Group #36447418 — subgroups

Subgroups — #36447418

Order 22784.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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	#26028964	C₂	1	1	yes	maximal, characteristic
2	#26019013	C₂	1	2	yes	maximal, complemented
3	#26018887	C₂	1	1	yes	maximal, characteristic
4	#26018162	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668019	C₄	1	2	yes	—
7	#19151676	—	2	2	no	—
8	D₄×#10623494	—	2	2	no	—
9	#19151655	C₂²	1	2	yes	—
10	#19151622	C₂²	1	1	yes	characteristic
11	#19151565	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151350	—	2	2	no	—
14	#19151348	—	2	2	no	—
15	#19151338	—	2	2	no	—
16	#532×C₈₉	C₂²	1	1	yes	characteristic
17	#432×C₈₉	C₂²	1	2	yes	—
18	#329×C₈₉	—	2	2	no	—
19	#15668019	—	4	4	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	C₂×C₄	1	2	yes	—
22	C₂²×#10623494	—	2	2	no	—
23	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₂²⋊#10623494	—	2	4	no	—
28	C₂²⋊#10623494	—	2	2	no	—
29	#15667923	—	2	2	no	—
30	C₄×#10623494	D₄	1	2	yes	—
31	C₄×#10623494	D₄	1	2	yes	—
32	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₈₉	Q₈	1	1	yes	characteristic
34	#127×C₈₉	Q₈	1	1	yes	characteristic
35	#124×C₈₉	D₄	1	2	yes	—
36	#121×C₈₉	—	2	2	no	—
37	C₄×D₄×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	—	2	2	no	—
40	C₂²⋊C₈×C₈₉	—	2	2	no	—
41	#97×C₈₉	—	2	2	no	—
42	#13083558	—	4	4	no	—
43	#13083558	—	4	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	4	1	no	—
47	C₂×#10623494	—	4	4	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
51	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
52	#41×C₈₉	—	4	2	no	—
53	#41×C₈₉	—	4	2	no	—
54	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₈₉	—	2	2	no	—
56	C₂×D₄×C₈₉	—	2	2	no	—
57	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	—	2	2	no	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	#41	1	2	yes	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₂²⋊C₄×C₈₉	—	2	4	no	—
64	C₄²×C₈₉	C₂×D₄	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	#117	1	1	yes	characteristic
69	Q₈×C₈₉	—	4	2	no	—
70	D₄×C₈₉	—	4	2	no	—
71	D₄×C₈₉	—	4	4	no	—
72	D₄×C₈₉	—	4	4	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	—	4	2	no	—
78	C₂×C₄×C₈₉	—	4	2	no	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#121	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₈₉	—	8	2	no	—
84	#10623494	—	2	1	no	—
85	#10623494	—	2	2	no	—
86	#10623494	—	4	2	no	—
87	#10623494	—	8	2	no	—
88	#10623494	#100	1	2	yes	—
89	C₂²×C₈₉	—	2	1	no	—
90	C₂²×C₈₉	—	2	2	no	—
91	C₂²×C₈₉	—	4	2	no	—
92	C₂²×C₈₉	—	4	4	no	—
93	C₂²×C₈₉	#380	1	1	yes	characteristic
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	—	2	2	no	—
97	C₄×C₈₉	—	4	2	no	—
98	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
99	C₄×C₈₉	#413	1	1	yes	characteristic
100	#8630	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	—	4	1	no	—
102	C₂×C₈₉	—	8	2	no	—
103	C₂×C₈₉	#1800	1	1	yes	characteristic
104	C₂×C₈₉	#1814	1	2	yes	—
105	#2931	—	89	1	no	—
106	#1568	—	89	2	no	—
107	#1536	—	89	1	no	—
108	#1536	D₈₉	1	1	yes	characteristic
109	#1307	—	89	2	no	—
110	C₈₉	#8630	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	—	89	2	no	—
112	#532	C₂×D₈₉	1	1	yes	characteristic
113	#436	—	178	2	no	—
114	C₈×D₄	—	178	2	no	—
115	#432	—	89	2	no	—
116	#432	C₂×D₈₉	1	2	yes	—
117	#423	—	89	1	no	—
118	#408	—	89	2	no	—
119	C₂×C₄×C₈	—	89	1	no	—
120	#335	—	178	2	no	—
121	#334	—	178	2	no	—
122	#329	—	2	2	no	—
123	#329	—	178	2	no	—
124	C₂×#41	C₄×D₈₉	1	2	yes	—
125	#131	—	356	4	no	—
126	C₂×M₁₆	—	89	2	no	—
127	C₂×M₁₆	—	178	2	no	—
128	C₂²×C₈	—	89	1	no	—
129	C₂²×C₈	—	89	1	no	—
130	C₂²×C₈	—	178	2	no	—
131	C₄⋊Q₈	Dic₁₇₈	1	1	yes	characteristic
132	#127	Dic₁₇₈	1	1	yes	characteristic
133	#124	C₂²⋊D₈₉	1	2	yes	—
134	#121	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	178	2	no	—
139	C₄⋊C₈	—	178	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	178	4	no	—
142	C₂²⋊C₈	—	178	2	no	—
143	#97	—	2	2	no	—
144	#97	—	178	2	no	—
145	C₄×C₈	—	89	2	no	—
146	C₄×C₈	—	89	2	no	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	#13083553	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	#13083553	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
155	C₂²×C₄	#13083573	1	2	yes	—
156	M₁₆	—	356	4	no	—
157	M₁₆	—	356	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	89	1	no	—
160	C₂×C₈	—	89	2	no	—
161	C₂×C₈	—	89	1	no	—
162	C₂×C₈	—	356	4	no	—
163	C₂×C₈	—	356	1	no	—
164	C₂×C₈	—	356	2	no	—
165	C₂×C₈	—	178	2	no	—
166	C₂×C₈	—	178	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	D₄×D₈₉	1	2	yes	—
170	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#15667968	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
186	C₂×C₄	#15667984	1	2	yes	—
187	C₂×C₄	#15667989	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	89	2	no	—
190	C₈	—	356	2	no	—
191	C₈	—	178	1	no	—
192	C₈	—	178	2	no	—
193	C₈	—	712	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#19151456	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₈₉	1	1	yes	characteristic
204	C₄	#19151579	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#26020242	1	1	yes	characteristic
208	C₂	#26020304	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028964

