Group #36447351 — subgroups

Subgroups — #36447351

Order 22784.

398 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#19151722	C₂	1	1	yes	maximal, characteristic
2	#26020245	C₂	1	2	yes	maximal
3	#26019007	C₂	1	2	yes	maximal
4	#26018127	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1292	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	#19151722	—	2	2	no	—
8	#19151722	—	2	2	no	—
9	#19151622	C₂²	1	2	yes	—
10	C₂×#15667941	C₄	1	2	yes	—
11	C₂×#15667930	—	2	2	no	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
14	#19151366	—	2	2	no	—
15	#19151348	—	2	2	no	—
16	#19151348	—	2	2	no	—
17	#423×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
22	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₇₈	—	2	2	no	—
24	C₄×Dic₁₇₈	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
26	C₂×#13083553	Q₈	1	2	yes	—
27	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	#15667941	—	2	4	no	—
31	#15667930	—	4	4	no	—
32	C₄×#10623494	D₄	1	2	yes	—
33	C₄×#10623494	D₄	1	2	yes	—
34	C₈×Dic₈₉	—	2	2	no	—
35	C₈×Dic₈₉	—	2	2	no	—
36	C₂×M₁₆×C₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	—	2	2	no	—
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₂×Dic₁₇₈	—	2	2	no	—
43	C₂×Dic₁₇₈	—	2	1	no	—
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×Dic₁₇₈	SD₁₆	1	4	yes	—
46	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
48	#13083558	—	4	2	no	—
49	#13083558	—	4	2	no	—
50	C₂×#10623494	—	2	2	no	—
51	C₂×#10623494	—	4	1	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
55	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊Dic₈₉	—	2	2	no	—
57	#13083553	—	2	4	no	—
58	C₄×Dic₈₉	—	2	2	no	—
59	C₄×Dic₈₉	—	2	2	no	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	M₁₆×C₈₉	—	4	2	no	—
64	M₁₆×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	2	2	no	—
66	C₂×C₈×C₈₉	—	4	2	no	—
67	C₂×C₈×C₈₉	—	4	2	no	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₄²×C₈₉	#41	1	2	yes	—
70	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	4	1	no	—
74	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#121	1	2	yes	—
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
81	C₂×Dic₈₉	—	2	2	no	—
82	C₂×Dic₈₉	—	2	2	no	—
83	C₂×Dic₈₉	—	2	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	#106	1	4	yes	—
86	Dic₁₇₈	—	2	4	no	—
87	Dic₁₇₈	—	2	4	no	—
88	C₈×C₈₉	—	4	2	no	—
89	C₈×C₈₉	—	8	2	no	—
90	#10623494	—	2	1	no	—
91	#10623494	—	2	2	no	—
92	#10623494	—	8	2	no	—
93	#10623494	#101	1	2	yes	complemented
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	#380	1	1	yes	characteristic
97	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
98	C₂²×C₈₉	#427	1	1	yes	characteristic
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	2	no	—
101	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
102	C₄×C₈₉	#413	1	1	yes	characteristic
103	C₄×C₈₉	#465	1	2	yes	—
104	C₄×C₈₉	#465	1	2	yes	—
105	Dic₈₉	—	2	4	no	—
106	Dic₈₉	—	4	2	no	—
107	#8615	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	—	2	1	no	—
109	C₂×C₈₉	—	2	1	no	—
110	C₂×C₈₉	#1801	1	1	yes	characteristic
