Group #11990257 — subgroups

Subgroups — #11990257

Order 1280.

516 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 183 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	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64766	C₂	1	2	yes	maximal
3	#10600184	C₂	1	2	yes	maximal
4	#10598125	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
7	C₅×#391	C₄	1	2	yes	—
8	C₅×#389	—	2	2	no	—
9	C₅×#384	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₄⋊C₄	—	2	2	no	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#168	C₂²	1	4	yes	—
13	C₂×#3935	—	2	2	no	—
14	#64766	C₄	1	8	yes	complemented
15	#64744	—	2	2	no	—
16	#8232	—	5	1	no	maximal, Sylow
17	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
18	C₅×C₄⋊Q₈	—	2	2	no	—
19	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
20	C₄×C₅×Q₈	—	4	4	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂²×#168	—	2	4	no	—
32	C₅×#95	—	2	4	no	—
33	C₄⋊#168	—	2	4	no	—
34	C₄⋊#168	C₂×C₄	1	8	yes	—
35	#3935	—	4	4	no	—
36	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
37	#1767	—	5	2	no	—
38	C₂×#333	—	5	2	no	—
39	#1296	—	5	2	no	—
40	C₂×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₈	—	4	2	no	—
42	C₂×C₅×Q₈	—	4	2	no	—
43	C₂×C₅×Q₈	C₄²	1	4	yes	—
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	—	2	2	no	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₅×C₄⋊C₄	—	2	8	no	—
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₅×C₄⋊C₄	—	4	4	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	—	2	4	no	—
60	C₄²×C₅	—	4	1	no	—
61	C₄²×C₅	—	4	4	no	—
62	C₄²×C₅	C₂²×C₄	1	2	yes	—
63	C₄²×C₅	#41	1	2	yes	—
64	C₂×#168	—	2	8	no	—
65	C₂×#168	—	4	4	no	—
66	C₂×#168	—	4	4	no	—
67	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
68	C₂×C₄×Q₈	Dic₅	1	2	yes	—
69	C₂×C₄⋊C₈	—	5	4	no	—
70	C₂×#97	—	10	2	no	—
71	#391	Dic₅	1	2	yes	—
72	#389	—	2	2	no	—
73	#384	C₂×D₅	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	2	2	no	—
75	C₄³	C₂×D₅	1	1	yes	characteristic
76	#336	—	10	2	no	—
77	#333	—	5	8	no	—
78	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₅×Q₈	—	4	4	no	—
80	C₅×Q₈	—	4	4	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	—	4	2	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	—	4	4	no	—
94	C₂×C₄×C₅	#100	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅	#117	1	1	yes	characteristic
97	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₅	#137	1	2	yes	—
100	C₂×C₄×C₅	#137	1	4	yes	—
101	#168	—	4	8	no	—
102	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
103	C₂²×C₈	—	10	4	no	—
104	C₄⋊Q₈	—	2	2	no	—
105	C₄⋊Q₈	C₄×D₅	1	4	yes	—
106	C₄×Q₈	—	4	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
114	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
115	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
116	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
117	C₄⋊C₈	—	5	8	no	—
118	C₄⋊C₈	—	10	4	no	—
119	#97	—	20	4	no	—
120	#95	—	2	4	no	—
121	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₅	C₂×#100	1	1	yes	characteristic
123	C₂²×C₅	#413	1	1	yes	characteristic
124	C₂²×C₅	#417	1	2	yes	—
125	C₂²×C₅	#419	1	2	yes	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	2	1	no	—
128	C₄×C₅	—	2	1	no	—
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	—	4	1	no	—
131	C₄×C₅	—	4	4	no	—
132	C₄×C₅	—	8	2	no	—
133	C₄×C₅	#440	1	4	yes	—
134	C₄×C₅	#441	1	4	yes	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	C₄×Dic₅	1	4	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
142	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
146	C₂×C₈	—	10	8	no	—
147	C₂×C₈	—	20	4	no	—
148	C₂×C₈	—	20	4	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×C₄×D₅	1	2	yes	—
161	C₄²	#709	1	2	yes	—
162	C₂×C₅	#1558	1	1	yes	characteristic
163	C₂×C₅	#1559	1	2	yes	—
164	C₂×C₅	#1560	1	1	yes	characteristic
165	C₂×C₅	C₄×#100	1	1	yes	characteristic
166	C₂×C₅	#1763	1	2	yes	—
167	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
168	Q₈	—	4	4	no	—
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	#3965	1	2	yes	—
184	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
185	C₂×C₄	#4072	1	1	yes	characteristic
186	C₂×C₄	C₄×C₂²⋊D₅	1	4	yes	—
187	C₂×C₄	#4088	1	4	yes	—
188	C₂×C₄	#4095	1	2	yes	—
189	C₂×C₄	#4097	1	2	yes	—
190	C₈	—	20	8	no	—
191	C₅	#8232	1	1	yes	characteristic, complemented, Sylow
192	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
193	C₂²	C₂×#3965	1	1	yes	characteristic
194	C₂²	#65514	1	2	yes	—
195	C₂²	#65521	1	2	yes	—
196	C₂²	#65526	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	8	2	no	—
204	C₄	#65315	1	4	yes	—
205	C₄	#65319	1	4	yes	—
206	C₂	C₄×#3965	1	1	yes	characteristic
207	C₂	#10602424	1	2	yes	—
208	C₂	#10602570	1	1	yes	characteristic
209	C₂	#10602603	1	1	yes	characteristic
210	C₂	#10602626	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#64766

