Group #36452033 — subgroups

Subgroups — #36452033

Order 22784.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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	#26029354	C₂	1	1	yes	maximal, characteristic
2	#26018990	C₂	1	1	yes	maximal, characteristic
3	#26018937	C₂	1	2	yes	maximal
4	#26018142	C₂	1	2	yes	maximal
5	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic
6	#19151729	—	2	2	no	—
7	Q₈×#10623494	—	2	2	no	—
8	#19151659	—	2	2	no	—
9	#19151651	C₂²	1	2	yes	—
10	#19151622	C₂²	1	2	yes	—
11	#19151344	—	2	2	no	—
12	#19151344	—	2	2	no	—
13	#19151344	—	2	2	no	—
14	#533×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	#491×C₈₉	C₄	1	4	yes	—
17	#487×C₈₉	—	2	2	no	—
18	#419×C₈₉	—	2	2	no	—
19	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₄⋊#10623494	—	2	4	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	D₄	1	2	yes	—
25	C₄⋊#10623494	D₄	1	2	yes	—
26	C₂²⋊#10623494	—	2	2	no	—
27	#15667923	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₄×#10623494	—	2	2	no	—
30	C₂²×Q₈×C₈₉	—	2	1	no	—
31	C₂×M₁₆×C₈₉	—	2	1	no	—
32	C₄⋊Q₈×C₈₉	—	2	2	no	—
33	C₄⋊Q₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
36	#125×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²⋊Q₈×C₈₉	—	2	2	no	—
38	C₄×Q₈×C₈₉	—	2	2	no	—
39	#117×C₈₉	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	#103×C₈₉	—	2	8	no	—
43	#97×C₈₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₉	—	2	1	no	—
45	#13083558	—	8	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×Q₈×C₈₉	—	2	2	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	4	4	no	—
52	C₂×Q₈×C₈₉	—	4	1	no	—
53	C₂²×C₄×C₈₉	—	4	1	no	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	#41	1	2	yes	—
56	M₁₆×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₄⋊C₄×C₈₉	—	2	4	no	—
59	C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₄⋊C₄×C₈₉	—	2	4	no	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
63	C₂²⋊C₄×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	2	no	—
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×D₄	1	2	yes	—
68	C₂³×C₈₉	#123	1	1	yes	characteristic
69	Q₈×C₈₉	—	4	4	no	—
70	Q₈×C₈₉	—	4	2	no	—
71	Q₈×C₈₉	—	4	2	no	—
72	Q₈×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	4	no	—
77	C₂×C₄×C₈₉	—	4	2	no	—
78	C₂×C₄×C₈₉	—	8	2	no	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#121	1	2	yes	—
83	C₈×C₈₉	—	4	2	no	—
84	#10623494	—	4	2	no	—
85	#10623494	—	4	2	no	—
86	#10623494	—	8	2	no	—
87	C₂²×C₈₉	—	2	1	no	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	#386	1	1	yes	characteristic
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	4	4	no	—
93	C₄×C₈₉	—	8	2	no	—
94	C₄×C₈₉	#454	1	2	yes	—
95	C₄×C₈₉	#471	1	2	yes	—
96	C₄×C₈₉	#473	1	2	yes	—
97	#9436	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	—	4	1	no	—
99	C₂×C₈₉	#1888	1	1	yes	characteristic
100	C₂×C₈₉	#1908	1	2	yes	—
101	#3139	D₈₉	1	1	yes	characteristic
102	#2986	—	89	1	no	—
103	#1561	—	89	1	no	—
104	#1549	—	89	2	no	—
105	#1299	—	89	2	no	—
106	C₈₉	#9436	1	1	yes	characteristic, complemented, Sylow
107	#533	C₂×D₈₉	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
109	#491	Dic₈₉	1	4	yes	—
110	#487	—	2	2	no	—
111	#446	—	178	2	no	—
112	C₈×Q₈	—	178	2	no	—
113	#433	—	178	2	no	—
114	#431	—	89	2	no	—
115	#423	—	89	2	no	—
116	#419	—	2	2	no	—
117	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
118	#332	—	178	2	no	—
119	#332	—	178	2	no	—
120	#332	—	178	2	no	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	178	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
127	C₄⋊Q₈	C₂²⋊D₈₉	1	2	yes	—
128	#125	C₂×Dic₈₉	1	2	yes	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	#117	C₂²⋊D₈₉	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊C₈	—	89	2	no	—
135	C₄⋊C₈	—	89	2	no	—
136	C₄⋊C₈	—	178	4	no	—
137	C₄⋊C₈	—	178	2	no	—
138	C₄⋊C₈	—	178	2	no	—
139	#103	—	2	8	no	—
140	C₂²⋊C₈	—	178	2	no	—
141	#97	—	178	2	no	—
142	#97	C₂×Dic₈₉	1	2	yes	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	178	2	no	—
145	C₄×C₈	—	178	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂²×C₄	#13083573	1	2	yes	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	712	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	356	2	no	—
157	C₂×C₈	—	178	2	no	—
158	C₂×C₈	—	178	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	C₂²⋊Dic₈₉	1	4	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
167	C₄²	D₄×D₈₉	1	2	yes	—
168	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂³	#15667993	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	2	no	—
172	Q₈	—	4	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
182	C₂×C₄	#15667980	1	1	yes	characteristic
183	C₂×C₄	#15667984	1	2	yes	—
184	C₈	—	4	2	no	—
185	C₈	—	356	2	no	—
186	C₈	—	356	2	no	—
187	C₈	—	712	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#19151478	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	2	no	—
195	C₄	#19151770	1	2	yes	—
196	C₄	#19151863	1	2	yes	—
197	C₄	#19151871	1	2	yes	—
198	C₂	—	4	1	no	—
199	C₂	#26020656	1	1	yes	characteristic
200	C₂	#26020769	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029354

