Group #36454043 — subgroups

Subgroups — #36454043

Order 22784.

442 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26021095	C₂	1	2	yes	maximal
3	#26018925	C₂	1	2	yes	maximal
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19152003	—	2	1	no	—
8	#19152003	—	2	1	no	—
9	#19152003	—	2	2	no	—
10	#19151651	C₂²	1	1	yes	characteristic
11	C₂×#15667946	C₄	1	4	yes	—
12	C₂×#15667930	—	2	4	no	—
13	C₄×#13083558	C₂²	1	1	yes	characteristic
14	#19151345	—	2	4	no	—
15	#19151320	—	2	1	no	—
16	C₈×#10623494	—	2	1	no	—
17	#423×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#335×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	C₂×#13083558	D₄	1	2	yes	—
22	#15668012	—	2	2	no	—
23	#15668012	—	2	2	no	—
24	#15668012	—	2	2	no	—
25	#15668012	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	#15667946	—	2	4	no	—
29	#15667946	—	2	4	no	—
30	C₄⋊#10623494	D₄	1	2	yes	—
31	#15667930	—	4	8	no	—
32	C₂²⋊#10623494	—	2	1	no	—
33	#15667923	—	2	1	no	—
34	C₄×#10623494	—	2	1	no	—
35	C₄×#10623494	—	2	1	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	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₂×Dic₁₇₈	—	4	2	no	—
45	C₂×Dic₁₇₈	—	4	2	no	—
46	C₂×Dic₁₇₈	—	4	4	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂²×Dic₈₉	—	2	4	no	—
49	#13083558	—	2	2	no	—
50	#13083558	D₈	1	4	yes	—
51	C₂×#10623494	—	2	2	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	2	4	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
60	M₁₆×C₈₉	—	4	4	no	—
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₈₉	#135	1	2	yes	—
80	C₂×Dic₈₉	—	4	4	no	—
81	C₂×Dic₈₉	—	4	8	no	—
82	Dic₁₇₈	—	8	4	no	—
83	Dic₁₇₈	—	8	2	no	—
84	Dic₁₇₈	—	8	2	no	—
85	C₈×C₈₉	—	2	2	no	—
86	C₈×C₈₉	—	8	2	no	—
87	C₈×C₈₉	#101	1	4	yes	complemented
88	#10623494	—	2	4	no	—
89	#10623494	—	4	2	no	—
90	C₂²×C₈₉	—	2	1	no	—
91	C₂²×C₈₉	—	2	1	no	—
92	C₂²×C₈₉	#390	1	1	yes	characteristic
93	C₂²×C₈₉	C₄×D₈	1	2	yes	—
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	2	1	no	—
96	C₄×C₈₉	C₂×#101	1	2	yes	—
97	C₄×C₈₉	#466	1	2	yes	—
98	C₄×C₈₉	#473	1	2	yes	—
99	C₄×C₈₉	#493	1	1	yes	characteristic
100	C₄×C₈₉	#495	1	1	yes	characteristic
101	Dic₈₉	—	8	4	no	—
102	#9799	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	2	2	no	—
104	C₂×C₈₉	#1966	1	1	yes	characteristic
105	C₂×C₈₉	#1970	1	1	yes	characteristic
106	C₂×C₈₉	#1979	1	1	yes	characteristic
107	C₂×#500	—	89	1	no	—
108	#1966	—	89	2	no	—
109	#1567	D₈₉	1	1	yes	characteristic
110	#1546	—	89	2	no	—
111	C₈×M₁₆	—	89	1	no	—
112	C₈₉	#9799	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	—	89	2	no	—
114	#500	—	178	1	no	—
115	#500	—	178	1	no	—
116	#500	—	178	2	no	—
117	#431	—	89	1	no	—
118	C₂×#107	—	89	4	no	—
119	#423	C₂×D₈₉	1	2	yes	—
