Group #36454098 — subgroups

Subgroups — #36454098

Order 22784.

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

normal subgroups only

This group has 434 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×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26021095	C₂	1	2	yes	maximal
3	#26018925	C₂	1	2	yes	maximal
4	#26018541	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19152003	—	2	1	no	—
8	#19152003	—	2	1	no	—
9	#19152003	—	2	2	no	—
10	#19151651	C₂²	1	1	yes	characteristic
11	C₂×#15667946	C₄	1	4	yes	—
12	C₂×#15667930	—	2	4	no	—
13	C₄×#13083558	C₂²	1	1	yes	characteristic
14	#19151345	—	2	4	no	—
15	#19151320	—	2	1	no	—
16	#19151319	—	2	1	no	—
17	#423×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#334×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	C₂×#13083558	D₄	1	2	yes	—
22	#15668012	—	2	2	no	—
23	#15668012	—	2	2	no	—
24	#15668012	—	2	2	no	—
25	#15668012	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	#15667946	—	2	4	no	—
29	#15667946	—	2	4	no	—
30	C₄⋊#10623494	D₄	1	2	yes	—
31	#15667930	—	4	8	no	—
32	C₂²⋊#10623494	—	2	1	no	—
33	#15667923	—	2	1	no	—
34	C₄×#10623494	—	2	1	no	—
35	C₄×#10623494	—	2	1	no	—
36	C₂×M₁₆×C₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	—	2	2	no	—
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₂×Dic₁₇₈	—	4	2	no	—
45	C₂×Dic₁₇₈	—	4	2	no	—
46	C₂×Dic₁₇₈	—	4	4	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂²×Dic₈₉	—	2	4	no	—
49	#13083558	—	2	4	no	—
50	C₂×#10623494	—	2	2	no	—
51	C₂×#10623494	—	2	2	no	—
52	C₄⋊Dic₈₉	—	2	4	no	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
59	M₁₆×C₈₉	—	4	4	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₉	#41	1	1	yes	characteristic
68	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	4	8	no	—
80	Dic₁₇₈	—	8	4	no	—
81	Dic₁₇₈	—	8	2	no	—
82	Dic₁₇₈	—	8	2	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	—	8	2	no	—
85	C₈×C₈₉	#101	1	4	yes	complemented
86	#10623494	—	4	2	no	—
87	#10623494	—	4	2	no	—
88	C₂²×C₈₉	—	2	1	no	—
89	C₂²×C₈₉	—	2	1	no	—
90	C₂²×C₈₉	#390	1	1	yes	characteristic
91	C₂²×C₈₉	#442	1	2	yes	—
92	C₄×C₈₉	—	2	1	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	C₂×#101	1	2	yes	—
95	C₄×C₈₉	#469	1	2	yes	—
96	C₄×C₈₉	#473	1	2	yes	—
97	C₄×C₈₉	#496	1	2	yes	—
98	Dic₈₉	—	8	4	no	—
99	#9808	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	2	no	—
101	C₂×C₈₉	#1966	1	1	yes	characteristic
102	C₂×C₈₉	#1972	1	1	yes	characteristic
103	C₂×C₈₉	#1979	1	1	yes	characteristic
104	C₂×#500	—	89	1	no	—
105	#1966	—	89	2	no	—
106	#1566	D₈₉	1	1	yes	characteristic
107	#1546	—	89	2	no	—
108	#1451	—	89	1	no	—
109	C₈₉	#9808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	89	2	no	—
111	#500	—	178	1	no	—
