Group #36473003 — subgroups

Subgroups — #36473003

Order 22784.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19152003	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667946	C₂	1	1	yes	maximal, characteristic
3	#26018344	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	#19152003	—	2	2	no	—
7	#19152003	C₄	1	4	yes	complemented
8	C₂×#15667946	—	2	1	no	—
9	C₂×#15667946	C₂²	1	2	yes	—
10	C₂×#15667946	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₈₉	—	2	1	no	—
12	#19151361	—	2	1	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₉	—	2	2	no	—
16	C₂²×Dic₁₇₈	—	4	1	no	—
17	#15668012	—	2	4	no	—
18	#15668012	—	4	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	4	1	no	—
22	C₂×C₄×Dic₈₉	—	4	1	no	—
23	#15667946	—	2	4	no	—
24	#15667946	—	2	2	no	—
25	#15667946	—	2	2	no	—
26	#15667946	C₂×C₄	1	4	yes	—
27	#15667946	Q₈	1	4	yes	complemented
28	#15667919	—	4	1	no	—
29	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	—	2	2	no	—
33	C₂×C₁₆×C₈₉	—	2	4	no	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₂×Dic₁₇₈	—	4	4	no	—
37	C₂×Dic₁₇₈	—	8	4	no	—
38	C₂²×Dic₈₉	—	4	2	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄⋊Dic₈₉	—	4	4	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄×Dic₈₉	—	8	2	no	—
46	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₈₉	#41	1	2	yes	—
55	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₈₉	SD₁₆	1	2	yes	—
57	C₁₆×C₈₉	—	4	2	no	—
58	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	#102	1	2	yes	—
61	C₂×C₄×C₈₉	#102	1	2	yes	—
62	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
63	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#137	1	2	yes	—
68	C₂×Dic₈₉	—	4	4	no	—
69	C₂×Dic₈₉	—	4	4	no	—
70	C₂×Dic₈₉	—	8	2	no	—
71	C₂×Dic₈₉	—	8	2	no	—
72	C₂×Dic₈₉	—	8	4	no	—
73	Dic₁₇₈	—	8	8	no	—
74	C₈×C₈₉	—	2	2	no	—
75	C₈×C₈₉	C₄×Q₈	1	2	yes	—
76	C₈×C₈₉	#124	1	1	yes	characteristic
77	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
79	C₂²×C₈₉	#474	1	2	yes	—
80	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
81	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₈₉	#509	1	2	yes	—
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	#358	1	4	yes	—
85	C₄×C₈₉	#391	1	1	yes	characteristic
86	C₄×C₈₉	#419	1	1	yes	characteristic
87	C₄×C₈₉	#475	1	2	yes	—
88	Dic₈₉	—	8	4	no	—
89	Dic₈₉	—	8	4	no	—
90	#13276	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	#1988	1	1	yes	characteristic
92	C₂×C₈₉	C₂×#358	1	1	yes	characteristic
93	C₂×C₈₉	#2140	1	1	yes	characteristic
94	C₂×C₈₉	#2226	1	2	yes	—
95	C₂×C₈₉	#2227	1	2	yes	—
96	C₂×#500	—	89	1	no	—
97	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
98	C₄×#107	—	89	1	no	—
99	#1380	—	89	4	no	—
