Group #36453920 — subgroups

Subgroups — #36453920

Order 22784.

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

normal subgroups only

This group has 506 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	#26020390	C₂	1	1	yes	maximal, characteristic, complemented
2	#26018924	C₂	1	2	yes	maximal, complemented
3	#26018920	C₂	1	2	yes	maximal
4	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#492×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	#19151622	C₂²	1	2	yes	—
8	#19151363	—	2	2	no	—
9	#19151344	—	2	4	no	—
10	#19151342	—	2	4	no	—
11	#19151320	—	2	2	no	—
12	C₂²×SD₁₆×C₈₉	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
15	#492×C₈₉	—	2	2	no	—
16	#492×C₈₉	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083558	—	2	2	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	D₄	1	4	yes	—
22	C₂²⋊#10623494	—	2	2	no	—
23	C₄×#10623494	—	2	2	no	—
24	C₂²×Q₈×C₈₉	—	2	1	no	—
25	C₂²×D₄×C₈₉	—	2	1	no	—
26	C₂²×D₄×C₈₉	—	2	2	no	—
27	C₂×SD₁₆×C₈₉	—	2	8	no	—
28	C₂×SD₁₆×C₈₉	—	4	8	no	—
29	C₂²×C₈×C₈₉	D₄	1	2	yes	—
30	C₄⋊Q₈×C₈₉	—	2	2	no	—
31	C₄⋊Q₈×C₈₉	—	2	1	no	—
32	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
33	#127×C₈₉	—	2	1	no	—
34	#127×C₈₉	—	2	2	no	—
35	#127×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	#13083558	—	4	4	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂⁴×C₈₉	—	4	1	no	—
44	C₂×Q₈×C₈₉	—	2	2	no	—
45	C₂×Q₈×C₈₉	—	4	1	no	—
46	C₂×Q₈×C₈₉	—	4	4	no	—
47	C₂×D₄×C₈₉	—	2	2	no	—
48	C₂×D₄×C₈₉	—	2	4	no	—
49	C₂×D₄×C₈₉	—	4	2	no	—
50	C₂×D₄×C₈₉	—	4	1	no	—
51	C₂×D₄×C₈₉	—	4	4	no	—
52	C₂×D₄×C₈₉	—	8	4	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	SD₁₆×C₈₉	—	4	16	no	—
57	C₂×C₈×C₈₉	—	2	4	no	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
59	C₄⋊C₄×C₈₉	—	2	4	no	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₂³×C₈₉	—	4	2	no	—
65	C₂³×C₈₉	—	8	4	no	—
66	C₂³×C₈₉	—	8	2	no	—
67	C₂³×C₈₉	#124	1	1	yes	characteristic
68	Q₈×C₈₉	—	4	4	no	—
69	Q₈×C₈₉	—	8	2	no	—
70	D₄×C₈₉	—	4	4	no	—
71	D₄×C₈₉	—	8	4	no	—
72	D₄×C₈₉	—	8	4	no	—
73	D₄×C₈₉	—	8	2	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	—	8	2	no	—
77	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₈₉	#121	1	2	yes	—
80	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
81	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
82	C₈×C₈₉	—	2	4	no	—
83	#10623494	—	4	4	no	—
84	#10623494	—	8	2	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	—	8	2	no	—
87	C₂²×C₈₉	—	8	4	no	—
88	C₂²×C₈₉	—	16	4	no	—
89	C₂²×C₈₉	#390	1	1	yes	characteristic
90	C₂²×C₈₉	#490	1	2	yes	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	8	2	no	—
93	C₄×C₈₉	#418	1	2	yes	—
94	C₄×C₈₉	#471	1	4	yes	—
95	C₄×C₈₉	#496	1	2	yes	—
96	#9777	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	2	no	—
98	C₂×C₈₉	—	16	2	no	—
99	C₂×C₈₉	#1961	1	1	yes	characteristic
100	C₂×C₈₉	#1977	1	2	yes	—
101	C₂×#492	D₈₉	1	1	yes	characteristic
102	#1837	—	89	1	no	—
103	#1546	—	89	2	no	—
104	#1545	—	89	2	no	—
105	#1452	—	89	1	no	—
106	C₈₉	#9777	1	1	yes	characteristic, complemented, Sylow
107	C₂²×SD₁₆	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂×#127	C₂×D₈₉	1	1	yes	characteristic
110	#492	—	2	2	no	—
111	#492	Dic₈₉	1	4	yes	—
112	#431	—	89	2	no	—
113	#423	—	89	2	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	#341	—	178	2	no	—
116	#332	—	178	4	no	—
117	#331	—	178	4	no	—
118	#322	—	178	2	no	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂²×D₄	—	2	1	no	—
121	C₂²×D₄	—	2	2	no	—
122	C₂×SD₁₆	—	2	8	no	—
123	C₂×SD₁₆	—	4	8	no	—
124	C₂×M₁₆	—	178	2	no	—
125	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
129	#127	—	2	2	no	—
130	#127	—	2	1	no	—
131	#127	C₂×Dic₈₉	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊C₈	—	89	4	no	—
135	C₄⋊C₈	—	178	2	no	—
136	C₄⋊C₈	—	178	2	no	—
137	C₂²⋊C₈	—	178	2	no	—
138	C₄×C₈	—	178	2	no	—
139	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
140	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	8	4	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₂²×C₄	#13083573	1	2	yes	—
154	SD₁₆	—	4	16	no	—
155	M₁₆	—	356	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	356	2	no	—
158	C₂×C₈	—	178	4	no	—
159	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
163	C₄²	D₄×D₈₉	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂³	—	4	2	no	—
166	C₂³	—	8	4	no	—
167	C₂³	—	8	2	no	—
168	C₂³	#15667999	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	8	2	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	4	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
180	C₂×C₄	#15667983	1	2	yes	—
181	C₂×C₄	#15668008	1	1	yes	characteristic
182	C₂×C₄	#15668038	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	356	4	no	—
185	C₈	—	712	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	8	4	no	—
188	C₂²	—	8	2	no	—
189	C₂²	—	16	4	no	—
190	C₂²	#19151501	1	1	yes	characteristic
191	C₂²	#19151962	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	8	2	no	—
194	C₄	#19151603	1	2	yes	—
195	C₄	#19151858	1	4	yes	—
196	C₄	#19151985	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	16	2	no	—
199	C₂	#26021077	1	1	yes	characteristic
200	C₂	#26021157	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020390