120	C₂×#101	—	178	4	no	—
121	C₄×M₁₆	—	89	1	no	—
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	#335	—	2	2	no	—
124	#332	—	178	4	no	—
125	#322	—	178	1	no	—
126	C₈²	—	178	1	no	—
127	C₂²×Q₈	—	178	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	89	2	no	—
130	C₂²×C₈	C₄×D₈₉	1	2	yes	—
131	C₄⋊Q₈	—	178	2	no	—
132	C₄⋊Q₈	—	178	2	no	—
133	C₄⋊Q₈	—	178	2	no	—
134	C₄⋊Q₈	—	178	2	no	—
135	C₂×C₄⋊C₄	—	89	4	no	—
136	C₂×C₄⋊C₄	—	178	2	no	—
137	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
138	#107	—	178	4	no	—
139	#107	—	178	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	89	2	no	—
143	#101	—	356	8	no	—
144	C₂²⋊C₈	—	178	1	no	—
145	#97	—	178	1	no	—
146	C₄×C₈	—	178	1	no	—
147	C₄×C₈	—	178	1	no	—
148	C₄×C₈	D₃₅₆	1	1	yes	characteristic
149	C₄×C₈	D₃₅₆	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
151	C₂×Q₈	—	356	2	no	—
152	C₂×Q₈	—	356	2	no	—
153	C₂×Q₈	—	356	4	no	—
154	C₂×Q₈	—	356	4	no	—
155	C₂²×C₄	—	178	4	no	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	#13083575	1	1	yes	characteristic
159	M₁₆	—	4	4	no	—
160	M₁₆	—	89	4	no	—
161	M₁₆	—	178	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	178	2	no	—
165	C₂×C₈	—	178	2	no	—
166	C₂×C₈	#13083551	1	2	yes	—
167	C₂×C₈	#13083551	1	2	yes	—
168	C₄⋊C₄	—	356	2	no	—
169	C₄⋊C₄	—	356	2	no	—
170	C₄⋊C₄	—	178	4	no	—
171	C₄⋊C₄	—	178	4	no	—
172	C₄²	D₄×D₈₉	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
174	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
175	C₄²	#13083575	1	1	yes	characteristic
176	C₂³	#15667974	1	1	yes	characteristic
177	Q₈	—	712	2	no	—
178	Q₈	—	712	2	no	—
179	Q₈	—	712	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	356	8	no	—
183	C₂×C₄	—	356	4	no	—
184	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
185	C₂×C₄	#15667974	1	1	yes	characteristic
186	C₂×C₄	#15667987	1	1	yes	characteristic
187	C₂×C₄	#15667987	1	1	yes	characteristic
188	C₂×C₄	#15668001	1	1	yes	characteristic
189	C₂×C₄	#15668008	1	1	yes	characteristic
190	C₂×C₄	D₈×D₈₉	1	2	yes	—
191	C₂×C₄	#15668035	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	8	2	no	—
194	C₈	—	356	2	no	—
195	C₈	—	178	4	no	—
196	C₈	#15667930	1	4	yes	complemented
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#19151499	1	1	yes	characteristic
200	C₂²	#19151680	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	712	4	no	—
204	C₄	C₂×#15667930	1	2	yes	—
205	C₄	#19151835	1	2	yes	—
206	C₄	#19151874	1	2	yes	—
207	C₄	#19151973	1	1	yes	characteristic
208	C₄	#19151983	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#26021095	1	1	yes	characteristic
211	C₂	#26021116	1	1	yes	characteristic
212	C₂	#26021165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152003