100	C₈₉	#13276	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	178	1	no	—
102	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
103	#500	—	89	4	no	—
104	#500	—	178	2	no	—
105	C₂×#107	—	89	2	no	—
106	C₂×#107	—	89	2	no	—
107	C₂×#107	—	178	1	no	—
108	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	178	1	no	—
110	C₄⋊C₁₆	—	2	2	no	—
111	#340	—	178	1	no	—
112	C₂²×Q₈	—	356	1	no	—
113	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
115	C₄⋊Q₈	—	356	2	no	—
116	C₄⋊Q₈	—	178	4	no	—
117	C₂×C₄⋊C₄	—	356	1	no	—
118	C₂×C₄⋊C₄	—	178	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄²	—	356	1	no	—
121	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	#107	—	89	4	no	—
125	#107	—	89	4	no	—
126	#107	—	178	4	no	—
127	#107	—	178	2	no	—
128	#107	—	178	2	no	—
129	C₄×C₈	C₄×D₈₉	1	2	yes	—
130	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
131	#95	—	356	1	no	—
132	C₂×Q₈	—	356	4	no	—
133	C₂×Q₈	—	712	4	no	—
134	C₂²×C₄	—	356	2	no	—
135	C₂²×C₄	—	356	2	no	—
136	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
137	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	#13083551	1	2	yes	—
141	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₈₉	1	2	yes	—
144	C₂×C₈	#13083573	1	2	yes	—
145	C₄⋊C₄	—	356	2	no	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	356	4	no	—
148	C₄⋊C₄	—	178	4	no	—
149	C₄⋊C₄	—	178	4	no	—
150	C₄²	—	712	2	no	—
151	C₄²	#13083551	1	2	yes	—
152	C₄²	#13083563	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
155	Q₈	—	712	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	356	4	no	—
158	C₂×C₄	—	356	4	no	—
159	C₂×C₄	—	712	2	no	—
160	C₂×C₄	—	712	2	no	—
161	C₂×C₄	—	712	4	no	—
162	C₂×C₄	#15667934	1	2	yes	—
163	C₂×C₄	#15667934	1	2	yes	—
164	C₂×C₄	#15667957	1	2	yes	—
165	C₂×C₄	Dic₇₁₂	1	2	yes	—
166	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
167	C₂×C₄	#15667992	1	2	yes	—
168	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
169	C₂×C₄	#15668045	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	#15667977	1	2	yes	—
172	C₈	#15667999	1	1	yes	characteristic
173	C₈	#15668010	1	1	yes	characteristic
174	C₂²	C₂×#15667934	1	1	yes	characteristic
175	C₂²	#19151879	1	2	yes	—
176	C₂²	C₂×#15667957	1	1	yes	characteristic
177	C₂²	C₂×Dic₇₁₂	1	1	yes	characteristic
178	C₂²	#19152036	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	712	4	no	—
181	C₄	—	712	4	no	—
182	C₄	#19151403	1	4	yes	—
183	C₄	#19151503	1	1	yes	characteristic
184	C₄	#19151608	1	1	yes	characteristic
185	C₄	#19151884	1	2	yes	—
186	C₂	#26021203	1	1	yes	characteristic
187	C₂	C₂×#19151403	1	1	yes	characteristic
188	C₂	#26021969	1	1	yes	characteristic
189	C₂	#26022400	1	2	yes	—
190	C₂	#26022405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152003

