Group #36472466 — subgroups

Subgroups — #36472466

Order 22784.

318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19151426	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151373	C₂	1	1	yes	maximal, characteristic
3	#26020983	C₂	1	2	yes	maximal
4	#26018343	C₂	1	2	yes	maximal
5	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
9	#19151494	—	2	2	no	—
10	#19151426	C₄	1	4	yes	complemented
11	#19151373	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	#366×C₈₉	—	2	2	no	—
16	C₂²×#10623494	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×#13083553	—	4	2	no	—
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	Q₈	1	2	yes	—
23	#15667946	—	2	2	no	—
24	#15667946	C₂×C₄	1	4	yes	—
25	C₈×Dic₈₉	—	2	1	no	—
26	C₈×Dic₈₉	—	2	1	no	—
27	#15667919	—	4	2	no	—
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
33	#107×C₈₉	—	2	2	no	—
34	#107×C₈₉	—	2	2	no	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
37	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
38	C₂×#10623494	—	2	1	no	—
39	C₂×#10623494	—	2	1	no	—
40	C₂×#10623494	Q₁₆	1	2	yes	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	#13083553	—	8	4	no	—
44	C₄×Dic₈₉	—	2	1	no	—
45	C₄×Dic₈₉	—	2	1	no	—
46	#13083549	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	4	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	4	2	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	8	2	no	—
60	C₂×C₄×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	—	8	2	no	—
62	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
68	C₂×Dic₈₉	—	2	1	no	—
69	C₂×Dic₈₉	—	2	1	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	C₂×Dic₈₉	—	4	4	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₈₉	#125	1	1	yes	characteristic
79	#10623494	—	2	2	no	—
80	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₈₉	#466	1	1	yes	characteristic
82	C₂²×C₈₉	#500	1	1	yes	characteristic
83	C₂²×C₈₉	#509	1	2	yes	—
84	C₂²×C₈₉	#510	1	2	yes	—
85	C₄×C₈₉	—	16	2	no	—
86	C₄×C₈₉	#389	1	1	yes	characteristic
87	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₈₉	#467	1	1	yes	characteristic
89	C₄×C₈₉	#499	1	1	yes	characteristic
90	Dic₈₉	—	4	1	no	—
91	Dic₈₉	—	4	1	no	—
92	Dic₈₉	—	8	4	no	—
93	#13167	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	#1942	1	1	yes	characteristic
95	C₂×C₈₉	#2175	1	2	yes	—
96	C₂×C₈₉	#2218	1	1	yes	characteristic
97	C₂×C₈₉	#2219	1	1	yes	characteristic
98	C₂×C₈₉	#2255	1	2	yes	—
99	C₂×#367	—	89	1	no	—
100	C₂×#366	D₈₉	1	1	yes	characteristic
101	C₂×#346	—	89	1	no	—
102	#1942	—	89	2	no	—
103	#1380	—	89	2	no	—
104	C₈₉	#13167	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	89	1	no	—
106	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
107	C₂×#107	—	89	2	no	—
108	C₂×#107	C₂×D₈₉	1	2	yes	—
109	C₂×C₄×C₈	—	89	1	no	—
110	#389	—	178	2	no	—
111	#367	—	89	4	no	—
112	#366	—	2	2	no	—
113	#346	—	178	2	no	—
114	#340	—	178	2	no	—
115	C₂²×C₈	—	89	1	no	—
116	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	356	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄²	—	89	1	no	—
121	C₂×C₁₆	—	89	2	no	—
122	C₂×C₁₆	—	178	1	no	—
123	C₂×C₁₆	—	178	1	no	—
124	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
125	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	89	4	no	—
130	#107	—	178	2	no	—
131	C₄×C₈	—	178	1	no	—
132	C₄×C₈	—	178	1	no	—
133	#95	—	356	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	89	1	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	—	356	2	no	—
138	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	—	89	2	no	—
140	C₂×C₈	—	178	1	no	—
141	C₂×C₈	—	178	1	no	—
142	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₂×C₈	#13083573	1	1	yes	characteristic
147	C₂×C₈	#13083575	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	356	2	no	—
151	C₄⋊C₄	—	178	4	no	—
152	C₄⋊C₄	—	712	4	no	—
153	C₄²	—	178	1	no	—
154	C₄²	—	178	1	no	—
155	C₁₆	—	178	2	no	—
156	C₁₆	#13083553	1	4	yes	complemented
157	C₂³	D₈×D₈₉	1	1	yes	characteristic
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	356	4	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	178	2	no	—
164	C₂×C₄	—	178	1	no	—
165	C₂×C₄	—	178	1	no	—
166	C₂×C₄	—	712	2	no	—
167	C₂×C₄	—	712	2	no	—
168	C₂×C₄	#15667974	1	1	yes	characteristic
169	C₂×C₄	#15667983	1	1	yes	characteristic
170	C₂×C₄	#15668011	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
172	C₂×C₄	#15668031	1	1	yes	characteristic
173	C₂×C₄	#15668035	1	1	yes	characteristic
174	C₈	—	178	2	no	—
175	C₈	C₂×#13083553	1	1	yes	characteristic
176	C₈	#15667977	1	1	yes	characteristic
177	C₈	#15667988	1	1	yes	characteristic
178	C₈	#15668003	1	1	yes	characteristic
179	C₂²	#19151680	1	1	yes	characteristic
180	C₂²	#19151831	1	1	yes	characteristic
181	C₂²	#19152002	1	1	yes	characteristic
182	C₂²	#19152032	1	2	yes	—
183	C₂²	#19152040	1	2	yes	—
184	C₄	—	16	2	no	—
185	C₄	—	356	1	no	—
186	C₄	—	356	1	no	—
187	C₄	—	712	4	no	—
188	C₄	#19151494	1	1	yes	characteristic
189	C₄	#19151692	1	1	yes	characteristic
190	C₄	#19151836	1	1	yes	characteristic
191	C₄	#19151999	1	1	yes	characteristic
192	C₂	#26020983	1	1	yes	characteristic
193	C₂	#26022115	1	2	yes	—
194	C₂	#26022352	1	1	yes	characteristic
195	C₂	#26022357	1	1	yes	characteristic
196	C₂	#26022536	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151426

