Group #36472990 — subgroups

Subgroups — #36472990

Order 22784.

334 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19152000	C₂	1	1	yes	maximal, characteristic
2	#26020131	C₂	1	1	yes	maximal, characteristic
3	#26018358	C₂	1	2	yes	maximal
4	#26018351	C₂	1	2	yes	maximal
5	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668004	—	2	1	no	—
7	#19152000	—	2	2	no	—
8	#19152000	C₄	1	4	yes	complemented
9	C₂×#15667948	C₂²	1	1	yes	characteristic
10	#19151636	C₂²	1	1	yes	characteristic
11	C₂×#15667946	—	2	1	no	—
12	C₂×#15667944	C₂²	1	2	yes	—
13	#19151553	—	2	1	no	—
14	#19151358	—	2	1	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#364×C₈₉	—	2	2	no	—
18	C₂×#13083558	—	2	1	no	—
19	#15668004	—	2	4	no	—
20	#15668004	—	4	2	no	—
21	#15667971	—	2	1	no	—
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×#13083553	—	4	2	no	—
24	#15667948	—	2	1	no	—
25	#15667948	Q₈	1	2	yes	—
26	#15667946	—	2	4	no	—
27	#15667946	Q₈	1	2	yes	—
28	#15667944	—	2	2	no	—
29	#15667944	—	2	1	no	—
30	#15667944	C₂×C₄	1	4	yes	—
31	#15667922	—	4	1	no	—
32	C₈×Dic₈₉	—	4	1	no	—
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	M₃₂×C₈₉	—	2	2	no	—
37	M₃₂×C₈₉	—	2	2	no	—
38	C₂×C₁₆×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	D₄	1	2	yes	—
41	C₂²×Dic₈₉	—	4	2	no	—
42	#13083558	—	2	2	no	—
43	#13083558	—	4	1	no	—
44	C₂×#10623494	—	4	1	no	—
45	C₄⋊Dic₈₉	—	2	4	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	#13083553	—	4	8	no	—
49	C₂²⋊Dic₈₉	—	4	1	no	—
50	C₄×Dic₈₉	—	4	1	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
63	C₄²×C₈₉	—	2	1	no	—
64	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₁₆×C₈₉	—	4	2	no	—
66	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×Dic₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₂×Dic₈₉	—	8	4	no	—
78	C₈×C₈₉	#103	1	2	yes	—
79	C₈×C₈₉	#103	1	2	yes	—
80	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₈₉	#124	1	1	yes	characteristic
83	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
84	#10623494	—	4	2	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	#418	1	1	yes	characteristic
87	C₂²×C₈₉	#476	1	1	yes	characteristic
88	C₂²×C₈₉	#477	1	1	yes	characteristic
89	C₄×C₈₉	—	4	2	no	—
90	C₄×C₈₉	#391	1	1	yes	characteristic
91	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
92	C₄×C₈₉	#476	1	1	yes	characteristic
93	C₄×C₈₉	#477	1	1	yes	characteristic
94	Dic₈₉	—	8	4	no	—
95	Dic₈₉	—	8	2	no	—
96	#13273	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	2	1	no	—
99	C₂×C₈₉	#1987	1	1	yes	characteristic
100	C₂×C₈₉	#2231	1	2	yes	—
101	C₂×#499	—	89	1	no	—
102	C₂×#364	D₈₉	1	1	yes	characteristic
103	#1772	—	89	1	no	—
104	#1386	—	89	2	no	—
105	#1383	—	89	2	no	—
