Group #36454113 — subgroups

Subgroups — #36454113

Order 22784.

402 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 135 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	#26021242	C₂	1	2	yes	maximal
2	#26019008	C₂	1	1	yes	maximal, characteristic
3	#26018937	C₂	1	2	yes	maximal
4	#26018541	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	2	yes	—
8	C₄×#13083558	C₂²	1	1	yes	characteristic
9	#19151367	—	2	4	no	—
10	#19151349	—	2	2	no	—
11	#19151344	—	2	4	no	—
12	#19151320	—	2	1	no	—
13	#19151319	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
15	#500×C₈₉	—	2	1	no	—
16	#500×C₈₉	—	2	1	no	—
17	#500×C₈₉	—	2	2	no	—
18	C₂×#107×C₈₉	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	D₄	1	2	yes	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	D₄	1	2	yes	—
25	C₂²⋊#10623494	—	2	1	no	—
26	#15667923	—	2	1	no	—
27	C₄×#10623494	—	2	1	no	—
28	C₄×#10623494	—	2	1	no	—
29	C₂²×Q₈×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	—	2	2	no	—
33	C₄⋊Q₈×C₈₉	—	2	2	no	—
34	C₄⋊Q₈×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	#107×C₈₉	—	2	4	no	—
39	#107×C₈₉	—	2	4	no	—
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	2	yes	—
43	#13083558	—	2	4	no	—
44	#13083558	—	4	4	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×Q₈×C₈₉	—	4	2	no	—
49	C₂×Q₈×C₈₉	—	4	2	no	—
50	C₂×Q₈×C₈₉	—	8	4	no	—
51	C₂²×C₄×C₈₉	—	2	4	no	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₄⋊C₄×C₈₉	—	2	4	no	—
59	C₄⋊C₄×C₈₉	—	2	4	no	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
67	Q₈×C₈₉	—	8	4	no	—
68	Q₈×C₈₉	—	8	4	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	4	4	no	—
72	C₂×C₄×C₈₉	—	8	4	no	—
73	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#137	1	4	yes	—
80	C₈×C₈₉	—	2	2	no	—
81	C₈×C₈₉	#101	1	4	yes	—
82	#10623494	—	4	2	no	—
83	#10623494	—	4	2	no	—
84	#10623494	—	8	2	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	—	2	1	no	—
87	C₂²×C₈₉	#390	1	1	yes	characteristic
88	C₂²×C₈₉	#441	1	2	yes	—
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	8	4	no	—
92	C₄×C₈₉	C₂×#101	1	2	yes	—
93	C₄×C₈₉	#470	1	2	yes	—
94	C₄×C₈₉	#472	1	2	yes	—
95	C₄×C₈₉	#497	1	2	yes	—
96	#9810	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	2	no	—
98	C₂×C₈₉	#1966	1	1	yes	characteristic
99	C₂×C₈₉	#1971	1	1	yes	characteristic
100	C₂×C₈₉	#1978	1	1	yes	characteristic
101	C₂×#500	D₈₉	1	1	yes	characteristic
102	#1997	—	89	2	no	—
103	#1566	—	89	1	no	—
104	#1549	—	89	2	no	—
105	#1451	—	89	1	no	—
106	C₈₉	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
108	#500	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	2	no	—
111	#431	—	89	1	no	—
112	C₂×#107	Dic₈₉	1	4	yes	—
113	#423	—	89	2	no	—
114	C₄×M₁₆	—	89	1	no	—
115	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
116	#343	—	178	4	no	—
117	#334	—	178	2	no	—
118	#332	—	178	4	no	—
119	#322	—	178	1	no	—
120	#322	—	178	1	no	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂×M₁₆	—	89	2	no	—
123	C₂×M₁₆	—	178	2	no	—
124	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
131	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	#107	—	2	4	no	—
133	#107	—	2	4	no	—
134	C₄⋊C₈	—	89	2	no	—
135	C₄⋊C₈	—	178	2	no	—
136	C₄⋊C₈	—	178	2	no	—
137	C₂²⋊C₈	—	178	1	no	—
138	#97	—	178	1	no	—
139	C₄×C₈	—	178	1	no	—
140	C₄×C₈	—	178	1	no	—
141	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
150	C₂²×C₄	#13083573	1	1	yes	characteristic
151	M₁₆	—	356	4	no	—
152	M₁₆	—	178	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	—	178	2	no	—
156	C₂×C₈	—	178	2	no	—
157	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄²	D₄×D₈₉	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
166	C₄²	#13083573	1	1	yes	characteristic
167	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	4	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
176	C₂×C₄	#15667983	1	1	yes	characteristic
177	C₂×C₄	#15667983	1	1	yes	characteristic
178	C₂×C₄	#15667999	1	1	yes	characteristic
179	C₂×C₄	#15668008	1	1	yes	characteristic
180	C₂×C₄	#15668044	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	356	2	no	—
183	C₈	—	356	2	no	—
184	C₈	—	712	2	no	—
185	C₈	#15667931	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#19151501	1	1	yes	characteristic
189	C₂²	#19151705	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	8	4	no	—
193	C₄	C₂×#15667931	1	2	yes	—
194	C₄	#19151853	1	2	yes	—
195	C₄	#19151865	1	2	yes	—
196	C₄	#19151991	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#26021097	1	1	yes	characteristic
199	C₂	#26021124	1	1	yes	characteristic
200	C₂	#26021163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021242