112	#500	—	178	1	no	—
113	#500	—	178	2	no	—
114	#431	—	89	1	no	—
115	C₂×#107	—	89	4	no	—
116	#423	C₂×D₈₉	1	2	yes	—
117	C₂×#101	—	178	4	no	—
118	C₄×M₁₆	—	89	1	no	—
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	#334	—	2	2	no	—
121	#332	—	178	4	no	—
122	#322	—	178	1	no	—
123	#322	—	178	1	no	—
124	C₂²×Q₈	—	178	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	89	2	no	—
127	C₂²×C₈	C₄×D₈₉	1	2	yes	—
128	C₄⋊Q₈	—	178	2	no	—
129	C₄⋊Q₈	—	178	2	no	—
130	C₄⋊Q₈	—	178	2	no	—
131	C₄⋊Q₈	—	178	2	no	—
132	C₂×C₄⋊C₄	—	89	4	no	—
133	C₂×C₄⋊C₄	—	178	2	no	—
134	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
135	#107	—	178	4	no	—
136	#107	—	178	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	89	2	no	—
140	#101	—	356	8	no	—
141	C₂²⋊C₈	—	178	1	no	—
142	#97	—	178	1	no	—
143	C₄×C₈	—	178	1	no	—
144	C₄×C₈	—	178	1	no	—
145	C₄×C₈	D₃₅₆	1	1	yes	characteristic
146	C₄×C₈	D₃₅₆	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
148	C₂×Q₈	—	356	2	no	—
149	C₂×Q₈	—	356	2	no	—
150	C₂×Q₈	—	356	4	no	—
151	C₂×Q₈	—	356	4	no	—
152	C₂²×C₄	—	178	4	no	—
153	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
155	C₂²×C₄	#13083575	1	1	yes	characteristic
156	M₁₆	—	4	4	no	—
157	M₁₆	—	178	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	178	2	no	—
161	C₂×C₈	—	178	2	no	—
162	C₂×C₈	#13083551	1	2	yes	—
163	C₂×C₈	#13083551	1	2	yes	—
164	C₄⋊C₄	—	356	2	no	—
165	C₄⋊C₄	—	356	2	no	—
166	C₄⋊C₄	—	178	4	no	—
167	C₄⋊C₄	—	178	4	no	—
168	C₄²	D₄×D₈₉	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
170	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
171	C₄²	#13083575	1	1	yes	characteristic
172	C₂³	#15667974	1	1	yes	characteristic
173	Q₈	—	712	2	no	—
174	Q₈	—	712	2	no	—
175	Q₈	—	712	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	356	8	no	—
179	C₂×C₄	—	356	4	no	—
180	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
181	C₂×C₄	#15667974	1	1	yes	characteristic
182	C₂×C₄	#15667987	1	1	yes	characteristic
183	C₂×C₄	#15667987	1	1	yes	characteristic
184	C₂×C₄	#15668001	1	1	yes	characteristic
185	C₂×C₄	#15668008	1	1	yes	characteristic
186	C₂×C₄	#15668039	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	8	2	no	—
189	C₈	—	356	2	no	—
190	C₈	—	356	2	no	—
191	C₈	#15667930	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#19151499	1	1	yes	characteristic
195	C₂²	#19151709	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	712	4	no	—
199	C₄	C₂×#15667930	1	2	yes	—
200	C₄	#19151852	1	2	yes	—
201	C₄	#19151874	1	2	yes	—
202	C₄	#19151984	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#26021095	1	1	yes	characteristic
205	C₂	#26021126	1	1	yes	characteristic
206	C₂	#26021165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152003