111	C₂×C₈₉	#1807	1	1	yes	characteristic
112	C₂×C₈₉	#1813	1	1	yes	characteristic
113	C₂×C₈×Q₈	—	89	1	no	—
114	#1801	—	89	2	no	—
115	#1566	—	89	2	no	—
116	#1292	—	89	1	no	—
117	#1292	D₈₉	1	1	yes	characteristic
118	C₈₉	#8615	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄×Q₈	—	89	1	no	—
120	C₈×Q₈	—	178	2	no	—
121	C₈×Q₈	—	178	2	no	—
122	#423	—	89	2	no	—
123	#423	C₂×D₈₉	1	2	yes	—
124	C₂×C₄⋊C₈	—	89	2	no	—
125	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₂×#101	—	178	2	no	—
127	C₂×C₄×C₈	—	89	1	no	—
128	C₂×C₄×C₈	—	89	2	no	—
129	#343	—	178	2	no	—
130	#334	—	178	2	no	—
131	#334	—	178	2	no	—
132	C₂²×Q₈	—	89	1	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	178	2	no	—
135	C₂²×C₈	—	89	1	no	—
136	C₂²×C₈	—	89	1	no	—
137	C₂²×C₈	C₄×D₈₉	1	2	yes	—
138	C₄×Q₈	—	178	2	no	—
139	C₄×Q₈	—	178	2	no	—
140	C₂×C₄⋊C₄	—	89	1	no	—
141	C₂×C₄⋊C₄	—	89	2	no	—
142	C₂×C₄²	—	89	2	no	—
143	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	178	4	no	—
148	C₄⋊C₈	—	178	2	no	—
149	C₄⋊C₈	—	178	2	no	—
150	#101	—	356	4	no	—
151	C₄×C₈	—	89	2	no	—
152	C₄×C₈	—	89	2	no	—
153	C₄×C₈	—	178	2	no	—
154	C₄×C₈	—	178	2	no	—
155	C₂×Q₈	—	89	4	no	—
156	C₂×Q₈	—	178	2	no	—
157	C₂×Q₈	—	178	1	no	—
158	C₂×Q₈	—	178	1	no	—
159	C₂²×C₄	—	89	2	no	—
160	C₂²×C₄	—	89	2	no	—
161	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
163	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	2	no	—
166	M₁₆	—	356	2	no	—
167	M₁₆	—	356	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	89	1	no	—
172	C₂×C₈	—	89	1	no	—
173	C₂×C₈	—	89	2	no	—
174	C₂×C₈	—	356	1	no	—
175	C₂×C₈	—	356	2	no	—
176	C₂×C₈	—	178	2	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	—	178	4	no	—
179	C₄²	—	178	2	no	—
180	C₄²	—	178	2	no	—
181	C₄²	D₄×D₈₉	1	2	yes	—
182	C₄²	#13083574	1	2	yes	—
183	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
184	Q₈	—	178	4	no	—
185	Q₈	—	178	4	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	89	4	no	—
190	C₂×C₄	—	356	2	no	—
191	C₂×C₄	—	178	2	no	—
192	C₂×C₄	—	178	2	no	—
193	C₂×C₄	—	178	2	no	—
194	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
195	C₂×C₄	#15667969	1	1	yes	characteristic
196	C₂×C₄	#15667985	1	2	yes	—
197	C₂×C₄	#15667990	1	2	yes	—
198	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
199	C₂×C₄	#15668031	1	1	yes	characteristic
200	C₂×C₄	#15668035	1	1	yes	characteristic
201	C₈	—	4	2	no	—
202	C₈	—	8	2	no	—
203	C₈	—	89	2	no	—
204	C₈	—	178	1	no	—
205	C₈	—	178	2	no	—
206	C₈	—	712	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#19151457	1	1	yes	characteristic
210	C₂²	#106×D₈₉	1	1	yes	characteristic
211	C₂²	#19151635	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	356	2	no	—
215	C₄	—	178	4	no	—
216	C₄	#101×D₈₉	1	1	yes	characteristic
217	C₄	#19151579	1	1	yes	characteristic
218	C₄	#19151826	1	2	yes	—
219	C₄	#19151830	1	2	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#26020247	1	1	yes	characteristic
223	C₂	#26020276	1	1	yes	characteristic
224	C₂	#26020301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151722