C₂

yes

maximal, characteristic

#26019013

C₂

yes

maximal, complemented

#26018887

C₂

yes

maximal, characteristic

#26018162

C₂

yes

maximal, complemented

C₈₉×#1536

C₂

yes

maximal, characteristic

C₂×#15668019

C₄

yes

—

#19151676

—

D₄×#10623494

—

#19151655

C₂²

yes

—

#19151622

C₂²

yes

characteristic

#19151565

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

—

—

—

#532×C₈₉

C₂²

yes

characteristic

#432×C₈₉

C₂²

yes

—

#329×C₈₉

—

#15668019

—

C₂×#13083558

—

C₂×#13083558

C₂×C₄

yes

—

C₂²×#10623494

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₂×#41×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

Q₈

yes

characteristic

#127×C₈₉

Q₈

yes

characteristic

#124×C₈₉

D₄

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

—

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₄⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#117

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

—

C₈×C₈₉

—

—

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#380

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

100

—

101

C₂×C₈₉

—

102

C₂×C₈₉

—

103

C₂×C₈₉

#1800

yes

characteristic

104

C₂×C₈₉

#1814

yes

—

105

#2931

—

106

#1568

—

107

#1536

—

108

#1536

D₈₉

yes

characteristic

109

#1307

—

110

C₈₉

#8630

yes

characteristic, complemented, Sylow

111

C₂×#131

—

112

#532

C₂×D₈₉

yes

characteristic

113

#436

—

178

—

114

C₈×D₄

—

178

—

115

#432

—

116

#432

C₂×D₈₉

yes

—

117

#423

—

118

#408

—

119

C₂×C₄×C₈

—

120

#335

—

178

—

121

#334

—

178

—

122

#329

—

123

#329

—

178

—

124

C₂×#41

C₄×D₈₉

yes

—

125

#131

—

356

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

178

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

178

—

131

C₄⋊Q₈

Dic₁₇₈

yes

characteristic

132

#127

Dic₁₇₈

yes

characteristic

133

#124

C₂²⋊D₈₉

yes

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

C₂²×D₈₉

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

178

—

139

C₄⋊C₈

—

178

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

178

—

142

C₂²⋊C₈

—

178

—

143

#97

—

144

#97

—

178

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#41

—

148

#41

—

149

C₂×Q₈

#13083553

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

#13083553

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

155

C₂²×C₄

#13083573

yes

—

156

M₁₆

—

356

—

157

M₁₆

—

356

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

356

—

163

C₂×C₈

—

356

—

164

C₂×C₈

—

356

—

165

C₂×C₈

—

178

—

166

C₂×C₈

—

178

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

D₄×D₈₉

yes

—

170

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

171

C₄²

C₂×Dic₁₇₈

yes

characteristic

172

C₂³

—

173

C₂³

#15667968

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

185

C₂×C₄

C₂×#13083553

yes

characteristic

186

C₂×C₄

#15667984

yes

—

187

C₂×C₄

#15667989

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

356

—

191

C₈

—

178

—

192

C₈

—

178

—

193

C₈

—

712

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#19151456

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₈₉

yes

characteristic

204

C₄

#19151579

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#26020242

yes

characteristic

208

C₂

#26020304

yes

—

	subgroup	G/H
center	C₂²	#19151456
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1536	C₂
socle	C₂²×C₈₉	#380
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1536	D₈₉
hypocenter	C₈₉	#8630
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#19151456

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1536

C₂

socle

C₂²×C₈₉

#380

prime radical

#532×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #26019013	C₂	semidirect
4. #26018162	C₂	semidirect
110. C₈₉	#8630	semidirect

H ≅ G/N

type

2. #26019013

C₂

semidirect

4. #26018162

C₂

semidirect

110. C₈₉

#8630

semidirect

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

subgroup	index
#26028964	2
#26019013	2
#26018887	2
#26018162	2
C₈₉×#1536	2
#8630	89

subgroup

index

C₈₉×#1536

Subgroups — #36447418

Conjugacy classes of subgroups

Complements

Maximal subgroups