C₂

yes

maximal, characteristic

#26021095

C₂

yes

maximal

#26018925

C₂

yes

maximal

C₈×#13083558

C₂

yes

maximal, characteristic

C₈₉×#1567

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#15667946

C₄

yes

—

C₂×#15667930

—

C₄×#13083558

C₂²

yes

—

—

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#335×C₈₉

—

C₂²×Dic₁₇₈

—

C₂×#13083558

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

#15667946

—

#15667946

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

D₈

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

characteristic

C₂³×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

C₄×D₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#101

yes

—

C₄×C₈₉

#466

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

#493

yes

characteristic

100

C₄×C₈₉

#495

yes

characteristic

101

Dic₈₉

—

102

#9799

—

maximal, Sylow

103

C₂×C₈₉

—

104

C₂×C₈₉

#1966

yes

characteristic

105

C₂×C₈₉

#1970

yes

characteristic

106

C₂×C₈₉

#1979

yes

characteristic

107

C₂×#500

—

108

#1966

—

109

#1567

D₈₉

yes

characteristic

110

#1546

—

111

C₈×M₁₆

—

112

C₈₉

#9799

yes

characteristic, complemented, Sylow

113

C₂×C₄⋊Q₈

—

114

#500

—

178

—

115

#500

—

178

—

116

#500

—

178

—

117

#431

—

118

C₂×#107

—

119

#423

C₂×D₈₉

yes

—

120

C₂×#101

—

178

—

121

C₄×M₁₆

—

122

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

123

#335

—

124

#332

—

178

—

125

#322

—

178

—

126

C₈²

—

178

—

127

C₂²×Q₈

—

178

—

128

C₂×M₁₆

—

129

C₂×M₁₆

—

130

C₂²×C₈

C₄×D₈₉

yes

—

131

C₄⋊Q₈

—

178

—

132

C₄⋊Q₈

—

178

—

133

C₄⋊Q₈

—

178

—

134

C₄⋊Q₈

—

178

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

178

—

137

C₂×C₄²

C₂²×D₈₉

yes

characteristic

138

#107

—

178

—

139

#107

—

178

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

#101

—

356

—

144

C₂²⋊C₈

—

178

—

145

#97

—

178

—

146

C₄×C₈

—

178

—

147

C₄×C₈

—

178

—

148

C₄×C₈

D₃₅₆

yes

characteristic

149

C₄×C₈

D₃₅₆

yes

characteristic

150

C₄×C₈

C₂²⋊D₈₉

yes

—

151

C₂×Q₈

—

356

—

152

C₂×Q₈

—

356

—

153

C₂×Q₈

—

356

—

154

C₂×Q₈

—

356

—

155

C₂²×C₄

—

178

—

156

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

157

C₂²×C₄

D₄×D₈₉

yes

characteristic

158

C₂²×C₄

#13083575

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

178

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

178

—

165

C₂×C₈

—

178

—

166

C₂×C₈

#13083551

yes

—

167

C₂×C₈

#13083551

yes

—

168

C₄⋊C₄

—

356

—

169

C₄⋊C₄

—

356

—

170

C₄⋊C₄

—

178

—

171

C₄⋊C₄

—

178

—

172

C₄²

D₄×D₈₉

yes

characteristic

173

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

174

C₄²

C₂×D₃₅₆

yes

characteristic

175

C₄²

#13083575

yes

characteristic

176

C₂³

#15667974

yes

characteristic

177

Q₈

—

712

—

178

Q₈

—

712

—

179

Q₈

—

712

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

356

—

183

C₂×C₄

—

356

—

184

C₂×C₄

C₂×#13083551

yes

characteristic

185

C₂×C₄

#15667974

yes

characteristic

186

C₂×C₄

#15667987

yes

characteristic

187

C₂×C₄

#15667987

yes

characteristic

188

C₂×C₄

#15668001

yes

characteristic

189

C₂×C₄

#15668008

yes

characteristic

190

C₂×C₄

D₈×D₈₉

yes

—

191

C₂×C₄

#15668035

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

356

—

195

C₈

—

178

—

196

C₈

#15667930

yes

complemented

197

C₂²

—

198

C₂²

—

199

C₂²

#19151499

yes

characteristic

200

C₂²

#19151680

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

712

—

204

C₄

C₂×#15667930

yes

—

205

C₄

#19151835

yes

—

206

C₄

#19151874

yes

—

207

C₄

#19151973

yes

characteristic

208

C₄

#19151983

yes

characteristic

209

C₂

—

210

C₂

#26021095

yes

characteristic

211

C₂

#26021116

yes

characteristic

212

C₂

#26021165

yes

characteristic

	subgroup	G/H
center	C₂²	#19151499
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1567	D₈₉
hypocenter	C₈₉	#9799
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂²

#19151499

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1567

C₂

socle

C₂²×C₈₉

#390

prime radical

C₂³×C₈₉

C₄×D₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
87. C₈×C₈₉	#101	semidirect
112. C₈₉	#9799	semidirect
196. C₈	#15667930	semidirect

H ≅ G/N

type

87. C₈×C₈₉

#101

semidirect

112. C₈₉

#9799

semidirect

196. C₈

#15667930

semidirect

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

subgroup	index
C₂×#19152003	2
#26021095	2
#26018925	2
C₈×#13083558	2
C₈₉×#1567	2
#9799	89

subgroup

index

C₈₉×#1567

Subgroups — #36454043

Conjugacy classes of subgroups

Complements

Maximal subgroups