Group #36454557 — subgroups

Subgroups — #36454557

Order 22784.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#26028615	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	#19151496	—	2	2	no	—
8	#19151496	C₄	1	4	yes	—
9	#19151468	C₂²	1	2	yes	—
10	#19151464	—	2	1	no	—
11	#19151353	—	2	2	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	—	2	1	no	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	#336×C₈₉	—	2	1	no	—
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	4	2	no	—
18	#15668004	—	4	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
22	C₂×#13083553	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	#15667919	—	2	4	no	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	2	yes	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	2	2	no	—
32	C₄⋊C₈×C₈₉	Q₈	1	4	yes	—
33	C₄×C₈×C₈₉	—	4	2	no	—
34	C₂²×Dic₈₉	—	2	4	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	4	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₄⋊Dic₈₉	—	4	4	no	—
41	#13083553	—	4	8	no	—
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	4	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	—	4	2	no	—
50	C₄²×C₈₉	—	2	4	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	—	4	2	no	—
53	C₄²×C₈₉	D₈	1	2	yes	—
54	C₄²×C₈₉	SD₁₆	1	2	yes	—
55	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
56	C₄²×C₈₉	#41	1	2	yes	—
57	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	#102	1	4	yes	—
65	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#135	1	4	yes	—
72	C₂×C₄×C₈₉	#137	1	2	yes	—
73	C₂×Dic₈₉	—	4	4	no	—
74	C₂×Dic₈₉	—	4	4	no	—
75	C₂×Dic₈₉	—	4	4	no	—
76	C₈×C₈₉	—	4	4	no	—
77	#10623494	—	8	2	no	—
78	C₂²×C₈₉	#391	1	1	yes	characteristic
79	C₂²×C₈₉	#413	1	2	yes	—
80	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
81	C₂²×C₈₉	#416	1	2	yes	—
82	C₂²×C₈₉	#419	1	1	yes	characteristic
83	C₄×C₈₉	—	2	4	no	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	#474	1	2	yes	—
87	C₄×C₈₉	#475	1	2	yes	—
88	C₄×C₈₉	#488	1	2	yes	—
89	C₄×C₈₉	#499	1	2	yes	—
90	Dic₈₉	—	8	4	no	—
91	#9899	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#1540	1	2	yes	—
93	C₂×C₈₉	#1554	1	2	yes	—
94	C₂×C₈₉	#1988	1	1	yes	characteristic
95	C₂×C₈₉	#1991	1	1	yes	characteristic
96	C₂×C₈₉	#1998	1	1	yes	characteristic
97	#2859	—	89	1	no	—
98	#1831	—	89	1	no	—
99	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
100	#1296	—	89	4	no	—
101	C₈₉	#9899	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	178	1	no	—
103	C₂×C₄⋊C₈	—	89	2	no	—
104	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
105	C₂×C₄×C₈	—	2	1	no	—
106	#389	—	89	4	no	—
107	#389	—	178	2	no	—
108	#384	—	89	2	no	—
109	#382	—	178	1	no	—
110	C₄³	C₂×D₈₉	1	1	yes	characteristic
111	#336	—	2	1	no	—
112	#336	—	178	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	178	2	no	—
115	#125	—	356	2	no	—
116	#125	—	356	2	no	—
117	C₂×C₄⋊C₄	—	89	4	no	—
118	C₂×C₄⋊C₄	—	178	4	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄⋊C₄	—	178	2	no	—
121	C₂×C₄²	C₄×D₈₉	1	2	yes	—
122	C₂×C₄²	D₃₅₆	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	178	4	no	—
128	C₄⋊C₈	Dic₁₇₈	1	4	yes	—
129	C₄×C₈	—	4	2	no	—
130	#95	—	178	4	no	—
131	C₂²×C₄	—	178	4	no	—
132	C₂²×C₄	#13083551	1	2	yes	—
133	C₂²×C₄	#13083551	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	356	2	no	—
141	C₂×C₈	—	356	4	no	—
142	C₄⋊C₄	—	356	4	no	—
143	C₄⋊C₄	—	356	2	no	—
144	C₄⋊C₄	—	356	2	no	—
145	C₄⋊C₄	—	356	4	no	—
146	C₄⋊C₄	—	356	8	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#13083559	1	2	yes	—
151	C₄²	#13083561	1	2	yes	—
152	C₄²	C₂×Dic₁₇₈	1	2	yes	—
153	C₄²	#13083574	1	2	yes	—
154	C₂³	C₂×#13083551	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	356	4	no	—
162	C₂×C₄	—	356	4	no	—
163	C₂×C₄	—	356	4	no	—
164	C₂×C₄	#15667932	1	4	yes	—
165	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
166	C₂×C₄	#15667991	1	2	yes	—
167	C₂×C₄	#15668000	1	1	yes	characteristic
168	C₂×C₄	#15668012	1	1	yes	characteristic
169	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
170	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
171	C₂×C₄	#15668034	1	4	yes	—
172	C₂×C₄	#15668042	1	2	yes	—
173	C₈	—	4	4	no	—
174	C₈	—	712	2	no	—
175	C₂²	#19151504	1	1	yes	characteristic
176	C₂²	#19151580	1	2	yes	—
177	C₂²	C₂×#15667932	1	1	yes	characteristic
178	C₂²	#19151595	1	2	yes	—
179	C₂²	#19151606	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	712	4	no	—
184	C₄	#19151878	1	2	yes	—
185	C₄	#19151883	1	2	yes	—
186	C₄	#19151953	1	2	yes	—
187	C₄	#19152000	1	2	yes	—
188	C₂	#26018902	1	2	yes	—
189	C₂	#26018960	1	2	yes	—
190	C₂	#26021204	1	1	yes	characteristic
191	C₂	#26021217	1	1	yes	characteristic
192	C₂	#26021245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028615