C₂

yes

maximal

#10600184

C₂

yes

maximal

#10598125

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄×C₅×Q₈

C₄

yes

—

C₅×#391

C₄

yes

—

C₅×#389

—

C₅×#384

C₂²

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

C₂²

yes

—

C₂×#3935

—

#64766

C₄

yes

—

—

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₅×#95

—

C₄⋊#168

—

C₄⋊#168

C₂×C₄

yes

—

#3935

—

C₄×C₄⋊Q₈

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₄²

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

—

C₄²×C₅

#41

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×C₄⋊Q₈

C₂×D₅

yes

characteristic

C₂×C₄×Q₈

Dic₅

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

#391

Dic₅

yes

—

#389

—

#384

C₂×D₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂×D₅

yes

characteristic

#336

—

#333

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#100

yes

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

100

C₂×C₄×C₅

#137

yes

—

101

#168

—

102

C₂²×Q₈

C₂×Dic₅

yes

—

103

C₂²×C₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

C₄×D₅

yes

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×Dic₅

yes

characteristic

113

C₂×C₄²

C₂×Dic₅

yes

characteristic

114

C₂×C₄²

C₂²⋊D₅

yes

—

115

C₂×C₄²

C₂²⋊D₅

yes

—

116

C₂×C₄²

C₂²×D₅

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#97

—

120

#95

—

121

C₂²×C₅

C₄×C₂²⋊C₄

yes

characteristic

122

C₂²×C₅

C₂×#100

yes

characteristic

123

C₂²×C₅

#413

yes

characteristic

124

C₂²×C₅

#417

yes

—

125

C₂²×C₅

#419

yes

—

126

C₄×C₅

—

127

C₄×C₅

—

128

C₄×C₅

—

129

C₄×C₅

—

130

C₄×C₅

—

131

C₄×C₅

—

132

C₄×C₅

—

133

C₄×C₅

#440

yes

—

134

C₄×C₅

#441

yes

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

C₄×Dic₅

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₅

yes

—

142

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

143

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

144

C₂²×C₄

C₂²×Dic₅

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×C₄×D₅

yes

—

161

C₄²

#709

yes

—

162

C₂×C₅

#1558

yes

characteristic

163

C₂×C₅

#1559

yes

—

164

C₂×C₅

#1560

yes

characteristic

165

C₂×C₅

C₄×#100

yes

characteristic

166

C₂×C₅

#1763

yes

—

167

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#3965

yes

—

184

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

185

C₂×C₄

#4072

yes

characteristic

186

C₂×C₄

C₄×C₂²⋊D₅

yes

—

187

C₂×C₄

#4088

yes

—

188

C₂×C₄

#4095

yes

—

189

C₂×C₄

#4097

yes

—

190

C₈

—

191

C₅

#8232

yes

characteristic, complemented, Sylow

192

C₂²

C₄×C₂²⋊Dic₅

yes

characteristic

193

C₂²

C₂×#3965

yes

characteristic

194

C₂²

#65514

yes

—

195

C₂²

#65521

yes

—

196

C₂²

#65526

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#65315

yes

—

205

C₄

#65319

yes

—

206

C₂

C₄×#3965

yes

characteristic

207

C₂

#10602424

yes

—

208

C₂

#10602570

yes

characteristic

209

C₂

#10602603

yes

characteristic

210

C₂

#10602626

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#8232
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₄×C₅×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

hypercenter

C₄×C₄⋊Q₈

D₅

hypocenter

C₅

#8232

LCS

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
14. #64766	C₄	semidirect
191. C₅	#8232	semidirect

H ≅ G/N

type

14. #64766

C₄

semidirect

191. C₅

#8232

semidirect

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₂×#64766	2
#10600184	2
#10598125	2
#8232	5

subgroup

index

C₄×C₅×C₄⋊Q₈

Subgroups — #11990257

Conjugacy classes of subgroups

Complements

Maximal subgroups