Group #36447427 — subgroups

Subgroups — #36447427

Order 22784.

388 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26028963	C₂	1	1	yes	maximal, characteristic
2	#26019013	C₂	1	2	yes	maximal, complemented
3	#26018886	C₂	1	2	yes	maximal
4	#26018163	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1307	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668020	C₄	1	2	yes	—
7	#19152148	C₂²	1	1	yes	characteristic
8	#19151675	—	2	2	no	—
9	#19151663	—	2	2	no	—
10	#19151655	C₂²	1	2	yes	—
11	#19151566	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151350	—	2	2	no	—
14	#19151348	—	2	2	no	—
15	#19151337	—	2	4	no	—
16	#432×C₈₉	C₂²	1	2	yes	—
17	#423×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083574	C₂×C₄	1	2	yes	—
19	#15668020	—	4	4	no	—
20	C₂×#13083557	—	2	2	no	—
21	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
22	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₈₉	—	2	2	no	—
24	#15668012	Q₈	1	1	yes	characteristic
25	#15668009	Q₈	1	1	yes	characteristic
26	#15668000	D₄	1	2	yes	—
27	#15667986	—	2	2	no	—
28	C₄×D₃₅₆	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₂²⋊#10623494	—	2	2	no	—
31	#15667924	—	2	4	no	—
32	#15667923	—	2	2	no	—
33	C₄×#10623494	D₄	1	2	yes	—
34	C₄×#10623494	D₄	1	2	yes	—
35	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₂²⋊C₈×C₈₉	—	2	2	no	—
40	#97×C₈₉	—	2	2	no	—
41	#13083574	—	4	2	no	—
42	#13083574	—	4	2	no	—
43	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
44	C₂×D₃₅₆	—	2	2	no	—
45	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
46	C₂×C₂²⋊D₈₉	—	2	2	no	—
47	C₂×C₄×D₈₉	—	2	2	no	—
48	#13083557	—	4	4	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₂×#10623494	—	4	1	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
54	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
55	C₈×D₈₉	—	4	4	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	#13083551	—	2	4	no	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	#41	1	2	yes	—
60	M₁₆×C₈₉	—	4	2	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
66	C₂³×C₈₉	#117	1	1	yes	characteristic
67	C₂²×D₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#121	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₂²⋊D₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	2	2	no	—
77	D₃₅₆	—	4	2	no	—
78	D₃₅₆	—	4	4	no	—
79	C₄×D₈₉	—	4	2	no	—
80	C₄×D₈₉	—	4	2	no	—
81	Dic₁₇₈	—	4	2	no	—
82	C₈×C₈₉	—	4	2	no	—
83	C₈×C₈₉	—	8	2	no	—
84	#10623494	—	2	1	no	—
85	#10623494	—	2	2	no	—
86	#10623494	—	8	2	no	—
87	#10623494	#100	1	2	yes	—
88	C₂²×C₈₉	—	2	1	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	#380	1	1	yes	characteristic
91	C₂×D₈₉	—	4	2	no	—
92	C₂×D₈₉	—	4	4	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
97	C₄×C₈₉	#413	1	1	yes	characteristic
98	Dic₈₉	—	4	2	no	—