106	C₈₉	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	178	1	no	—
108	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
109	#499	—	89	4	no	—
110	#499	—	178	2	no	—
111	C₂×#108	—	89	1	no	—
112	#427	—	89	1	no	—
113	C₂×#107	—	178	1	no	—
114	C₂×#106	—	89	2	no	—
115	#405	—	178	1	no	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#364	—	2	2	no	—
118	#339	—	178	1	no	—
119	C₂×M₁₆	—	178	1	no	—
120	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	#125	—	356	2	no	—
123	#125	—	178	4	no	—
124	#117	—	178	1	no	—
125	C₂×C₄⋊C₄	—	356	2	no	—
126	C₂×C₄⋊C₄	—	178	2	no	—
127	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	89	2	no	—
132	#108	—	178	1	no	—
133	#107	—	89	2	no	—
134	#107	—	178	4	no	—
135	#106	—	89	4	no	—
136	#106	—	178	2	no	—
137	#106	—	178	1	no	—
138	#97	—	356	1	no	—
139	C₄×C₈	—	356	1	no	—
140	C₄×C₈	C₄×D₈₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	356	2	no	—
144	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
145	M₁₆	—	356	1	no	—
146	M₁₆	—	178	2	no	—
147	C₂×C₈	—	356	1	no	—
148	C₂×C₈	#13083551	1	2	yes	—
149	C₂×C₈	#13083563	1	1	yes	characteristic
150	C₂×C₈	#13083563	1	1	yes	characteristic
151	C₂×C₈	Dic₃₅₆	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
156	C₂×C₈	#13083573	1	1	yes	characteristic
157	C₂×C₈	#13083573	1	1	yes	characteristic
158	C₄⋊C₄	—	356	8	no	—
159	C₄⋊C₄	—	356	2	no	—
160	C₄⋊C₄	—	178	2	no	—
161	C₄⋊C₄	—	178	4	no	—
162	C₂²⋊C₄	—	356	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	356	1	no	—
165	C₄²	#13083551	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	#15668040	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	356	2	no	—
171	C₂×C₄	—	356	4	no	—
172	C₂×C₄	—	712	4	no	—
173	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
174	C₂×C₄	#15667992	1	1	yes	characteristic
175	C₂×C₄	#15667992	1	1	yes	characteristic
176	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
178	C₂×C₄	#15668040	1	1	yes	characteristic
179	C₈	—	356	2	no	—
180	C₈	#15667937	1	2	yes	—
181	C₈	#15667937	1	2	yes	—
182	C₈	#15667977	1	1	yes	characteristic
183	C₈	#15667977	1	1	yes	characteristic
184	C₈	#15667999	1	1	yes	characteristic
185	C₈	#15668010	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#19151604	1	1	yes	characteristic
188	C₂²	#19151889	1	1	yes	characteristic
189	C₂²	#19151894	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	712	4	no	—
192	C₄	—	712	2	no	—
193	C₄	#19151503	1	1	yes	characteristic
194	C₄	C₂×#15667937	1	1	yes	characteristic
195	C₄	#19151889	1	1	yes	characteristic
196	C₄	#19151894	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#26021199	1	1	yes	characteristic
200	C₂	#26022429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic

