Group #36454456 — subgroups

Subgroups — #36454456

Order 22784.

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

normal subgroups only

This group has 434 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	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	#26018093	C₂	1	4	yes	maximal
4	C₄×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151353	—	2	2	no	—
8	#19151353	—	2	2	no	—
9	C₂×#125×C₈₉	C₄	1	1	yes	characteristic
10	#389×C₈₉	C₄	1	2	yes	—
11	#387×C₈₉	—	2	2	no	—
12	#382×C₈₉	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
14	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	—	2	2	no	—
17	C₂²×#10623494	—	2	2	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄×#10623494	—	2	2	no	—
20	C₄×#10623494	Q₈	1	4	yes	—
21	#125×C₈₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₈₉	—	2	2	no	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#95×C₈₉	—	2	2	no	—
33	#95×C₈₉	—	2	2	no	—
34	#95×C₈₉	—	2	2	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	—	4	4	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	4	no	—
39	C₂²×C₄×C₈₉	—	2	2	no	—
40	C₂²×C₄×C₈₉	—	2	2	no	—
41	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₈₉	—	2	8	no	—
47	C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₄⋊C₄×C₈₉	—	2	4	no	—
49	C₄⋊C₄×C₈₉	—	4	4	no	—
50	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	—	2	2	no	—
53	C₄²×C₈₉	—	2	4	no	—
54	C₄²×C₈₉	—	4	2	no	—
55	C₄²×C₈₉	—	4	4	no	—
56	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
57	C₄²×C₈₉	#41	1	2	yes	—
58	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	4	2	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	#131	1	2	yes	—
78	C₂×C₄×C₈₉	#135	1	4	yes	—
79	#10623494	—	4	4	no	—
80	#10623494	—	8	2	no	—
81	C₂²×C₈₉	#391	1	1	yes	characteristic
82	C₂²×C₈₉	#407	1	1	yes	characteristic
83	C₂²×C₈₉	#408	1	1	yes	characteristic
84	C₂²×C₈₉	#416	1	2	yes	—
85	C₂²×C₈₉	#421	1	2	yes	—
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	—	4	2	no	—
89	C₄×C₈₉	—	4	4	no	—
90	C₄×C₈₉	—	8	2	no	—
91	C₄×C₈₉	#446	1	4	yes	—
92	C₄×C₈₉	#488	1	2	yes	—
93	C₄×C₈₉	#499	1	2	yes	—
94	#9875	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	#1482	1	2	yes	—
96	C₂×C₈₉	#1488	1	2	yes	—
97	C₂×C₈₉	#1985	1	1	yes	characteristic
98	C₂×C₈₉	#1991	1	1	yes	characteristic
99	C₂×C₈₉	#1995	1	1	yes	characteristic
100	C₄×#125	D₈₉	1	1	yes	characteristic
101	#1831	—	89	1	no	—
102	#1744	—	89	1	no	—
103	#1279	—	89	4	no	—
104	C₈₉	#9875	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	Dic₈₉	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₂×C₄×C₈	—	89	2	no	—
108	#389	Dic₈₉	1	2	yes	—
109	#387	—	2	2	no	—
110	#382	Dic₈₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	2	no	—
112	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
113	C₄³	C₂×D₈₉	1	1	yes	characteristic
114	#336	—	178	2	no	—
115	#336	—	178	2	no	—
116	C₂²×C₈	—	178	2	no	—
117	C₂²×C₈	—	178	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊C₈	—	178	4	no	—
130	C₄×C₈	—	89	4	no	—
131	C₄×C₈	—	178	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
138	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂×C₈	—	356	4	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	356	4	no	—
145	C₂×C₈	—	178	4	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	#13083558	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	4	no	—
156	C₄²	Q₈×D₈₉	1	2	yes	—
157	C₄²	#13083575	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
173	C₂×C₄	#15667988	1	2	yes	—
174	C₂×C₄	#15668001	1	1	yes	characteristic
175	C₂×C₄	#15668010	1	1	yes	characteristic
176	C₂×C₄	C₂×#13083558	1	2	yes	—
177	C₂×C₄	#15668019	1	2	yes	—
178	C₂×C₄	#15668032	1	4	yes	—
179	C₈	—	356	4	no	—
180	C₈	—	712	2	no	—
181	C₂²	#19151505	1	1	yes	characteristic
182	C₂²	#19151562	1	1	yes	characteristic
183	C₂²	#19151565	1	1	yes	characteristic
184	C₂²	#19151594	1	2	yes	—
185	C₂²	#19151615	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	#19151729	1	4	yes	—
192	C₄	#19151955	1	2	yes	—
193	C₄	#19151997	1	2	yes	—
194	C₂	#26018660	1	2	yes	—
195	C₂	#26018686	1	2	yes	—
196	C₂	#26021193	1	1	yes	characteristic
197	C₂	#26021218	1	1	yes	characteristic
198	C₂	#26021234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020369