99	#8632	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	4	1	no	—
101	C₂×C₈₉	#1800	1	1	yes	characteristic
102	C₂×C₈₉	#1814	1	2	yes	—
103	D₈₉	—	8	2	no	—
104	#2931	—	89	1	no	—
105	#1568	—	89	2	no	—
106	#1536	—	89	2	no	—
107	#1307	—	89	1	no	—
108	#1307	D₈₉	1	1	yes	characteristic, complemented
109	C₈₉	#8632	1	1	yes	characteristic, complemented, Sylow
110	C₂×#131	—	89	2	no	—
111	#532	—	89	1	no	—
112	#436	—	178	2	no	—
113	C₈×D₄	—	178	2	no	—
114	#432	—	89	2	no	—
115	#432	C₂×D₈₉	1	2	yes	—
116	#423	C₂×D₈₉	1	1	yes	characteristic
117	#408	—	89	2	no	—
118	C₂×C₄×C₈	—	89	1	no	—
119	#335	—	178	2	no	—
120	#334	—	178	2	no	—
121	#329	—	178	4	no	—
122	C₂×#41	—	89	2	no	—
123	#131	—	356	4	no	—
124	C₂×M₁₆	—	178	2	no	—
125	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
126	C₂²×C₈	—	89	1	no	—
127	C₂²×C₈	—	89	1	no	—
128	C₂²×C₈	—	178	2	no	—
129	C₄⋊Q₈	—	89	1	no	—
130	#127	—	89	1	no	—
131	#124	—	89	2	no	—
132	#121	—	178	2	no	—
133	C₄×D₄	—	178	2	no	—
134	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	178	2	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	178	4	no	—
140	C₂²⋊C₈	—	178	2	no	—
141	#97	—	2	2	no	—
142	#97	—	178	2	no	—
143	C₄×C₈	—	89	2	no	—
144	C₄×C₈	—	89	2	no	—
145	#41	—	356	2	no	—
146	#41	—	356	2	no	—
147	C₂×Q₈	—	89	2	no	—
148	C₂×D₄	—	89	2	no	—
149	C₂×D₄	—	178	2	no	—
150	C₂×D₄	—	178	2	no	—
151	C₂²×C₄	—	178	2	no	—
152	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₄	#13083574	1	2	yes	—
154	M₁₆	—	4	2	no	—
155	M₁₆	—	356	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	89	1	no	—
159	C₂×C₈	—	89	2	no	—
160	C₂×C₈	—	89	1	no	—
161	C₂×C₈	—	356	4	no	—
162	C₂×C₈	—	356	1	no	—
163	C₂×C₈	—	356	2	no	—
164	C₂×C₈	—	178	2	no	—
165	C₄⋊C₄	—	178	2	no	—
166	C₂²⋊C₄	—	178	4	no	—
167	C₄²	D₄×D₈₉	1	1	yes	characteristic
168	C₄²	D₄×D₈₉	1	2	yes	—
169	C₄²	Q₈×D₈₉	1	1	yes	characteristic
170	C₂³	—	178	2	no	—
171	C₂³	#15667969	1	1	yes	characteristic
172	Q₈	—	356	2	no	—
173	D₄	—	356	2	no	—
174	D₄	—	356	4	no	—
175	D₄	—	356	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	356	2	no	—
181	C₂×C₄	—	178	2	no	—
182	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
183	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
184	C₂×C₄	#15667985	1	2	yes	—
185	C₂×C₄	#15667990	1	2	yes	—
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	—	89	2	no	—
189	C₈	—	178	1	no	—
190	C₈	—	178	2	no	—
191	C₈	—	712	2	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	356	2	no	—
195	C₂²	—	356	4	no	—
196	C₂²	#19151457	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	356	2	no	—
201	C₄	#100×D₈₉	1	1	yes	characteristic
202	C₄	#19151581	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	712	2	no	—
205	C₂	#26020243	1	1	yes	characteristic
206	C₂	#26020305	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028963

