Group #36447384 — subgroups

Subgroups — #36447384

Order 22784.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
2	#26019013	C₂	1	2	yes	maximal
3	#26018596	C₂	1	1	yes	maximal, characteristic
4	#26018121	C₂	1	2	yes	maximal
5	C₈₉×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10623494	—	2	2	no	—
7	Q₈×#10623494	—	2	2	no	—
8	#19151655	C₂²	1	2	yes	—
9	C₂×C₄⋊#10623494	C₄	1	2	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151350	—	2	2	no	—
14	#19151348	—	2	2	no	—
15	#19151324	—	2	2	no	—
16	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	#432×C₈₉	C₂²	1	2	yes	—
18	#324×C₈₉	—	2	2	no	—
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10623494	C₂×C₄	1	2	yes	—
21	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₂²⋊#10623494	—	2	2	no	—
26	#15667923	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₄×#10623494	D₄	1	2	yes	—
30	C₄×#10623494	D₄	1	2	yes	—
31	C₂²×Q₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×Q₈×C₈₉	—	2	2	no	—
33	C₄×Q₈×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₂²⋊C₈×C₈₉	—	2	2	no	—
40	#97×C₈₉	—	2	2	no	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₂×#10623494	—	4	1	no	—
44	C₂×#10623494	—	4	2	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
48	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₈₉	—	2	4	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	2	4	no	—
58	C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₄²×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	—	2	2	no	—
61	C₄²×C₈₉	C₂×D₄	1	2	yes	—
62	C₄²×C₈₉	#41	1	2	yes	—
63	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₈₉	—	2	8	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	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	4	1	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	#108	1	4	yes	—
74	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#121	1	2	yes	—
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₈₉	—	8	2	no	—
79	#10623494	—	2	1	no	—
80	#10623494	—	2	2	no	—
81	#10623494	—	4	2	no	—
82	#10623494	—	8	2	no	—
83	#10623494	#101	1	2	yes	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	#380	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×#108	1	2	yes	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	—	2	4	no	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	4	2	no	—
93	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
94	C₄×C₈₉	#413	1	1	yes	characteristic
95	#8623	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	#1801	1	1	yes	characteristic
98	C₂×C₈₉	#1810	1	1	yes	characteristic
99	C₂×C₈₉	#1812	1	1	yes	characteristic
100	C₂×C₈×Q₈	—	89	1	no	—
101	#1568	—	89	2	no	—
102	#1467	—	89	1	no	—
103	#1467	D₈₉	1	1	yes	characteristic
104	#1290	—	89	2	no	—
105	C₈₉	#8623	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
107	C₈×Q₈	—	178	2	no	—
108	C₈×Q₈	—	178	2	no	—
109	#432	—	89	2	no	—
110	#432	C₂×D₈₉	1	2	yes	—
111	C₂×C₄⋊C₈	—	89	2	no	—
112	C₂×C₄⋊C₈	—	89	1	no	—
113	C₂×C₄×C₈	—	89	1	no	—
114	C₂×C₄×C₈	—	89	2	no	—
115	#335	—	178	2	no	—
116	#334	—	178	2	no	—
117	#324	—	2	2	no	—
118	#324	—	178	2	no	—
119	C₂²×Q₈	Dic₁₇₈	1	1	yes	characteristic
120	C₂²×C₈	—	89	1	no	—
121	C₂²×C₈	—	89	1	no	—
122	C₂²×C₈	—	89	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₇₈	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
127	C₂×C₄²	C₄×D₈₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	178	4	no	—
131	C₄⋊C₈	—	178	2	no	—
132	C₄⋊C₈	—	178	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	178	2	no	—
135	#97	—	2	2	no	—
136	#97	—	178	2	no	—
137	C₄×C₈	—	89	2	no	—
138	C₄×C₈	—	89	2	no	—
139	C₄×C₈	—	178	2	no	—
140	C₄×C₈	—	178	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	#13083553	1	2	yes	—
144	C₂²×C₄	#13083553	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	89	1	no	—
150	C₂×C₈	—	89	1	no	—
151	C₂×C₈	—	89	2	no	—
152	C₂×C₈	—	356	2	no	—
153	C₂×C₈	—	356	1	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	—	178	2	no	—
156	C₂×C₈	—	178	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₈₉	1	2	yes	—
162	C₄²	#13083573	1	2	yes	—
163	C₂³	C₂×#13083553	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#15667947	1	4	yes	—
174	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₄	#15667968	1	1	yes	characteristic
176	C₂×C₄	#15667984	1	2	yes	—
177	C₂×C₄	#15667989	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	89	2	no	—
180	C₈	—	356	2	no	—
181	C₈	—	178	1	no	—
182	C₈	—	178	2	no	—
183	C₈	—	712	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#19151456	1	1	yes	characteristic
187	C₂²	C₂×#15667947	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₈₉	1	1	yes	characteristic
194	C₄	#19151581	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#26020246	1	1	yes	characteristic
197	C₂	#26020288	1	1	yes	characteristic
198	C₂	#26020296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×#10623494

