Group #36453514 — subgroups

Subgroups — #36453514

Order 22784.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#26020105	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151344	C₂	1	4	yes	maximal
4	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667923	—	2	1	no	—
8	#19151353	—	2	1	no	—
9	#19151353	—	2	2	no	—
10	#19151344	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₈₉	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
13	#384×C₈₉	—	2	1	no	—
14	#384×C₈₉	—	2	1	no	—
15	#384×C₈₉	C₄	1	4	yes	—
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	D₄	1	4	yes	—
22	#15667923	—	4	2	no	—
23	C₂²×Q₈×C₈₉	—	2	2	no	—
24	C₄⋊Q₈×C₈₉	—	2	4	no	—
25	C₄⋊Q₈×C₈₉	—	2	4	no	—
26	C₄⋊Q₈×C₈₉	—	4	8	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	2	yes	—
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	4	no	—
41	C₂×Q₈×C₈₉	—	4	2	no	—
42	C₂×Q₈×C₈₉	—	4	2	no	—
43	C₂×Q₈×C₈₉	—	4	8	no	—
44	C₂²×C₄×C₈₉	—	2	4	no	—
45	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₈₉	—	2	8	no	—
51	C₄⋊C₄×C₈₉	—	4	8	no	—
52	C₄⋊C₄×C₈₉	—	4	8	no	—
53	C₄⋊C₄×C₈₉	—	4	4	no	—
54	C₄²×C₈₉	—	2	4	no	—
55	C₄²×C₈₉	—	2	4	no	—
56	C₄²×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	SD₁₆	1	2	yes	—
58	C₄²×C₈₉	Q₁₆	1	2	yes	—
59	C₄²×C₈₉	C₂×D₄	1	2	yes	—
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₈₉	—	8	4	no	—
63	Q₈×C₈₉	—	8	4	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	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	4	4	no	—
70	C₂×C₄×C₈₉	—	4	8	no	—
71	C₂×C₄×C₈₉	#101	1	4	yes	—
72	C₂×C₄×C₈₉	#103	1	4	yes	—
73	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₈₉	#121	1	2	yes	—
75	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₈₉	#137	1	2	yes	—
81	#10623494	—	4	4	no	—
82	#10623494	—	8	2	no	—
83	C₂²×C₈₉	#390	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×#101	1	2	yes	—
85	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
86	C₂²×C₈₉	#418	1	2	yes	—
87	C₂²×C₈₉	#419	1	1	yes	characteristic
88	C₄×C₈₉	—	2	4	no	—
89	C₄×C₈₉	—	2	4	no	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	8	4	no	—
92	C₄×C₈₉	#461	1	2	yes	—
93	C₄×C₈₉	#462	1	2	yes	—
94	C₄×C₈₉	#490	1	2	yes	—
95	C₄×C₈₉	#496	1	2	yes	—
96	#9703	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	#1549	1	2	yes	—
98	C₂×C₈₉	#1556	1	2	yes	—
99	C₂×C₈₉	#1961	1	1	yes	characteristic
100	C₂×C₈₉	#1966	1	1	yes	characteristic
101	C₂×C₈₉	#1981	1	1	yes	characteristic
102	#2868	D₈₉	1	1	yes	characteristic
103	#1831	—	89	1	no	—
104	#1767	—	89	1	no	—
105	C₂×#332	—	89	4	no	—
106	C₈₉	#9703	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
109	C₂×C₄⋊C₈	—	89	2	no	—
110	C₂×C₄⋊C₈	—	89	2	no	—
111	C₂×#97	—	178	1	no	—
112	#384	—	2	1	no	—
113	#384	—	2	1	no	—
114	#384	Dic₈₉	1	4	yes	—
115	C₄³	C₂×D₈₉	1	1	yes	characteristic
116	#336	—	178	2	no	—
117	#336	—	178	1	no	—
118	#332	—	178	8	no	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	—	178	2	no	—
121	C₂²×C₈	—	178	2	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
129	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊C₈	—	89	4	no	—
135	C₄⋊C₈	—	178	4	no	—
136	C₄⋊C₈	—	178	2	no	—
137	#97	—	356	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₈	—	356	2	no	—
148	C₂×C₈	—	356	2	no	—
149	C₂×C₈	—	356	2	no	—
150	C₂×C₈	—	356	4	no	—
151	C₂×C₈	—	178	4	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	4	no	—
159	C₄²	#13083561	1	2	yes	—
160	C₄²	#13083564	1	2	yes	—
161	C₄²	D₄×D₈₉	1	2	yes	—
162	C₄²	#13083573	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#15667931	1	4	yes	—
174	C₂×C₄	#15667936	1	4	yes	—
175	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
176	C₂×C₄	#15667983	1	2	yes	—
177	C₂×C₄	#15667999	1	1	yes	characteristic
178	C₂×C₄	#15668008	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
180	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
181	C₂×C₄	#15668038	1	4	yes	—
182	C₂×C₄	#15668043	1	2	yes	—
183	C₈	—	356	4	no	—
184	C₈	—	712	2	no	—
185	C₂²	#19151501	1	1	yes	characteristic
186	C₂²	C₂×#15667931	1	2	yes	—
187	C₂²	C₂×#15667936	1	1	yes	characteristic
188	C₂²	#19151603	1	2	yes	—
189	C₂²	#19151607	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	#19151800	1	2	yes	—
195	C₄	#19151805	1	2	yes	—
196	C₄	#19151962	1	2	yes	—
197	C₄	#19151985	1	2	yes	—
198	C₂	#26018937	1	2	yes	—
199	C₂	#26018969	1	2	yes	—
200	C₂	#26021077	1	1	yes	characteristic
201	C₂	#26021097	1	1	yes	characteristic
202	C₂	#26021176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020369