C₂

yes

maximal, characteristic

#26021095

C₂

yes

maximal

#26018925

C₂

yes

maximal

#26018541

C₂

yes

maximal, characteristic

C₈₉×#1566

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#15667946

C₄

yes

—

C₂×#15667930

—

C₄×#13083558

C₂²

yes

—

—

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#334×C₈₉

—

C₂²×Dic₁₇₈

—

C₂×#13083558

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

#15667946

—

#15667946

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

characteristic

C₂³×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#442

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#101

yes

—

C₄×C₈₉

#469

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

#496

yes

—

Dic₈₉

—

#9808

—

maximal, Sylow

100

C₂×C₈₉

—

101

C₂×C₈₉

#1966

yes

characteristic

102

C₂×C₈₉

#1972

yes

characteristic

103

C₂×C₈₉

#1979

yes

characteristic

104

C₂×#500

—

105

#1966

—

106

#1566

D₈₉

yes

characteristic

107

#1546

—

108

#1451

—

109

C₈₉

#9808

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#500

—

178

—

112

#500

—

178

—

113

#500

—

178

—

114

#431

—

115

C₂×#107

—

116

#423

C₂×D₈₉

yes

—

117

C₂×#101

—

178

—

118

C₄×M₁₆

—

119

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

120

#334

—

121

#332

—

178

—

122

#322

—

178

—

123

#322

—

178

—

124

C₂²×Q₈

—

178

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

C₄×D₈₉

yes

—

128

C₄⋊Q₈

—

178

—

129

C₄⋊Q₈

—

178

—

130

C₄⋊Q₈

—

178

—

131

C₄⋊Q₈

—

178

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

178

—

134

C₂×C₄²

C₂²×D₈₉

yes

characteristic

135

#107

—

178

—

136

#107

—

178

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

#101

—

356

—

141

C₂²⋊C₈

—

178

—

142

#97

—

178

—

143

C₄×C₈

—

178

—

144

C₄×C₈

—

178

—

145

C₄×C₈

D₃₅₆

yes

characteristic

146

C₄×C₈

D₃₅₆

yes

characteristic

147

C₄×C₈

C₂²⋊D₈₉

yes

—

148

C₂×Q₈

—

356

—

149

C₂×Q₈

—

356

—

150

C₂×Q₈

—

356

—

151

C₂×Q₈

—

356

—

152

C₂²×C₄

—

178

—

153

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

154

C₂²×C₄

D₄×D₈₉

yes

characteristic

155

C₂²×C₄

#13083575

yes

characteristic

156

M₁₆

—

157

M₁₆

—

178

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

178

—

161

C₂×C₈

—

178

—

162

C₂×C₈

#13083551

yes

—

163

C₂×C₈

#13083551

yes

—

164

C₄⋊C₄

—

356

—

165

C₄⋊C₄

—

356

—

166

C₄⋊C₄

—

178

—

167

C₄⋊C₄

—

178

—

168

C₄²

D₄×D₈₉

yes

characteristic

169

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

170

C₄²

C₂×D₃₅₆

yes

characteristic

171

C₄²

#13083575

yes

characteristic

172

C₂³

#15667974

yes

characteristic

173

Q₈

—

712

—

174

Q₈

—

712

—

175

Q₈

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

356

—

180

C₂×C₄

C₂×#13083551

yes

characteristic

181

C₂×C₄

#15667974

yes

characteristic

182

C₂×C₄

#15667987

yes

characteristic

183

C₂×C₄

#15667987

yes

characteristic

184

C₂×C₄

#15668001

yes

characteristic

185

C₂×C₄

#15668008

yes

characteristic

186

C₂×C₄

#15668039

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

356

—

190

C₈

—

356

—

191

C₈

#15667930

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#19151499

yes

characteristic

195

C₂²

#19151709

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

712

—

199

C₄

C₂×#15667930

yes

—

200

C₄

#19151852

yes

—

201

C₄

#19151874

yes

—

202

C₄

#19151984

yes

—

203

C₂

—

204

C₂

#26021095

yes

characteristic

205

C₂

#26021126

yes

characteristic

206

C₂

#26021165

yes

characteristic

	subgroup	G/H
center	C₂²	#19151499
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#9808
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#19151499

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1566

C₂

socle

C₂²×C₈₉

#390

prime radical

C₂³×C₈₉

C₄×D₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
85. C₈×C₈₉	#101	semidirect
109. C₈₉	#9808	semidirect
191. C₈	#15667930	semidirect

H ≅ G/N

type

85. C₈×C₈₉

#101

semidirect

109. C₈₉

#9808

semidirect

191. C₈

#15667930

semidirect

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

subgroup	index
C₂×#19152003	2
#26021095	2
#26018925	2
#26018541	2
C₈₉×#1566	2
#9808	89

subgroup

index

C₈₉×#1566

Subgroups — #36454098

Conjugacy classes of subgroups

Complements

Maximal subgroups