C₂

yes

maximal, characteristic, complemented

#26018924

C₂

yes

maximal, complemented

#26018920

C₂

yes

maximal

#26018544

C₂

yes

maximal, characteristic, complemented

C₂×#492×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

—

—

—

—

—

C₂²×SD₁₆×C₈₉

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₂×#127×C₈₉

C₂²

yes

characteristic

#492×C₈₉

—

#492×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

—

—

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂²×Q₈×C₈₉

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

—

C₂×SD₁₆×C₈₉

—

C₂×SD₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

#127×C₈₉

—

#127×C₈₉

—

#127×C₈₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

—

C₂⁴×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

SD₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#124

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×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

#127

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#490

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#418

yes

—

C₄×C₈₉

#471

yes

—

C₄×C₈₉

#496

yes

—

#9777

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1961

yes

characteristic

100

C₂×C₈₉

#1977

yes

—

101

C₂×#492

D₈₉

yes

characteristic

102

#1837

—

103

#1546

—

104

#1545

—

105

#1452

—

106

C₈₉

#9777

yes

characteristic, complemented, Sylow

107

C₂²×SD₁₆

—

108

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

109

C₂×#127

C₂×D₈₉

yes

characteristic

110

#492

—

111

#492

Dic₈₉

yes

—

112

#431

—

113

#423

—

114

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

115

#341

—

178

—

116

#332

—

178

—

117

#331

—

178

—

118

#322

—

178

—

119

C₂²×Q₈

—

120

C₂²×D₄

—

121

C₂²×D₄

—

122

C₂×SD₁₆

—

123

C₂×SD₁₆

—

124

C₂×M₁₆

—

178

—

125

C₂²×C₈

C₂²⋊D₈₉

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×Dic₈₉

yes

—

129

#127

—

130

#127

—

131

#127

C₂×Dic₈₉

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₈₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

178

—

136

C₄⋊C₈

—

178

—

137

C₂²⋊C₈

—

178

—

138

C₄×C₈

—

178

—

139

C₄×C₈

C₂×Dic₈₉

yes

—

140

C₄×C₈

C₂²⋊D₈₉

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

153

C₂²×C₄

#13083573

yes

—

154

SD₁₆

—

155

M₁₆

—

356

—

156

C₂×C₈

—

157

C₂×C₈

—

356

—

158

C₂×C₈

—

178

—

159

C₂×C₈

C₂²⋊Dic₈₉

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

C₂²×Dic₈₉

yes

characteristic

163

C₄²

D₄×D₈₉

yes

—

164

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#15667999

yes

characteristic

169

Q₈

—

170

Q₈

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

179

C₂×C₄

D₄×Dic₈₉

yes

—

180

C₂×C₄

#15667983

yes

—

181

C₂×C₄

#15668008

yes

characteristic

182

C₂×C₄

#15668038

yes

—

183

C₈

—

184

C₈

—

356

—

185

C₈

—

712

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#19151501

yes

characteristic

191

C₂²

#19151962

yes

—

192

C₄

—

193

C₄

—

194

C₄

#19151603

yes

—

195

C₄

#19151858

yes

—

196

C₄

#19151985

yes

—

197

C₂

—

198

C₂

—

199

C₂

#26021077

yes

characteristic

200

C₂

#26021157

yes

—

	subgroup	G/H
center	C₂²	#19151501
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#492×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂×#127×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#492	D₈₉
hypocenter	C₈₉	#9777
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

subgroup

G/H

center

C₂²

#19151501

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#492×C₈₉

C₂

socle

C₂²×C₈₉

#390

prime radical

C₂×#127×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

N	H ≅ G/N	type
1. #26020390	C₂	semidirect
2. #26018924	C₂	semidirect
4. #26018544	C₂	semidirect
106. C₈₉	#9777	semidirect

H ≅ G/N

type

1. #26020390

C₂

semidirect

2. #26018924

C₂

semidirect

4. #26018544

C₂

semidirect

106. C₈₉

#9777

semidirect

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

subgroup	index
#26020390	2
#26018924	2
#26018920	2
#26018544	2
C₂×#492×C₈₉	2
#9777	89

subgroup

index

C₂×#492×C₈₉

Subgroups — #36453920

Conjugacy classes of subgroups

Complements

Maximal subgroups