C₂

yes

maximal, characteristic

#26020105

C₂

yes

maximal, characteristic

C₂×#19151344

C₂

yes

maximal

C₈₉×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄⋊#10623494

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₈₉

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

—

#384×C₈₉

—

#384×C₈₉

—

#384×C₈₉

C₄

yes

—

C₄³×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

—

#15667923

—

C₂²×Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×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

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₈₉

—

Q₈×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

#101

yes

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

C₂×#101

yes

—

C₂²×C₈₉

C₂×#103

yes

characteristic

C₂²×C₈₉

#418

yes

—

C₂²×C₈₉

#419

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#461

yes

—

C₄×C₈₉

#462

yes

—

C₄×C₈₉

#490

yes

—

C₄×C₈₉

#496

yes

—

#9703

—

maximal, Sylow

C₂×C₈₉

#1549

yes

—

C₂×C₈₉

#1556

yes

—

C₂×C₈₉

#1961

yes

characteristic

100

C₂×C₈₉

#1966

yes

characteristic

101

C₂×C₈₉

#1981

yes

characteristic

102

#2868

D₈₉

yes

characteristic

103

#1831

—

104

#1767

—

105

C₂×#332

—

106

C₈₉

#9703

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊Q₈

C₂×D₈₉

yes

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

—

111

C₂×#97

—

178

—

112

#384

—

113

#384

—

114

#384

Dic₈₉

yes

—

115

C₄³

C₂×D₈₉

yes

characteristic

116

#336

—

178

—

117

#336

—

178

—

118

#332

—

178

—

119

C₂²×Q₈

—

120

C₂²×C₈

—

178

—

121

C₂²×C₈

—

178

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

—

129

C₂×C₄²

C₂×Dic₈₉

yes

—

130

C₂×C₄²

C₂²⋊D₈₉

yes

—

131

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

132

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

133

C₂×C₄²

C₂²×D₈₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

178

—

136

C₄⋊C₈

—

178

—

137

#97

—

356

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

143

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

144

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

147

C₂×C₈

—

356

—

148

C₂×C₈

—

356

—

149

C₂×C₈

—

356

—

150

C₂×C₈

—

356

—

151

C₂×C₈

—

178

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#13083561

yes

—

160

C₄²

#13083564

yes

—

161

C₄²

D₄×D₈₉

yes

—

162

C₄²

#13083573

yes

—

163

C₂³

C₂×C₂²⋊Dic₈₉

yes

characteristic

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#15667931

yes

—

174

C₂×C₄

#15667936

yes

—

175

C₂×C₄

D₄×Dic₈₉

yes

—

176

C₂×C₄

#15667983

yes

—

177

C₂×C₄

#15667999

yes

characteristic

178

C₂×C₄

#15668008

yes

characteristic

179

C₂×C₄

C₂×#13083561

yes

characteristic

180

C₂×C₄

C₂×#13083564

yes

characteristic

181

C₂×C₄

#15668038

yes

—

182

C₂×C₄

#15668043

yes

—

183

C₈

—

356

—

184

C₈

—

712

—

185

C₂²

#19151501

yes

characteristic

186

C₂²

C₂×#15667931

yes

—

187

C₂²

C₂×#15667936

yes

characteristic

188

C₂²

#19151603

yes

—

189

C₂²

#19151607

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#19151800

yes

—

195

C₄

#19151805

yes

—

196

C₄

#19151962

yes

—

197

C₄

#19151985

yes

—

198

C₂

#26018937

yes

—

199

C₂

#26018969

yes

—

200

C₂

#26021077

yes

characteristic

201

C₂

#26021097

yes

characteristic

202

C₂

#26021176

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2868	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2868	D₈₉
hypocenter	C₈₉	#9703
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#2868

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
106. C₈₉	#9703	semidirect

H ≅ G/N

type

106. C₈₉

#9703

semidirect

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

subgroup	index
#26020369	2
#26020105	2
C₂×#19151344	2
C₈₉×#2868	2
#9703	89

subgroup

index

C₈₉×#2868

Subgroups — #36453514

Conjugacy classes of subgroups

Complements

Maximal subgroups