	1	no	—
192	C₂×C₈	—	178	1	no	—
193	C₂×C₈	D₄×D₈₉	1	2	yes	—
194	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
195	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
196	C₂×C₈	#13083575	1	1	yes	characteristic
197	C₁₆	—	2	2	no	—
198	C₁₆	—	356	1	no	—
199	C₂³	—	178	1	no	—
200	C₂³	#15668035	1	1	yes	characteristic
201	Q₈	—	356	4	no	—
202	Q₈	—	356	4	no	—
203	Q₈	—	178	2	no	—
204	Q₈	—	712	2	no	—
205	Q₈	—	712	2	no	—
206	D₄	—	356	2	no	—
207	D₄	—	178	2	no	—
208	C₂×C₄	—	356	1	no	—
209	C₂×C₄	—	356	1	no	—
210	C₂×C₄	—	356	4	no	—
211	C₂×C₄	—	178	1	no	—
212	C₂×C₄	—	712	2	no	—
213	C₂×C₄	—	712	2	no	—
214	C₂×C₄	#15667982	1	2	yes	—
215	C₂×C₄	#15667987	1	1	yes	characteristic
216	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
217	C₂×C₄	#15668039	1	2	yes	—
218	C₈	—	4	4	no	—
219	C₈	—	356	1	no	—
220	C₈	#15667982	1	1	yes	characteristic
221	C₈	#15667987	1	2	yes	—
222	C₈	#15668009	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	356	1	no	—
225	C₂²	—	356	2	no	—
226	C₂²	#19151746	1	2	yes	—
227	C₂²	#19151835	1	1	yes	characteristic
228	C₄	—	356	1	no	—
229	C₄	—	712	4	no	—
230	C₄	#19151508	1	1	yes	characteristic
231	C₄	#19151733	1	2	yes	—
232	C₄	#19151852	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	712	1	no	—
235	C₂	#26021251	1	1	yes	characteristic
236	C₂	#26022378	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₂