C₂

yes

maximal, characteristic

#26019998

C₂

yes

maximal, characteristic

#26018093

C₂

yes

maximal

C₄×#125×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

#19151353

—

#19151353

—

C₂×#125×C₈₉

C₄

yes

characteristic

#389×C₈₉

C₄

yes

—

#387×C₈₉

—

#382×C₈₉

C₄

yes

characteristic

C₄×C₄⋊C₄×C₈₉

—

C₄×C₄⋊C₄×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

—

—

—

—

Q₈

yes

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#95×C₈₉

—

#95×C₈₉

—

#95×C₈₉

—

—

—

—

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

M₁₆

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#135

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#421

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#446

yes

—

C₄×C₈₉

#488

yes

—

C₄×C₈₉

#499

yes

—

#9875

—

maximal, Sylow

C₂×C₈₉

#1482

yes

—

C₂×C₈₉

#1488

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

100

D₈₉

yes

characteristic

101

#1831

—

102

#1744

—

103

#1279

—

104

C₈₉

#9875

yes

characteristic, complemented, Sylow

105

C₂×#125

Dic₈₉

yes

characteristic

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

#389

Dic₈₉

yes

—

109

#387

—

110

#382

Dic₈₉

yes

characteristic

111

C₄×C₄⋊C₄

—

112

C₄×C₄⋊C₄

C₂×D₈₉

yes

—

113

C₄³

C₂×D₈₉

yes

characteristic

114

#336

—

178

—

115

#336

—

178

—

116

C₂²×C₈

—

178

—

117

C₂²×C₈

—

178

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

—

122

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

125

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

126

C₂×C₄²

C₂²⋊D₈₉

yes

—

127

C₂×C₄²

C₂²⋊D₈₉

yes

—

128

C₂×C₄²

C₂²×D₈₉

yes

characteristic

129

C₄⋊C₈

—

178

—

130

C₄×C₈

—

131

C₄×C₈

—

178

—

132

#95

—

133

#95

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

138

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

139

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₂×C₈

—

356

—

143

C₂×C₈

—

356

—

144

C₂×C₈

—

356

—

145

C₂×C₈

—

178

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

#13083558

yes

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

Q₈×D₈₉

yes

—

157

C₄²

#13083575

yes

—

158

C₂³

C₂×C₂²⋊Dic₈₉

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

Q₈×Dic₈₉

yes

—

173

C₂×C₄

#15667988

yes

—

174

C₂×C₄

#15668001

yes

characteristic

175

C₂×C₄

#15668010

yes

characteristic

176

C₂×C₄

C₂×#13083558

yes

—

177

C₂×C₄

#15668019

yes

—

178

C₂×C₄

#15668032

yes

—

179

C₈

—

356

—

180

C₈

—

712

—

181

C₂²

#19151505

yes

characteristic

182

C₂²

#19151562

yes

characteristic

183

C₂²

#19151565

yes

characteristic

184

C₂²

#19151594

yes

—

185

C₂²

#19151615

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#19151729

yes

—

192

C₄

#19151955

yes

—

193

C₄

#19151997

yes

—

194

C₂

#26018660

yes

—

195

C₂

#26018686

yes

—

196

C₂

#26021193

yes

characteristic

197

C₂

#26021218

yes

characteristic

198

C₂

#26021234

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#125×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#125	D₈₉
hypocenter	C₈₉	#9875
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×#125×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
104. C₈₉	#9875	semidirect

H ≅ G/N

type

104. C₈₉

#9875

semidirect

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

subgroup	index
#26020369	2
#26019998	2
#26018093	2
C₄×#125×C₈₉	2
#9875	89

subgroup

index

C₄×#125×C₈₉

Subgroups — #36454456

Conjugacy classes of subgroups

Complements

Maximal subgroups