C₂

yes

maximal, characteristic

#26020369

C₂

yes

maximal, characteristic

#26018136

C₂

yes

maximal

C₄×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂×C₄⋊#10623494

C₂²

yes

—

#19151496

—

#19151496

C₄

yes

—

#19151468

C₂²

yes

—

#19151464

—

#19151353

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

#15668004

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083553

—

C₄⋊#10623494

—

#15667919

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#391

yes

characteristic

C₂²×C₈₉

#413

yes

—

C₂²×C₈₉

C₂×#102

yes

characteristic

C₂²×C₈₉

#416

yes

—

C₂²×C₈₉

#419

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#475

yes

—

C₄×C₈₉

#488

yes

—

C₄×C₈₉

#499

yes

—

Dic₈₉

—

#9899

—

maximal, Sylow

C₂×C₈₉

#1540

yes

—

C₂×C₈₉

#1554

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

C₄×C₄⋊C₈

D₈₉

yes

characteristic

100

#1296

—

101

C₈₉

#9899

yes

characteristic, complemented, Sylow

102

C₂×#125

—

178

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

105

C₂×C₄×C₈

—

106

#389

—

107

#389

—

178

—

108

#384

—

109

#382

—

178

—

110

C₄³

C₂×D₈₉

yes

characteristic

111

#336

—

112

#336

—

178

—

113

C₂²×C₈

—

114

C₂²×C₈

—

178

—

115

#125

—

356

—

116

#125

—

356

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

178

—

119

C₂×C₄⋊C₄

—

178

—

120

C₂×C₄⋊C₄

—

178

—

121

C₂×C₄²

C₄×D₈₉

yes

—

122

C₂×C₄²

D₃₅₆

yes

—

123

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

124

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

125

C₂×C₄²

C₂²×D₈₉

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

178

—

128

C₄⋊C₈

Dic₁₇₈

yes

—

129

C₄×C₈

—

130

#95

—

178

—

131

C₂²×C₄

—

178

—

132

C₂²×C₄

#13083551

yes

—

133

C₂²×C₄

#13083551

yes

—

134

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

136

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

356

—

141

C₂×C₈

—

356

—

142

C₄⋊C₄

—

356

—

143

C₄⋊C₄

—

356

—

144

C₄⋊C₄

—

356

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

356

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#13083559

yes

—

151

C₄²

#13083561

yes

—

152

C₄²

C₂×Dic₁₇₈

yes

—

153

C₄²

#13083574

yes

—

154

C₂³

C₂×#13083551

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

356

—

162

C₂×C₄

—

356

—

163

C₂×C₄

—

356

—

164

C₂×C₄

#15667932

yes

—

165

C₂×C₄

C₄×Dic₁₇₈

yes

—

166

C₂×C₄

#15667991

yes

—

167

C₂×C₄

#15668000

yes

characteristic

168

C₂×C₄

#15668012

yes

characteristic

169

C₂×C₄

C₂×#13083559

yes

characteristic

170

C₂×C₄

C₂×#13083561

yes

characteristic

171

C₂×C₄

#15668034

yes

—

172

C₂×C₄

#15668042

yes

—

173

C₈

—

174

C₈

—

712

—

175

C₂²

#19151504

yes

characteristic

176

C₂²

#19151580

yes

—

177

C₂²

C₂×#15667932

yes

characteristic

178

C₂²

#19151595

yes

—

179

C₂²

#19151606

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

712

—

184

C₄

#19151878

yes

—

185

C₄

#19151883

yes

—

186

C₄

#19151953

yes

—

187

C₄

#19152000

yes

—

188

C₂

#26018902

yes

—

189

C₂

#26018960

yes

—

190

C₂

#26021204

yes

characteristic

191

C₂

#26021217

yes

characteristic

192

C₂

#26021245

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#9899
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×C₄⋊C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₄×C₄⋊C₈

D₈₉

hypocenter

C₈₉

#9899

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
101. C₈₉	#9899	semidirect

H ≅ G/N

type

101. C₈₉

#9899

semidirect

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

subgroup	index
#26028615	2
#26020369	2
#26018136	2
C₄×C₄⋊C₈×C₈₉	2
#9899	89

subgroup

index

C₄×C₄⋊C₈×C₈₉

Subgroups — #36454557

Conjugacy classes of subgroups

Complements

Maximal subgroups