C₂

yes

maximal, characteristic

#26018990

C₂

yes

maximal, characteristic

#26018937

C₂

yes

maximal

#26018142

C₂

yes

maximal

C₈₉×#3139

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#19151622

C₂²

yes

—

#19151344

—

#19151344

—

#19151344

—

#533×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#491×C₈₉

C₄

yes

—

#487×C₈₉

—

#419×C₈₉

—

C₄×M₁₆×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₈₉

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#125×C₈₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

#97×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#123

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#386

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#454

yes

—

C₄×C₈₉

#471

yes

—

C₄×C₈₉

#473

yes

—

#9436

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1888

yes

characteristic

100

C₂×C₈₉

#1908

yes

—

101

#3139

D₈₉

yes

characteristic

102

#2986

—

103

#1561

—

104

#1549

—

105

#1299

—

106

C₈₉

#9436

yes

characteristic, complemented, Sylow

107

#533

C₂×D₈₉

yes

characteristic

108

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

109

#491

Dic₈₉

yes

—

110

#487

—

111

#446

—

178

—

112

C₈×Q₈

—

178

—

113

#433

—

178

—

114

#431

—

115

#423

—

116

#419

—

117

C₄×M₁₆

C₂×D₈₉

yes

characteristic

118

#332

—

178

—

119

#332

—

178

—

120

#332

—

178

—

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

178

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

C₂×Dic₈₉

yes

—

127

C₄⋊Q₈

C₂²⋊D₈₉

yes

—

128

#125

C₂×Dic₈₉

yes

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

C₂²⋊D₈₉

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₈₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

178

—

137

C₄⋊C₈

—

178

—

138

C₄⋊C₈

—

178

—

139

#103

—

140

C₂²⋊C₈

—

178

—

141

#97

—

178

—

142

#97

C₂×Dic₈₉

yes

—

143

C₄×C₈

—

144

C₄×C₈

—

178

—

145

C₄×C₈

—

178

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

152

C₂²×C₄

#13083573

yes

—

153

M₁₆

—

154

M₁₆

—

712

—

155

C₂×C₈

—

156

C₂×C₈

—

356

—

157

C₂×C₈

—

178

—

158

C₂×C₈

—

178

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

C₂²⋊Dic₈₉

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

C₂²×Dic₈₉

yes

characteristic

167

C₄²

D₄×D₈₉

yes

—

168

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

169

C₂³

#15667993

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

181

C₂×C₄

D₄×Dic₈₉

yes

—

182

C₂×C₄

#15667980

yes

characteristic

183

C₂×C₄

#15667984

yes

—

184

C₈

—

185

C₈

—

356

—

186

C₈

—

356

—

187

C₈

—

712

—

188

C₂²

—

189

C₂²

—

190

C₂²

#19151478

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#19151770

yes

—

196

C₄

#19151863

yes

—

197

C₄

#19151871

yes

—

198

C₂

—

199

C₂

#26020656

yes

characteristic

200

C₂

#26020769

yes

—

	subgroup	G/H
center	C₂²	#19151478
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#3139	C₂
socle	C₂²×C₈₉	#386
prime radical	C₂³×C₈₉	#123
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#3139	D₈₉
hypocenter	C₈₉	#9436
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

subgroup

G/H

center

C₂²

#19151478

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#3139

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

N	H ≅ G/N	type
106. C₈₉	#9436	semidirect

H ≅ G/N

type

106. C₈₉

#9436

semidirect

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

subgroup	index
#26029354	2
#26018990	2
#26018937	2
#26018142	2
C₈₉×#3139	2
#9436	89

subgroup

index

C₈₉×#3139

Subgroups — #36452033

Conjugacy classes of subgroups

Complements

Maximal subgroups