Group #36454288 — subgroups

Subgroups — #36454288

Order 22784.

510 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 195 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×#19151996	C₂	1	1	yes	maximal, characteristic
2	#26021213	C₂	1	2	yes	maximal
3	#26021054	C₂	1	2	yes	maximal
4	#26020412	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19151996	—	2	2	no	—
8	#19151996	C₄	1	4	yes	complemented
9	C₂×#15667946	—	2	2	no	—
10	C₂×#15667944	—	2	2	no	—
11	C₂×#15667937	—	2	4	no	—
12	#19151468	C₂²	1	1	yes	characteristic
13	#19151468	C₂²	1	1	yes	characteristic
14	#19151361	—	2	4	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	—	2	2	no	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668012	—	2	2	no	—
21	#15668012	—	2	2	no	—
22	#15668012	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	#15667946	—	4	4	no	—
30	#15667944	—	2	8	no	—
31	#15667937	—	4	8	no	—
32	C₂²×C₈×C₈₉	D₄	1	4	yes	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	—	2	2	no	—
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	C₂×Dic₁₇₈	—	2	4	no	—
40	C₂×Dic₁₇₈	—	4	4	no	—
41	C₂×Dic₁₇₈	—	4	4	no	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	2	no	—
44	C₂²×Dic₈₉	—	4	1	no	—
45	C₂²×Dic₈₉	—	4	1	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	C₄⋊Dic₈₉	—	2	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	—	4	4	no	—
53	C₄⋊Dic₈₉	—	4	4	no	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₈₉	—	2	4	no	—
57	C₂×C₈×C₈₉	—	2	4	no	—
58	C₂×C₈×C₈₉	—	2	4	no	—
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
60	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
61	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
62	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₉	#41	1	2	yes	—
65	C₂³×C₈₉	#127	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₈₉	#121	1	2	yes	—
69	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×C₄×C₈₉	#137	1	4	yes	—
74	C₂×Dic₈₉	—	4	4	no	—
75	C₂×Dic₈₉	—	8	2	no	—
76	C₂×Dic₈₉	—	8	2	no	—
77	C₂×Dic₈₉	—	8	1	no	—
78	C₂×Dic₈₉	—	8	1	no	—
79	C₂×Dic₈₉	—	8	2	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	Dic₁₇₈	—	4	8	no	—
82	Dic₁₇₈	—	8	4	no	—
83	C₈×C₈₉	—	4	4	no	—
84	C₈×C₈₉	#103	1	8	yes	complemented
85	C₂²×C₈₉	#390	1	1	yes	characteristic
86	C₂²×C₈₉	#492	1	1	yes	characteristic
87	C₂²×C₈₉	#494	1	1	yes	characteristic
88	C₂²×C₈₉	#496	1	2	yes	—
89	C₂²×C₈₉	#497	1	2	yes	—
90	C₄×C₈₉	C₂×#103	1	2	yes	—
91	C₄×C₈₉	#440	1	4	yes	—
92	C₄×C₈₉	#468	1	4	yes	—
93	C₄×C₈₉	#487	1	2	yes	—
94	Dic₈₉	—	8	4	no	—
95	Dic₈₉	—	16	1	no	—
96	Dic₈₉	—	16	1	no	—
97	#9841	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	#1715	1	1	yes	characteristic
99	C₂×C₈₉	#1716	1	1	yes	characteristic
100	C₂×C₈₉	#1723	1	2	yes	—
101	C₂×C₈₉	#1956	1	1	yes	characteristic
102	C₂×C₈₉	#1973	1	2	yes	—
103	C₂×#498	—	89	1	no	—
104	#1990	—	89	2	no	—
105	#1956	—	89	2	no	—