#26020131

C₂

yes

maximal, characteristic

#26018358

C₂

yes

maximal

#26018351

C₂

yes

maximal

C₂×#364×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668004

—

#19152000

—

#19152000

C₄

yes

complemented

C₂×#15667948

C₂²

yes

characteristic

#19151636

C₂²

yes

characteristic

C₂×#15667946

—

C₂×#15667944

C₂²

yes

—

#19151553

—

#19151358

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#364×C₈₉

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

#15667948

—

#15667948

Q₈

yes

—

#15667946

—

#15667946

Q₈

yes

—

#15667944

—

#15667944

—

#15667944

C₂×C₄

yes

—

#15667922

—

C₈×Dic₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

C₂×C₈×C₈₉

yes

C₂×C₈×C₈₉

yes

C₄²×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#103

yes

—

C₈×C₈₉

#103

yes

—

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#13273

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1987

yes

characteristic

100

C₂×C₈₉

#2231

yes

—

101

C₂×#499

—

102

C₂×#364

D₈₉

yes

characteristic

103

#1772

—

104

#1386

—

105

#1383

—

106

C₈₉

#13273

yes

characteristic, complemented, Sylow

107

C₂×#125

—

178

—

108

C₂×M₃₂

C₂×D₈₉

yes

—

109

#499

—

110

#499

—

178

—

111

C₂×#108

—

112

#427

—

113

C₂×#107

—

178

—

114

C₂×#106

—

115

#405

—

178

—

116

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

117

#364

—

118

#339

—

178

—

119

C₂×M₁₆

—

178

—

120

C₂²×C₈

D₃₅₆

yes

characteristic

121

C₂²×C₈

C₂²×D₈₉

yes

characteristic

122

#125

—

356

—

123

#125

—

178

—

124

#117

—

178

—

125

C₂×C₄⋊C₄

—

356

—

126

C₂×C₄⋊C₄

—

178

—

127

C₂×C₄²

D₃₅₆

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

178

—

133

#107

—

134

#107

—

178

—

135

#106

—

136

#106

—

178

—

137

#106

—

178

—

138

#97

—

356

—

139

C₄×C₈

—

356

—

140

C₄×C₈

C₄×D₈₉

yes

—

141

C₄×C₈

C₂²⋊D₈₉

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

356

—

144

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

145

M₁₆

—

356

—

146

M₁₆

—

178

—

147

C₂×C₈

—

356

—

148

C₂×C₈

#13083551

yes

—

149

C₂×C₈

#13083563

yes

characteristic

150

C₂×C₈

#13083563

yes

characteristic

151

C₂×C₈

Dic₃₅₆

yes

—

152

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

153

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

154

C₂×C₈

Q₈×D₈₉

yes

characteristic

155

C₂×C₈

Q₈×D₈₉

yes

characteristic

156

C₂×C₈

#13083573

yes

characteristic

157

C₂×C₈

#13083573

yes

characteristic

158

C₄⋊C₄

—

356

—

159

C₄⋊C₄

—

356

—

160

C₄⋊C₄

—

178

—

161

C₄⋊C₄

—

178

—

162

C₂²⋊C₄

—

356

—

163

C₄²

—

164

C₄²

—

356

—

165

C₄²

#13083551

yes

—

166

C₁₆

—

167

C₂³

#15668040

yes

characteristic

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

356

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

712

—

173

C₂×C₄

C₂×#13083551

yes

characteristic

174

C₂×C₄

#15667992

yes

characteristic

175

C₂×C₄

#15667992

yes

characteristic

176

C₂×C₄

C₂×#13083563

yes

characteristic

177

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

178

C₂×C₄

#15668040

yes

characteristic

179

C₈

—

356

—

180

C₈

#15667937

yes

—

181

C₈

#15667937

yes

—

182

C₈

#15667977

yes

characteristic

183

C₈

#15667977

yes

characteristic

184

C₈

#15667999

yes

characteristic

185

C₈

#15668010

yes

characteristic

186

C₂²

—

187

C₂²

#19151604

yes

characteristic

188

C₂²

#19151889

yes

characteristic

189

C₂²

#19151894

yes

characteristic

190

C₄

—

191

C₄

—

712

—

192

C₄

—

712

—

193

C₄

#19151503

yes

characteristic

194

C₄

C₂×#15667937

yes

characteristic

195

C₄

#19151889

yes

characteristic

196

C₄

#19151894

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#26021199

yes

characteristic

200

C₂

#26022429

yes

—

	subgroup	G/H
center	C₂²	#19151889
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#13273
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#19151889

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#364×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #19152000	C₄	semidirect
106. C₈₉	#13273	semidirect

H ≅ G/N

type

8. #19152000

C₄

semidirect

106. C₈₉

#13273

semidirect

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

subgroup	index
C₂×#19152000	2
#26020131	2
#26018358	2
#26018351	2
C₂×#364×C₈₉	2
#13273	89

subgroup

index

C₂×#364×C₈₉

Subgroups — #36472990

Conjugacy classes of subgroups

Complements

Maximal subgroups