C₂

yes

maximal, characteristic

C₂×#19151373

C₂

yes

maximal, characteristic

#26020983

C₂

yes

maximal

#26018343

C₂

yes

maximal

C₂×#366×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151494

—

#19151426

C₄

yes

complemented

#19151373

—

#19151360

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#366×C₈₉

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

D₄

yes

—

—

Q₈

yes

—

#15667946

—

#15667946

C₂×C₄

yes

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

#15667919

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

Q₈

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

D₈

yes

—

—

Q₁₆

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

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

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₄⋊C₄

yes

complemented

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₄×D₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#510

yes

—

C₄×C₈₉

—

C₄×C₈₉

#389

yes

characteristic

C₄×C₈₉

C₄×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#367

—

100

C₂×#366

D₈₉

yes

characteristic

101

C₂×#346

—

102

#1942

—

103

#1380

—

104

C₈₉

#13167

yes

characteristic, complemented, Sylow

105

C₂²×C₁₆

—

106

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

107

C₂×#107

—

108

C₂×#107

C₂×D₈₉

yes

—

109

C₂×C₄×C₈

—

110

#389

—

178

—

111

#367

—

112

#366

—

113

#346

—

178

—

114

#340

—

178

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂²×D₈₉

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

356

—

119

C₂×C₄⋊C₄

—

178

—

120

C₂×C₄²

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

178

—

123

C₂×C₁₆

—

178

—

124

C₂×C₁₆

Dic₁₇₈

yes

—

125

C₂×C₁₆

C₄×D₈₉

yes

—

126

C₂×C₁₆

C₂²⋊D₈₉

yes

—

127

#107

—

128

#107

—

129

#107

—

130

#107

—

178

—

131

C₄×C₈

—

178

—

132

C₄×C₈

—

178

—

133

#95

—

356

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

356

—

138

C₂²×C₄

D₄×D₈₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

178

—

141

C₂×C₈

—

178

—

142

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

143

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

144

C₂×C₈

Q₈×D₈₉

yes

characteristic

145

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

146

C₂×C₈

#13083573

yes

characteristic

147

C₂×C₈

#13083575

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

178

—

152

C₄⋊C₄

—

712

—

153

C₄²

—

178

—

154

C₄²

—

178

—

155

C₁₆

—

178

—

156

C₁₆

#13083553

yes

complemented

157

C₂³

D₈×D₈₉

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

356

—

162

C₂×C₄

—

178

—

163

C₂×C₄

—

178

—

164

C₂×C₄

—

178

—

165

C₂×C₄

—

178

—

166

C₂×C₄

—

712

—

167

C₂×C₄

—

712

—

168

C₂×C₄

#15667974

yes

characteristic

169

C₂×C₄

#15667983

yes

characteristic

170

C₂×C₄

#15668011

yes

characteristic

171

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

172

C₂×C₄

#15668031

yes

characteristic

173

C₂×C₄

#15668035

yes

characteristic

174

C₈

—

178

—

175

C₈

C₂×#13083553

yes

characteristic

176

C₈

#15667977

yes

characteristic

177

C₈

#15667988

yes

characteristic

178

C₈

#15668003

yes

characteristic

179

C₂²

#19151680

yes

characteristic

180

C₂²

#19151831

yes

characteristic

181

C₂²

#19152002

yes

characteristic

182

C₂²

#19152032

yes

—

183

C₂²

#19152040

yes

—

184

C₄

—

185

C₄

—

356

—

186

C₄

—

356

—

187

C₄

—

712

—

188

C₄

#19151494

yes

characteristic

189

C₄

#19151692

yes

characteristic

190

C₄

#19151836

yes

characteristic

191

C₄

#19151999

yes

characteristic

192

C₂

#26020983

yes

characteristic

193

C₂

#26022115

yes

—

194

C₂

#26022352

yes

characteristic

195

C₂

#26022357

yes

characteristic

196

C₂

#26022536

yes

—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	#13167
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#366×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
10. #19151426	C₄	semidirect
57. C₁₆×C₈₉	C₄⋊C₄	semidirect
104. C₈₉	#13167	semidirect
156. C₁₆	#13083553	semidirect

H ≅ G/N

type

10. #19151426

C₄

semidirect

57. C₁₆×C₈₉

C₄⋊C₄

semidirect

104. C₈₉

#13167

semidirect

156. C₁₆

#13083553

semidirect

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

subgroup	index
C₂×#19151426	2
C₂×#19151373	2
#26020983	2
#26018343	2
C₂×#366×C₈₉	2
#13167	89

subgroup

index

C₂×#366×C₈₉

Subgroups — #36472466

Conjugacy classes of subgroups

Complements

Maximal subgroups