106	#1845	—	89	1	no	—
107	C₂×#322	D₈₉	1	1	yes	characteristic
108	C₈₉	#9841	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	—	89	2	no	—
110	#498	—	89	4	no	—
111	#498	—	178	2	no	—
112	C₂×#107	—	178	2	no	—
113	C₂×#106	—	178	2	no	—
114	C₂×#103	—	178	4	no	—
115	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#384	—	89	1	no	—
118	#384	—	89	1	no	—
119	#340	—	178	4	no	—
120	#322	—	2	2	no	—
121	C₂²×Q₈	—	178	2	no	—
122	C₂²×C₈	D₃₅₆	1	4	yes	—
123	C₂²×C₈	D₃₅₆	1	2	yes	—
124	C₄⋊Q₈	—	89	4	no	—
125	C₄⋊Q₈	—	178	2	no	—
126	C₄⋊Q₈	—	178	2	no	—
127	C₄⋊Q₈	—	178	2	no	—
128	C₂×C₄⋊C₄	—	178	2	no	—
129	C₂×C₄⋊C₄	—	178	2	no	—
130	C₂×C₄⋊C₄	—	178	1	no	—
131	C₂×C₄⋊C₄	—	178	1	no	—
132	C₂×C₄⋊C₄	—	178	2	no	—
133	C₂×C₄⋊C₄	—	178	2	no	—
134	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
135	#107	—	356	4	no	—
136	#106	—	178	8	no	—
137	#103	—	356	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₈₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
142	C₂×Q₈	—	356	2	no	—
143	C₂×Q₈	—	356	4	no	—
144	C₂×Q₈	—	356	4	no	—
145	C₂×Q₈	—	178	4	no	—
146	C₂²×C₄	—	356	2	no	—
147	C₂²×C₄	—	356	1	no	—
148	C₂²×C₄	—	356	1	no	—
149	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
150	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	#13083551	1	4	yes	—
155	C₂×C₈	#13083563	1	4	yes	—
156	C₂×C₈	Dic₃₅₆	1	4	yes	—
157	C₄⋊C₄	—	356	2	no	—
158	C₄⋊C₄	—	356	2	no	—
159	C₄⋊C₄	—	356	2	no	—
160	C₄⋊C₄	—	356	2	no	—
161	C₄⋊C₄	—	356	4	no	—
162	C₄⋊C₄	—	356	4	no	—
163	C₄⋊C₄	—	178	4	no	—
164	C₄⋊C₄	—	178	4	no	—
165	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
167	C₄²	#13083574	1	2	yes	—
168	C₂³	#15668009	1	1	yes	characteristic
169	Q₈	—	356	8	no	—
170	Q₈	—	712	4	no	—
171	C₂×C₄	—	356	4	no	—
172	C₂×C₄	—	712	1	no	—
173	C₂×C₄	—	712	1	no	—
174	C₂×C₄	—	712	2	no	—
175	C₂×C₄	—	712	2	no	—
176	C₂×C₄	—	712	2	no	—
177	C₂×C₄	—	712	2	no	—
178	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
179	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
180	C₂×C₄	#15667986	1	2	yes	—
181	C₂×C₄	#15668000	1	1	yes	characteristic
182	C₂×C₄	C₂×#13083563	1	2	yes	—
183	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
184	C₂×C₄	#15668040	1	4	yes	—
185	C₂×C₄	#15668045	1	4	yes	—
186	C₈	—	4	4	no	—
187	C₈	#15667937	1	8	yes	complemented
188	C₂²	#19151500	1	1	yes	characteristic
189	C₂²	#19151971	1	1	yes	characteristic
190	C₂²	#19151978	1	1	yes	characteristic
191	C₂²	#19151987	1	2	yes	—
192	C₂²	#19151992	1	2	yes	—
193	C₄	—	1424	1	no	—
194	C₄	—	1424	1	no	—
195	C₄	—	712	4	no	—
196	C₄	C₂×#15667937	1	2	yes	—
197	C₄	#19151700	1	4	yes	—
198	C₄	#19151844	1	4	yes	—
199	C₄	#19151949	1	2	yes	—
200	C₂	#26019880	1	1	yes	characteristic
201	C₂	#26019885	1	1	yes	characteristic
202	C₂	#26019929	1	2	yes	—
203	C₂	#26021054	1	1	yes	characteristic
204	C₂	#26021133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151996

