Group #36451971 — subgroups

Subgroups — #36451971

Order 22784.

422 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26023433	C₂	1	1	yes	maximal, characteristic
2	#26018978	C₂	1	1	yes	maximal, characteristic
3	#26018803	C₂	1	2	yes	maximal
4	#26018117	C₂	1	2	yes	maximal
5	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic
6	#19151655	C₂²	1	2	yes	—
7	#19151505	—	2	2	no	—
8	#19151484	—	2	4	no	—
9	#19151466	C₂²	1	2	yes	—
10	#19151347	—	2	2	no	—
11	#19151335	—	2	4	no	—
12	#533×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	#491×C₈₉	—	2	2	no	—
15	#487×C₈₉	C₄	1	4	yes	complemented
16	#419×C₈₉	—	2	2	no	—
17	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄×Dic₈₉	—	2	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₂²⋊#10623494	—	2	2	no	—
22	#15667923	—	2	2	no	—
23	#15667919	—	2	4	no	—
24	#15667919	—	2	2	no	—
25	#15667919	—	2	2	no	—
26	#15667919	D₄	1	4	yes	—
27	C₂²×Q₈×C₈₉	D₄	1	2	yes	—
28	C₂×M₁₆×C₈₉	—	2	1	no	—
29	C₄⋊Q₈×C₈₉	—	2	2	no	—
30	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
33	#125×C₈₉	—	2	1	no	—
34	C₂²⋊Q₈×C₈₉	—	2	2	no	—
35	C₄×Q₈×C₈₉	—	2	2	no	—
36	#117×C₈₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	#103×C₈₉	—	2	8	no	—
40	#97×C₈₉	—	2	1	no	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₈₉	—	2	4	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	—	8	2	no	—
45	C₄×Dic₈₉	—	8	2	no	—
46	C₂×Q₈×C₈₉	—	2	2	no	—
47	C₂×Q₈×C₈₉	—	2	2	no	—
48	C₂×Q₈×C₈₉	—	4	2	no	—
49	C₂×Q₈×C₈₉	—	4	2	no	—
50	C₂×Q₈×C₈₉	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	M₁₆×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₄⋊C₄×C₈₉	—	2	4	no	—
57	C₄⋊C₄×C₈₉	—	2	4	no	—
58	C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₂²⋊C₄×C₈₉	—	4	1	no	—
62	C₄²×C₈₉	—	4	1	no	—
63	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	2	yes	—
66	C₂³×C₈₉	#120	1	1	yes	characteristic
67	Q₈×C₈₉	—	4	4	no	—
68	Q₈×C₈₉	—	8	2	no	—
69	Q₈×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	#121	1	2	yes	—
79	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
80	C₂×Dic₈₉	—	4	8	no	—
81	C₂×Dic₈₉	—	4	4	no	—
82	C₈×C₈₉	—	4	2	no	—
83	#10623494	—	8	2	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	#386	1	1	yes	characteristic
86	C₂²×C₈₉	#455	1	2	yes	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	4	4	no	—
92	C₄×C₈₉	—	8	2	no	—
93	Dic₈₉	—	8	4	no	—
94	#9426	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	—	2	2	no	—
96	C₂×C₈₉	#1885	1	1	yes	characteristic
97	C₂×C₈₉	#1905	1	2	yes	—
98	#3139	D₈₉	1	1	yes	characteristic
99	#2381	—	89	1	no	—
100	#1558	—	89	1	no	—
101	#1515	—	89	2	no	—
102	#1288	—	89	2	no	—
103	C₈₉	#9426	1	1	yes	characteristic, complemented, Sylow
104	#533	C₂×D₈₉	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
106	#491	—	2	2	no	—
107	#487	Dic₈₉	1	4	yes	complemented
108	#432	—	89	2	no	—
109	#419	—	2	2	no	—
110	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
111	#391	—	178	2	no	—
112	#387	—	178	4	no	—
113	#383	—	89	2	no	—
114	#333	—	178	2	no	—
115	#328	—	178	4	no	—
116	C₂²×Q₈	C₂²⋊D₈₉	1	2	yes	—
117	C₂×M₁₆	—	2	1	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
120	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
121	C₄⋊Q₈	C₂²⋊D₈₉	1	2	yes	—
122	#125	—	2	1	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	#117	—	2	1	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	178	2	no	—
128	C₂×C₄²	—	178	2	no	—
129	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
130	C₄⋊C₈	—	178	2	no	—
131	#103	—	2	8	no	—
132	C₂²⋊C₈	—	178	2	no	—
133	#97	—	2	1	no	—
134	#97	—	178	2	no	—
135	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
136	#95	—	89	4	no	—
137	#95	—	178	4	no	—
138	#95	—	178	2	no	—
139	#95	—	178	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	C₂²⋊Dic₈₉	1	4	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	178	4	no	—
147	C₂²×C₄	D₄×D₈₉	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	356	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	712	2	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	712	2	no	—
161	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₄²	#13083573	1	2	yes	—
164	C₂³	#15667980	1	1	yes	characteristic
165	Q₈	—	4	4	no	—
166	Q₈	—	8	2	no	—
167	Q₈	—	8	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	356	8	no	—
175	C₂×C₄	—	356	4	no	—
176	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
177	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
178	C₂×C₄	#15667984	1	2	yes	—
179	C₂×C₄	#15667993	1	1	yes	characteristic
180	C₈	—	4	2	no	—
181	C₈	—	712	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	#19151478	1	1	yes	characteristic
184	C₂²	#19151775	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	—	712	4	no	—
192	C₂	—	2	2	no	—
193	C₂	#26020641	1	1	yes	characteristic
194	C₂	#26020748	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26023433