C₂

yes

maximal, characteristic

C₄×#15667946

C₂

yes

maximal, characteristic

#26018344

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668012

—

#19152003

—

#19152003

C₄

yes

complemented

C₂×#15667946

—

C₂×#15667946

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₄×C₄⋊Dic₈₉

—

#19151361

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²×Dic₁₇₈

—

#15668012

—

#15668012

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄×Dic₈₉

—

—

—

—

C₂×C₄

yes

—

#15667946

Q₈

yes

complemented

#15667919

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

SD₃₂

yes

—

C₂×C₄×C₈₉

Q₃₂

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₄×Q₈

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

C₂×SD₃₂

yes

characteristic

C₂²×C₈₉

C₂×Q₃₂

yes

characteristic

C₂²×C₈₉

#509

yes

—

C₄×C₈₉

—

C₄×C₈₉

#358

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2227

yes

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₈₉

yes

characteristic

C₄×#107

—

#1380

—

100

C₈₉

#13276

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

178

—

102

C₂²×C₁₆

C₂×D₈₉

yes

—

103

#500

—

104

#500

—

178

—

105

C₂×#107

—

106

C₂×#107

—

107

C₂×#107

—

178

—

108

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

109

C₄×C₄⋊C₄

—

178

—

110

C₄⋊C₁₆

—

111

#340

—

178

—

112

C₂²×Q₈

—

356

—

113

C₂²×C₈

D₃₅₆

yes

characteristic

114

C₂²×C₈

C₂²×D₈₉

yes

characteristic

115

C₄⋊Q₈

—

356

—

116

C₄⋊Q₈

—

178

—

117

C₂×C₄⋊C₄

—

356

—

118

C₂×C₄⋊C₄

—

178

—

119

C₂×C₄⋊C₄

—

178

—

120

C₂×C₄²

—

356

—

121

C₂×C₄²

D₃₅₆

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

#107

—

125

#107

—

126

#107

—

178

—

127

#107

—

178

—

128

#107

—

178

—

129

C₄×C₈

C₄×D₈₉

yes

—

130

C₄×C₈

C₂²⋊D₈₉

yes

—

131

#95

—

356

—

132

C₂×Q₈

—

356

—

133

C₂×Q₈

—

712

—

134

C₂²×C₄

—

356

—

135

C₂²×C₄

—

356

—

136

C₂²×C₄

D₇₁₂

yes

characteristic

137

C₂²×C₄

D₇₁₂

yes

characteristic

138

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

#13083551

yes

—

141

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

142

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

143

C₂×C₈

Q₈×D₈₉

yes

—

144

C₂×C₈

#13083573

yes

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄²

—

712

—

151

C₄²

#13083551

yes

—

152

C₄²

#13083563

yes

—

153

C₁₆

—

154

C₂³

C₂×D₇₁₂

yes

characteristic

155

Q₈

—

712

—

156

C₂×C₄

—

157

C₂×C₄

—

356

—

158

C₂×C₄

—

356

—

159

C₂×C₄

—

712

—

160

C₂×C₄

—

712

—

161

C₂×C₄

—

712

—

162

C₂×C₄

#15667934

yes

—

163

C₂×C₄

#15667934

yes

—

164

C₂×C₄

#15667957

yes

—

165

C₂×C₄

Dic₇₁₂

yes

—

166

C₂×C₄

C₂×#13083551

yes

characteristic

167

C₂×C₄

#15667992

yes

—

168

C₂×C₄

C₂×#13083563

yes

characteristic

169

C₂×C₄

#15668045

yes

—

170

C₈

—

171

C₈

#15667977

yes

—

172

C₈

#15667999

yes

characteristic

173

C₈

#15668010

yes

characteristic

174

C₂²

C₂×#15667934

yes

characteristic

175

C₂²

#19151879

yes

—

176

C₂²

C₂×#15667957

yes

characteristic

177

C₂²

C₂×Dic₇₁₂

yes

characteristic

178

C₂²

#19152036

yes

—

179

C₄

—

180

C₄

—

712

—

181

C₄

—

712

—

182

C₄

#19151403

yes

—

183

C₄

#19151503

yes

characteristic

184

C₄

#19151608

yes

characteristic

185

C₄

#19151884

yes

—

186

C₂

#26021203

yes

characteristic

187

C₂

C₂×#19151403

yes

characteristic

188

C₂

#26021969

yes

characteristic

189

C₂

#26022400

yes

—

190

C₂

#26022405

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#13276
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₇₁₂

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×C₄⋊C₁₆×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₈₉

hypocenter

C₈₉

#13276

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #19152003	C₄	semidirect
27. #15667946	Q₈	semidirect
100. C₈₉	#13276	semidirect

H ≅ G/N

type

7. #19152003

C₄

semidirect

27. #15667946

Q₈

semidirect

100. C₈₉

#13276

semidirect

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

subgroup	index
C₂×#19152003	2
C₄×#15667946	2
#26018344	2
C₂×C₄⋊C₁₆×C₈₉	2
#13276	89

subgroup

index

C₂×C₄⋊C₁₆×C₈₉

Subgroups — #36473003

Conjugacy classes of subgroups

Complements

Maximal subgroups