C₂

yes

maximal

#26019008

C₂

yes

maximal, characteristic

#26018937

C₂

yes

maximal

#26018541

C₂

yes

maximal, characteristic

C₂×#500×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

characteristic

#19151622

C₂²

yes

—

C₄×#13083558

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

—

#500×C₈₉

—

#500×C₈₉

—

#500×C₈₉

—

C₂×#107×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

C₂×#13083558

D₄

yes

—

C₄⋊#10623494

—

C₄⋊#10623494

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

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

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

yes

—

C₈×C₈₉

—

C₈×C₈₉

#101

yes

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#441

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#101

yes

—

C₄×C₈₉

#470

yes

—

C₄×C₈₉

#472

yes

—

C₄×C₈₉

#497

yes

—

#9810

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

100

C₂×C₈₉

#1978

yes

characteristic

101

C₂×#500

D₈₉

yes

characteristic

102

#1997

—

103

#1566

—

104

#1549

—

105

#1451

—

106

C₈₉

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

C₂×D₈₉

yes

—

108

#500

—

109

#500

—

110

#500

—

111

#431

—

112

C₂×#107

Dic₈₉

yes

—

113

#423

—

114

C₄×M₁₆

—

115

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

116

#343

—

178

—

117

#334

—

178

—

118

#332

—

178

—

119

#322

—

178

—

120

#322

—

178

—

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

178

—

124

C₂²×C₈

C₂×Dic₈₉

yes

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

—

131

C₂×C₄²

C₂²×D₈₉

yes

characteristic

132

#107

—

133

#107

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

178

—

136

C₄⋊C₈

—

178

—

137

C₂²⋊C₈

—

178

—

138

#97

—

178

—

139

C₄×C₈

—

178

—

140

C₄×C₈

—

178

—

141

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

142

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

143

C₄×C₈

C₂²⋊D₈₉

yes

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

149

C₂²×C₄

D₄×D₈₉

yes

characteristic

150

C₂²×C₄

#13083573

yes

characteristic

151

M₁₆

—

356

—

152

M₁₆

—

178

—

153

C₂×C₈

—

154

C₂×C₈

—

356

—

155

C₂×C₈

—

178

—

156

C₂×C₈

—

178

—

157

C₂×C₈

C₂²⋊Dic₈₉

yes

—

158

C₂×C₈

C₂²⋊Dic₈₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

D₄×D₈₉

yes

characteristic

164

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

165

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

166

C₄²

#13083573

yes

characteristic

167

C₂³

D₄×Dic₈₉

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

175

C₂×C₄

D₄×Dic₈₉

yes

characteristic

176

C₂×C₄

#15667983

yes

characteristic

177

C₂×C₄

#15667983

yes

characteristic

178

C₂×C₄

#15667999

yes

characteristic

179

C₂×C₄

#15668008

yes

characteristic

180

C₂×C₄

#15668044

yes

—

181

C₈

—

182

C₈

—

356

—

183

C₈

—

356

—

184

C₈

—

712

—

185

C₈

#15667931

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#19151501

yes

characteristic

189

C₂²

#19151705

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

C₂×#15667931

yes

—

194

C₄

#19151853

yes

—

195

C₄

#19151865

yes

—

196

C₄

#19151991

yes

—

197

C₂

—

198

C₂

#26021097

yes

characteristic

199

C₂

#26021124

yes

characteristic

200

C₂

#26021163

yes

characteristic

	subgroup	G/H
center	C₂²	#19151501
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	#9810
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#19151501

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#500×C₈₉

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 ◁ C₂×#500

N	H ≅ G/N	type
106. C₈₉	#9810	semidirect

H ≅ G/N

type

106. C₈₉

#9810

semidirect

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

subgroup	index
#26021242	2
#26019008	2
#26018937	2
#26018541	2
C₂×#500×C₈₉	2
#9810	89

subgroup

index

C₂×#500×C₈₉

Subgroups — #36454113

Conjugacy classes of subgroups

Complements

Maximal subgroups