C₂

yes

maximal, characteristic

#26018978

C₂

yes

maximal, characteristic

#26018803

C₂

yes

maximal

#26018117

C₂

yes

maximal

C₈₉×#3139

C₂

yes

maximal, characteristic

#19151655

C₂²

yes

—

#19151505

—

#19151484

—

#19151466

C₂²

yes

—

#19151347

—

#19151335

—

#533×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#491×C₈₉

—

#487×C₈₉

C₄

yes

complemented

#419×C₈₉

—

C₄×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄×Dic₈₉

—

—

—

—

—

—

—

D₄

yes

—

C₂²×Q₈×C₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#120

yes

characteristic

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

#121

yes

—

C₂×C₄×C₈₉

#123

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#386

yes

characteristic

C₂²×C₈₉

#455

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

#9426

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1885

yes

characteristic

C₂×C₈₉

#1905

yes

—

#3139

D₈₉

yes

characteristic

#2381

—

100

#1558

—

101

#1515

—

102

#1288

—

103

C₈₉

#9426

yes

characteristic, complemented, Sylow

104

#533

C₂×D₈₉

yes

characteristic

105

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

106

#491

—

107

#487

Dic₈₉

yes

complemented

108

#432

—

109

#419

—

110

C₄×M₁₆

C₂×D₈₉

yes

characteristic

111

#391

—

178

—

112

#387

—

178

—

113

#383

—

114

#333

—

178

—

115

#328

—

178

—

116

C₂²×Q₈

C₂²⋊D₈₉

yes

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

C₂×Dic₈₉

yes

—

120

C₄⋊Q₈

C₂×Dic₈₉

yes

—

121

C₄⋊Q₈

C₂²⋊D₈₉

yes

—

122

#125

—

123

C₂²⋊Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

178

—

128

C₂×C₄²

—

178

—

129

C₂×C₄²

C₂²×D₈₉

yes

characteristic

130

C₄⋊C₈

—

178

—

131

#103

—

132

C₂²⋊C₈

—

178

—

133

#97

—

134

#97

—

178

—

135

C₄×C₈

C₂×Dic₈₉

yes

—

136

#95

—

137

#95

—

178

—

138

#95

—

178

—

139

#95

—

178

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

C₂²⋊Dic₈₉

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

—

178

—

147

C₂²×C₄

D₄×D₈₉

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

356

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

712

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

712

—

161

C₄²

C₂²×Dic₈₉

yes

characteristic

162

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₄²

#13083573

yes

—

164

C₂³

#15667980

yes

characteristic

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

356

—

175

C₂×C₄

—

356

—

176

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

177

C₂×C₄

D₄×Dic₈₉

yes

—

178

C₂×C₄

#15667984

yes

—

179

C₂×C₄

#15667993

yes

characteristic

180

C₈

—

181

C₈

—

712

—

182

C₂²

—

183

C₂²

#19151478

yes

characteristic

184

C₂²

#19151775

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

712

—

192

C₂

—

193

C₂

#26020641

yes

characteristic

194

C₂

#26020748

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	#120
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#3139	D₈₉
hypocenter	C₈₉	#9426
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
15. #487×C₈₉	C₄	semidirect
103. C₈₉	#9426	semidirect
107. #487	Dic₈₉	semidirect

H ≅ G/N

type

15. #487×C₈₉

C₄

semidirect

103. C₈₉

#9426

semidirect

107. #487

Dic₈₉

semidirect

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

subgroup	index
#26023433	2
#26018978	2
#26018803	2
#26018117	2
C₈₉×#3139	2
#9426	89

subgroup

index

C₈₉×#3139

Subgroups — #36451971

Conjugacy classes of subgroups

Complements

Maximal subgroups