C₂

yes

maximal, characteristic

#26021213

C₂

yes

maximal

#26021054

C₂

yes

maximal

#26020412

C₂

yes

maximal, characteristic

C₂×#322×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

#19151996

—

#19151996

C₄

yes

—

—

—

C₂²

yes

characteristic

#19151468

C₂²

yes

characteristic

#19151361

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#322×C₈₉

—

C₂²×Dic₁₇₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

#15667944

—

#15667937

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#497

yes

—

C₄×C₈₉

C₂×#103

yes

—

C₄×C₈₉

#440

yes

—

C₄×C₈₉

#468

yes

—

C₄×C₈₉

#487

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

100

C₂×C₈₉

#1723

yes

—

101

C₂×C₈₉

#1956

yes

characteristic

102

C₂×C₈₉

#1973

yes

—

103

C₂×#498

—

104

#1990

—

105

#1956

—

106

#1845

—

107

C₂×#322

D₈₉

yes

characteristic

108

C₈₉

#9841

yes

characteristic, complemented, Sylow

109

C₂×C₄⋊Q₈

—

110

#498

—

111

#498

—

178

—

112

C₂×#107

—

178

—

113

C₂×#106

—

178

—

114

C₂×#103

—

178

—

115

C₂×C₄×C₈

C₂×D₈₉

yes

—

116

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

117

#384

—

118

#384

—

119

#340

—

178

—

120

#322

—

121

C₂²×Q₈

—

178

—

122

C₂²×C₈

D₃₅₆

yes

—

123

C₂²×C₈

D₃₅₆

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

178

—

126

C₄⋊Q₈

—

178

—

127

C₄⋊Q₈

—

178

—

128

C₂×C₄⋊C₄

—

178

—

129

C₂×C₄⋊C₄

—

178

—

130

C₂×C₄⋊C₄

—

178

—

131

C₂×C₄⋊C₄

—

178

—

132

C₂×C₄⋊C₄

—

178

—

133

C₂×C₄⋊C₄

—

178

—

134

C₂×C₄²

C₂²×D₈₉

yes

characteristic

135

#107

—

356

—

136

#106

—

178

—

137

#103

—

356

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₈₉

yes

—

141

C₄×C₈

C₂²⋊D₈₉

yes

—

142

C₂×Q₈

—

356

—

143

C₂×Q₈

—

356

—

144

C₂×Q₈

—

356

—

145

C₂×Q₈

—

178

—

146

C₂²×C₄

—

356

—

147

C₂²×C₄

—

356

—

148

C₂²×C₄

—

356

—

149

C₂²×C₄

C₂×D₃₅₆

yes

—

150

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#13083551

yes

—

155

C₂×C₈

#13083563

yes

—

156

C₂×C₈

Dic₃₅₆

yes

—

157

C₄⋊C₄

—

356

—

158

C₄⋊C₄

—

356

—

159

C₄⋊C₄

—

356

—

160

C₄⋊C₄

—

356

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

178

—

164

C₄⋊C₄

—

178

—

165

C₄²

C₂×C₄×D₈₉

yes

characteristic

166

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

167

C₄²

#13083574

yes

—

168

C₂³

#15668009

yes

characteristic

169

Q₈

—

356

—

170

Q₈

—

712

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

712

—

173

C₂×C₄

—

712

—

174

C₂×C₄

—

712

—

175

C₂×C₄

—

712

—

176

C₂×C₄

—

712

—

177

C₂×C₄

—

712

—

178

C₂×C₄

C₂×#13083551

yes

characteristic

179

C₂×C₄

C₄×D₃₅₆

yes

—

180

C₂×C₄

#15667986

yes

—

181

C₂×C₄

#15668000

yes

characteristic

182

C₂×C₄

C₂×#13083563

yes

—

183

C₂×C₄

C₂×Dic₃₅₆

yes

—

184

C₂×C₄

#15668040

yes

—

185

C₂×C₄

#15668045

yes

—

186

C₈

—

187

C₈

#15667937

yes

complemented

188

C₂²

#19151500

yes

characteristic

189

C₂²

#19151971

yes

characteristic

190

C₂²

#19151978

yes

characteristic

191

C₂²

#19151987

yes

—

192

C₂²

#19151992

yes

—

193

C₄

—

1424

—

194

C₄

—

1424

—

195

C₄

—

712

—

196

C₄

C₂×#15667937

yes

—

197

C₄

#19151700

yes

—

198

C₄

#19151844

yes

—

199

C₄

#19151949

yes

—

200

C₂

#26019880

yes

characteristic

201

C₂

#26019885

yes

characteristic

202

C₂

#26019929

yes

—

203

C₂

#26021054

yes

characteristic

204

C₂

#26021133

yes

—

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9841
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#15668009

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#322×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
8. #19151996	C₄	semidirect
84. C₈×C₈₉	#103	semidirect
108. C₈₉	#9841	semidirect
187. C₈	#15667937	semidirect

H ≅ G/N

type

8. #19151996

C₄

semidirect

84. C₈×C₈₉

#103

semidirect

108. C₈₉

#9841

semidirect

187. C₈

#15667937

semidirect

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

subgroup	index
C₂×#19151996	2
#26021213	2
#26021054	2
#26020412	2
C₂×#322×C₈₉	2
#9841	89

subgroup

index

C₂×#322×C₈₉

Subgroups — #36454288

Conjugacy classes of subgroups

Complements

Maximal subgroups