C₂

yes

maximal, characteristic

#26020245

C₂

yes

maximal

#26019007

C₂

yes

maximal

#26018127

C₂

yes

maximal, characteristic

C₈₉×#1292

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₇₈

C₂²

yes

—

—

C₂²

yes

—

C₂×#15667941

C₄

yes

—

C₂×#15667930

—

C₂×C₄×#10623494

C₂²

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

—

#19151366

—

#19151348

—

#19151348

—

#423×C₈₉

C₂²

yes

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

D₄

yes

characteristic

C₂×#13083558

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

—

C₂×C₄⋊Dic₈₉

D₄

yes

characteristic

C₂×#13083553

Q₈

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

SD₁₆

yes

—

C₂²×Dic₈₉

C₄⋊C₄

yes

—

C₂²×Dic₈₉

C₄⋊C₄

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#106

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

100

C₄×C₈₉

—

101

C₄×C₈₉

C₂×#101

yes

characteristic

102

C₄×C₈₉

#413

yes

characteristic

103

C₄×C₈₉

#465

yes

—

104

C₄×C₈₉

#465

yes

—

105

Dic₈₉

—

106

Dic₈₉

—

107

#8615

—

maximal, Sylow

108

C₂×C₈₉

—

109

C₂×C₈₉

—

110

C₂×C₈₉

#1801

yes

characteristic

111

C₂×C₈₉

#1807

yes

characteristic

112

C₂×C₈₉

#1813

yes

characteristic

113

C₂×C₈×Q₈

—

114

#1801

—

115

#1566

—

116

#1292

—

117

#1292

D₈₉

yes

characteristic

118

C₈₉

#8615

yes

characteristic, complemented, Sylow

119

C₂×C₄×Q₈

—

120

C₈×Q₈

—

178

—

121

C₈×Q₈

—

178

—

122

#423

—

123

#423

C₂×D₈₉

yes

—

124

C₂×C₄⋊C₈

—

125

C₂×C₄⋊C₈

C₂×D₈₉

yes

characteristic

126

C₂×#101

—

178

—

127

C₂×C₄×C₈

—

128

C₂×C₄×C₈

—

129

#343

—

178

—

130

#334

—

178

—

131

#334

—

178

—

132

C₂²×Q₈

—

133

C₂×M₁₆

—

134

C₂×M₁₆

—

178

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

C₄×D₈₉

yes

—

138

C₄×Q₈

—

178

—

139

C₄×Q₈

—

178

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

—

143

C₂×C₄²

C₂²×D₈₉

yes

characteristic

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

178

—

148

C₄⋊C₈

—

178

—

149

C₄⋊C₈

—

178

—

150

#101

—

356

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

178

—

154

C₄×C₈

—

178

—

155

C₂×Q₈

—

156

C₂×Q₈

—

178

—

157

C₂×Q₈

—

178

—

158

C₂×Q₈

—

178

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

162

C₂²×C₄

D₄×D₈₉

yes

characteristic

163

C₂²×C₄

Q₈×D₈₉

yes

characteristic

164

M₁₆

—

165

M₁₆

—

166

M₁₆

—

356

—

167

M₁₆

—

356

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

356

—

175

C₂×C₈

—

356

—

176

C₂×C₈

—

178

—

177

C₄⋊C₄

—

178

—

178

C₄⋊C₄

—

178

—

179

C₄²

—

178

—

180

C₄²

—

178

—

181

C₄²

D₄×D₈₉

yes

—

182

C₄²

#13083574

yes

—

183

C₂³

C₄⋊C₄×D₈₉

yes

characteristic

184

Q₈

—

178

—

185

Q₈

—

178

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

356

—

191

C₂×C₄

—

178

—

192

C₂×C₄

—

178

—

193

C₂×C₄

—

178

—

194

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

195

C₂×C₄

#15667969

yes

characteristic

196

C₂×C₄

#15667985

yes

—

197

C₂×C₄

#15667990

yes

—

198

C₂×C₄

SD₁₆×D₈₉

yes

—

199

C₂×C₄

#15668031

yes

characteristic

200

C₂×C₄

#15668035

yes

characteristic

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

178

—

205

C₈

—

178

—

206

C₈

—

712

—

207

C₂²

—

208

C₂²

—

209

C₂²

#19151457

yes

characteristic

210

C₂²

#106×D₈₉

yes

characteristic

211

C₂²

#19151635

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

356

—

215

C₄

—

178

—

216

C₄

#101×D₈₉

yes

characteristic

217

C₄

#19151579

yes

characteristic

218

C₄

#19151826

yes

—

219

C₄

#19151830

yes

—

220

C₂

—

221

C₂

—

222

C₂

#26020247

yes

characteristic

223

C₂

#26020276

yes

characteristic

224

C₂

#26020301

yes

characteristic

	subgroup	G/H
center	C₂²	#19151457
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1292	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1292	D₈₉
hypocenter	C₈₉	#8615
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

subgroup

G/H

center

C₂²

#19151457

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1292

C₂

socle

C₂²×C₈₉

#380

prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

N	H ≅ G/N	type
93. #10623494	#101	semidirect
118. C₈₉	#8615	semidirect

H ≅ G/N

type

93. #10623494

#101

semidirect

118. C₈₉

#8615

semidirect

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

subgroup	index
C₂×#19151722	2
#26020245	2
#26019007	2
#26018127	2
C₈₉×#1292	2
#8615	89

subgroup

index

C₈₉×#1292

Subgroups — #36447351

Conjugacy classes of subgroups

Complements

Maximal subgroups