C₂

yes

maximal, characteristic

#26019013

C₂

yes

maximal

#26018596

C₂

yes

maximal, characteristic

#26018121

C₂

yes

maximal

C₈₉×#1467

C₂

yes

maximal, characteristic

Q₈×#10623494

—

Q₈×#10623494

—

#19151655

C₂²

yes

—

C₂×C₄⋊#10623494

C₄

yes

—

C₂×C₄⋊#10623494

C₂²

yes

characteristic

C₂×C₄×#10623494

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₈₉

C₂²

yes

characteristic

#432×C₈₉

C₂²

yes

—

#324×C₈₉

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

C₂²×#10623494

C₂×C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₂²×Q₈×C₈₉

Q₈

yes

characteristic

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

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

—

C₂×C₄×C₈₉

#108

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₈×C₈₉

—

—

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#380

yes

characteristic

C₂²×C₈₉

C₂×#108

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

100

C₂×C₈×Q₈

—

101

#1568

—

102

#1467

—

103

#1467

D₈₉

yes

characteristic

104

#1290

—

105

C₈₉

#8623

yes

characteristic, complemented, Sylow

106

C₂×C₄×Q₈

C₂×D₈₉

yes

characteristic

107

C₈×Q₈

—

178

—

108

C₈×Q₈

—

178

—

109

#432

—

110

#432

C₂×D₈₉

yes

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

#335

—

178

—

116

#334

—

178

—

117

#324

—

118

#324

—

178

—

119

C₂²×Q₈

Dic₁₇₈

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

Dic₁₇₈

yes

characteristic

126

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

127

C₂×C₄²

C₄×D₈₉

yes

—

128

C₂×C₄²

C₂²×D₈₉

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

178

—

131

C₄⋊C₈

—

178

—

132

C₄⋊C₈

—

178

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

178

—

135

#97

—

136

#97

—

178

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

178

—

140

C₄×C₈

—

178

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

#13083553

yes

—

144

C₂²×C₄

#13083553

yes

—

145

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

147

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

356

—

153

C₂×C₈

—

356

—

154

C₂×C₈

—

356

—

155

C₂×C₈

—

178

—

156

C₂×C₈

—

178

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₈₉

yes

—

162

C₄²

#13083573

yes

—

163

C₂³

C₂×#13083553

yes

characteristic

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#15667947

yes

—

174

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

175

C₂×C₄

#15667968

yes

characteristic

176

C₂×C₄

#15667984

yes

—

177

C₂×C₄

#15667989

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

356

—

181

C₈

—

178

—

182

C₈

—

178

—

183

C₈

—

712

—

184

C₂²

—

185

C₂²

—

186

C₂²

#19151456

yes

characteristic

187

C₂²

C₂×#15667947

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₈₉

yes

characteristic

194

C₄

#19151581

yes

characteristic

195

C₂

—

196

C₂

#26020246

yes

characteristic

197

C₂

#26020288

yes

characteristic

198

C₂

#26020296

yes

characteristic

	subgroup	G/H
center	C₂²	#19151456
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1467	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1467	D₈₉
hypocenter	C₈₉	#8623
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#19151456

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1467

C₂

socle

C₂²×C₈₉

#380

prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
105. C₈₉	#8623	semidirect

H ≅ G/N

type

105. C₈₉

#8623

semidirect

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

subgroup	index
C₂×Q₈×#10623494	2
#26019013	2
#26018596	2
#26018121	2
C₈₉×#1467	2
#8623	89

subgroup

index

C₈₉×#1467

Subgroups — #36447384

Conjugacy classes of subgroups

Complements

Maximal subgroups