C₂

yes

maximal, characteristic

#26019013

C₂

yes

maximal, complemented

#26018886

C₂

yes

maximal

#26018163

C₂

yes

maximal, characteristic, complemented

C₈₉×#1307

C₂

yes

maximal, characteristic, complemented

C₂×#15668020

C₄

yes

—

#19152148

C₂²

yes

—

—

C₂²

yes

—

#19151566

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

—

—

—

#432×C₈₉

C₂²

yes

—

#423×C₈₉

C₂²

yes

characteristic

C₂×#13083574

C₂×C₄

yes

—

#15668020

—

C₂×#13083557

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂×C₈×D₈₉

—

#15668012

Q₈

yes

characteristic

#15668009

Q₈

yes

characteristic

#15668000

D₄

yes

—

#15667986

—

C₄×D₃₅₆

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₄⋊C₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₄⋊C₄

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#117

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#380

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#8632

—

maximal, Sylow

100

C₂×C₈₉

—

101

C₂×C₈₉

#1800

yes

characteristic

102

C₂×C₈₉

#1814

yes

—

103

D₈₉

—

104

#2931

—

105

#1568

—

106

#1536

—

107

#1307

—

108

#1307

D₈₉

yes

characteristic, complemented

109

C₈₉

#8632

yes

characteristic, complemented, Sylow

110

C₂×#131

—

111

#532

—

112

#436

—

178

—

113

C₈×D₄

—

178

—

114

#432

—

115

#432

C₂×D₈₉

yes

—

116

#423

C₂×D₈₉

yes

characteristic

117

#408

—

118

C₂×C₄×C₈

—

119

#335

—

178

—

120

#334

—

178

—

121

#329

—

178

—

122

C₂×#41

—

123

#131

—

356

—

124

C₂×M₁₆

—

178

—

125

C₂×M₁₆

C₄×D₈₉

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

178

—

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

178

—

133

C₄×D₄

—

178

—

134

C₂×C₄²

C₂²×D₈₉

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

178

—

138

C₂²⋊C₈

—

139

C₂²⋊C₈

—

178

—

140

C₂²⋊C₈

—

178

—

141

#97

—

142

#97

—

178

—

143

C₄×C₈

—

144

C₄×C₈

—

145

#41

—

356

—

146

#41

—

356

—

147

C₂×Q₈

—

148

C₂×D₄

—

149

C₂×D₄

—

178

—

150

C₂×D₄

—

178

—

151

C₂²×C₄

—

178

—

152

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

153

C₂²×C₄

#13083574

yes

—

154

M₁₆

—

155

M₁₆

—

356

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

356

—

162

C₂×C₈

—

356

—

163

C₂×C₈

—

356

—

164

C₂×C₈

—

178

—

165

C₄⋊C₄

—

178

—

166

C₂²⋊C₄

—

178

—

167

C₄²

D₄×D₈₉

yes

characteristic

168

C₄²

D₄×D₈₉

yes

—

169

C₄²

Q₈×D₈₉

yes

characteristic

170

C₂³

—

178

—

171

C₂³

#15667969

yes

characteristic

172

Q₈

—

356

—

173

D₄

—

356

—

174

D₄

—

356

—

175

D₄

—

356

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

356

—

180

C₂×C₄

—

356

—

181

C₂×C₄

—

178

—

182

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

183

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

184

C₂×C₄

#15667985

yes

—

185

C₂×C₄

#15667990

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

178

—

190

C₈

—

178

—

191

C₈

—

712

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

356

—

195

C₂²

—

356

—

196

C₂²

#19151457

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

356

—

201

C₄

#100×D₈₉

yes

characteristic

202

C₄

#19151581

yes

characteristic

203

C₂

—

204

C₂

—

712

—

205

C₂

#26020243

yes

characteristic

206

C₂

#26020305

yes

—

	subgroup	G/H
center	C₂²	#19151457
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1307	C₂
socle	C₂²×C₈₉	#380
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1307	D₈₉
hypocenter	C₈₉	#8632
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#19151457

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1307

C₂

C₂²×C₈₉

C₂²

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
2. #26019013	C₂	semidirect
4. #26018163	C₂	semidirect
5. C₈₉×#1307	C₂	semidirect
108. #1307	D₈₉	semidirect
109. C₈₉	#8632	semidirect

H ≅ G/N

type

2. #26019013

C₂

semidirect

4. #26018163

C₂

semidirect

5. C₈₉×#1307

C₂

semidirect

108. #1307

D₈₉

semidirect

109. C₈₉

#8632

semidirect

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

subgroup	index
#26028963	2
#26019013	2
#26018886	2
#26018163	2
C₈₉×#1307	2
#8632	89

subgroup

index

C₈₉×#1307

Subgroups — #36447427

Conjugacy classes of subgroups

Complements

Maximal subgroups