C₂

yes

maximal, characteristic, complemented

C₂×#19151874

C₂

yes

maximal

C₂×#19151413

C₂

yes

maximal, complemented

#26021865

C₂

yes

maximal, characteristic

C₈₉×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#15668045

—

C₂²×Dic₃₅₆

C₂²

yes

—

C₂×#15668018

C₂²

yes

characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

—

C₂×#15667952

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

C₂×#13083574

D₄

yes

characteristic

C₂²×Dic₁₇₈

—

#15668045

—

#15668045

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

D₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#13083557

—

C₂×C₈×D₈₉

—

Dic₇₁₂

—

#15667952

—

C₂×#13083549

—

#15667930

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#108×C₈₉

—

#108×C₈₉

D₄

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₈

yes

characteristic

C₂×D₃₅₆

D₈

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

—

—

—

—

C₈×D₈₉

—

C₈×D₈₉

—

#13083549

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

#41

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

#135

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

#120

yes

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

100

C₈×C₈₉

#120

yes

characteristic

101

C₈×C₈₉

#121

yes

—

102

C₈×C₈₉

#127

yes

characteristic

103

#10623494

—

104

C₂²×C₈₉

—

105

C₂²×C₈₉

#449

yes

—

106

C₂²×C₈₉

#466

yes

characteristic

107

C₂×D₈₉

—

108

C₂×D₈₉

—

109

C₄×C₈₉

#392

yes

characteristic

110

C₄×C₈₉

#447

yes

—

111

C₄×C₈₉

#469

yes

characteristic

112

Dic₈₉

—

113

Dic₈₉

—

114

#13315

—

maximal, Sylow

115

C₂×C₈₉

—

116

C₂×C₈₉

#1999

yes

characteristic

117

C₂×C₈₉

#2222

yes

—

118

D₈₉

—

119

C₂²×Q₃₂

—

120

C₂×#473

—

121

C₂×#362

—

122

#2113

—

123

#1381

D₈₉

yes

characteristic, complemented

124

C₈₉

#13315

yes

characteristic, complemented, Sylow

125

C₂×#137

—

178

—

126

C₂²×Q₁₆

—

127

C₂×#131

—

128

C₂×Q₃₂

—

178

—

129

C₂×Q₃₂

—

178

—

130

C₂×M₃₂

—

131

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

132

#473

—

178

—

133

#473

—

178

—

134

C₂×#108

C₂×D₈₉

yes

—

135

C₂×#101

—

178

—

136

#362

—

178

—

137

C₂²⋊C₁₆

—

178

—

138

C₂×#41

—

139

C₂²×Q₈

—

178

—

140

#137

—

356

—

141

#137

—

178

—

142

C₂×Q₁₆

—

143

C₂×Q₁₆

—

356

—

144

C₂×Q₁₆

—

178

—

145

C₂×Q₁₆

—

178

—

146

C₂×Q₁₆

—

178

—

147

C₂×Q₁₆

—

178

—

148

C₂×Q₁₆

—

178

—

149

C₂×SD₁₆

—

356

—

150

#131

—

178

—

151

#131

—

178

—

152

C₂×M₁₆

—

153

C₂×M₁₆

—

154

C₂×M₁₆

—

178

—

155

C₂²×C₈

—

178

—

156

C₂²×C₈

C₂²×D₈₉

yes

characteristic

157

Q₃₂

—

178

—

158

M₃₂

—

178

—

159

C₂×C₁₆

—

160

C₂×C₁₆

—

178

—

161

C₂×C₁₆

D₃₅₆

yes

—

162

#108

—

163

#108

D₃₅₆

yes

—

164

#101

—

178

—

165

#41

—

178

—

166

#41

—

178

—

167

C₂×Q₈

—

168

C₂×Q₈

—

356

—

169

C₂×Q₈

—

356

—

170

C₂×Q₈

—

356

—

171

C₂×Q₈

—

356

—

172

C₂×Q₈

—

356

—

173

C₂×Q₈

—

178

—

174

C₂×D₄

—

175

C₂×D₄

—

178

—

176

C₂²×C₄

—

356

—

177

C₂²×C₄

—

178

—

178

C₂²×C₄

D₄×D₈₉

yes

characteristic

179

Q₁₆

—

356

—

180

Q₁₆

—

356

—

181

Q₁₆

—

356

—

182

Q₁₆

—

178

—

183

Q₁₆

—

178

—

184

SD₁₆

—

356

—

185

M₁₆

—

186

M₁₆

—

187

M₁₆

—

356

—

188

M₁₆

—

178

—

189

C₂×C₈

—

190

C₂×C₈

—

356

—

191

C₂×C₈

—

356

—

192

C₂×C₈

—

178

—

193

C₂×C₈

D₄×D₈₉

yes

—

194

C₂×C₈

C₂×D₃₅₆

yes

characteristic

195

C₂×C₈

C₂×D₃₅₆

yes

—

196

C₂×C₈

#13083575

yes

characteristic

197

C₁₆

—

198

C₁₆

—

356

—

199

C₂³

—

178

—

200

C₂³

#15668035

yes

characteristic

201

Q₈

—

356

—

202

Q₈

—

356

—

203

Q₈

—

178

—

204

Q₈

—

712

—

205

Q₈

—

712

—

206

D₄

—

356

—

207

D₄

—

178

—

208

C₂×C₄

—

356

—

209

C₂×C₄

—

356

—

210

C₂×C₄

—

356

—

211

C₂×C₄

—

178

—

212

C₂×C₄

—

712

—

213

C₂×C₄

—

712

—

214

C₂×C₄

#15667982

yes

—

215

C₂×C₄

#15667987

yes

characteristic

216

C₂×C₄

D₈×D₈₉

yes

characteristic

217

C₂×C₄

#15668039

yes

—

218

C₈

—

219

C₈

—

356

—

220

C₈

#15667982

yes

characteristic

221

C₈

#15667987

yes

—

222

C₈

#15668009

yes

characteristic

223

C₂²

—

224

C₂²

—

356

—

225

C₂²

—

356

—

226

C₂²

#19151746

yes

—

227

C₂²

#19151835

yes

characteristic

228

C₄

—

356

—

229

C₄

—

712

—

230

C₄

#19151508

yes

characteristic

231

C₄

#19151733

yes

—

232

C₄

#19151852

yes

characteristic

233

C₂

—

234

C₂

—

712

—

235

C₂

#26021251

yes

characteristic

236

C₂

#26022378

yes

—

	subgroup	G/H
center	C₂²	#19151835
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1381	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂×#13083574	D₄
hypercenter	#1381	D₈₉
hypocenter	C₈₉	#13315
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#19151835

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1381

C₂

C₂²×C₈₉

D₄

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂²×Dic₇₁₂	C₂	semidirect
3. C₂×#19151413	C₂	semidirect
5. C₈₉×#1381	C₂	semidirect
123. #1381	D₈₉	semidirect
124. C₈₉	#13315	semidirect

H ≅ G/N

type

1. C₂²×Dic₇₁₂

C₂

semidirect

3. C₂×#19151413

C₂

semidirect

5. C₈₉×#1381

C₂

semidirect

123. #1381

D₈₉

semidirect

124. C₈₉

#13315

semidirect

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

subgroup	index
C₂²×Dic₇₁₂	2
C₂×#19151874	2
C₂×#19151413	2
#26021865	2
C₈₉×#1381	2
#13315	89

subgroup

index

C₂²×Dic₇₁₂

C₈₉×#1381

Subgroups — #36473193

Conjugacy classes of subgroups

Complements

Maximal subgroups