Group #12003739 — subgroups

Subgroups — #12003739

Order 1280.

494 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₅×#2551	C₂	1	1	yes	maximal, characteristic
2	#10600369	C₂	1	1	yes	maximal, characteristic
3	#10600260	C₂	1	2	yes	maximal
4	#10598125	C₂	1	1	yes	maximal, characteristic
5	#10598110	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	—	2	2	no	—
7	C₅×#395	—	2	2	no	—
8	C₅×#389	—	2	2	no	—
9	C₅×#384	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×#3940	—	2	2	no	—
12	C₂×#3940	—	2	2	no	—
13	#65224	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	#64749	—	2	2	no	—
17	#10574	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	4	4	no	—
20	C₅×C₄⋊Q₈	—	4	4	no	—
21	C₂×C₅×C₄⋊C₄	—	2	4	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
24	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
25	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
26	C₂×C₄²×C₅	—	2	2	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	1	no	—
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂²×#168	—	2	1	no	—
31	C₂²×#168	—	2	2	no	—
32	C₅×#95	—	2	2	no	—
33	C₅×#95	—	2	2	no	—
34	#3940	—	4	4	no	—
35	#3940	—	4	4	no	—
36	C₄⋊#168	—	2	2	no	—
37	C₄⋊#168	Q₈	1	4	yes	complemented
38	C₄×#168	—	2	1	no	—
39	C₄×#168	Q₈	1	2	yes	complemented
40	#2551	D₅	1	1	yes	characteristic
41	#1948	—	5	2	no	—
42	#1844	—	5	1	no	—
43	#1296	—	5	1	no	—
44	#1279	—	5	2	no	—
45	C₂×C₅×Q₈	—	4	8	no	—
46	C₂×C₅×Q₈	—	8	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂²×Dic₅	—	4	1	no	—
53	C₅×C₄⋊C₄	—	2	2	no	—
54	C₅×C₄⋊C₄	—	2	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₅×C₄⋊C₄	—	4	8	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	SD₁₆	1	4	yes	—
60	C₅×C₄⋊C₄	SD₁₆	1	4	yes	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	—	4	2	no	—
63	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	2	yes	—
65	C₄²×C₅	#41	1	1	yes	characteristic
66	C₄⋊Dic₅	—	4	2	no	—
67	C₄⋊Dic₅	—	4	2	no	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	—	2	2	no	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	—	2	4	no	—
73	C₂×#168	—	4	2	no	—
74	C₂×#168	—	4	2	no	—
75	C₂×C₄⋊Q₈	—	2	2	no	—
76	C₂×#106	—	10	2	no	—
77	C₂×#106	—	10	2	no	—
78	C₂×C₄⋊C₈	—	5	2	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	#395	—	2	2	no	—
81	#389	—	2	2	no	—
82	#384	—	5	1	no	—
83	#384	C₂×D₅	1	1	yes	characteristic
84	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
85	#340	—	10	2	no	—
86	C₂³×C₅	#120	1	1	yes	characteristic
87	C₅×Q₈	—	8	8	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₅	#137	1	2	yes	—
106	C₂×Dic₅	—	8	2	no	—
107	C₂×Dic₅	—	8	1	no	—
108	#168	—	2	4	no	—
109	#168	—	4	4	no	—
110	C₂²×Q₈	—	2	2	no	—
111	C₂²×C₈	—	10	1	no	—
112	C₂²×C₈	—	10	2	no	—
113	C₄⋊Q₈	—	4	4	no	—
114	C₄⋊Q₈	—	4	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	10	2	no	—
118	C₂×C₄⋊C₄	—	10	1	no	—
119	C₂×C₄⋊C₄	C₂²⋊D₅	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²⋊D₅	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²⋊D₅	1	2	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
124	#106	—	20	4	no	—
125	#106	—	20	4	no	—
126	C₄⋊C₈	—	5	4	no	—
127	C₄⋊C₈	—	10	2	no	—
128	C₄×C₈	—	5	2	no	—
129	C₄×C₈	—	10	1	no	—
130	#95	—	2	2	no	—
131	#95	—	2	2	no	—
132	C₂²×C₅	#395	1	1	yes	characteristic
133	C₂²×C₅	#448	1	2	yes	—
134	C₂²×C₅	#450	1	2	yes	—
135	C₂²×C₅	#453	1	1	yes	characteristic
136	C₂²×C₅	#456	1	1	yes	characteristic
137	C₄×C₅	—	2	1	no	—
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	4	4	no	—
140	C₄×C₅	—	4	4	no	—
141	C₄×C₅	—	8	2	no	—
142	C₄×C₅	#475	1	2	yes	—
143	C₄×C₅	#476	1	2	yes	—
144	C₄×C₅	#498	1	2	yes	—
145	C₄×C₅	#501	1	2	yes	—
146	Dic₅	—	16	1	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂×Q₈	—	8	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	20	1	no	—
153	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
155	C₂×C₈	—	10	2	no	—
156	C₂×C₈	—	10	2	no	—
157	C₂×C₈	—	10	2	no	—
158	C₂×C₈	—	10	4	no	—
159	C₂×C₈	—	20	2	no	—
160	C₂×C₈	—	20	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	20	2	no	—
168	C₄⋊C₄	—	20	2	no	—
169	C₄⋊C₄	—	20	2	no	—
170	C₄⋊C₄	#687	1	4	yes	—
171	C₄⋊C₄	#688	1	4	yes	—
172	C₄²	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	Q₈×D₅	1	1	yes	characteristic
175	C₄²	Q₈×D₅	1	2	yes	—
176	C₄²	#713	1	1	yes	characteristic
177	C₂×C₅	#1623	1	1	yes	characteristic
178	C₂×C₅	#1629	1	1	yes	characteristic
179	C₂×C₅	#1651	1	2	yes	—
180	C₂×C₅	#2058	1	2	yes	—
181	C₂×C₅	#2060	1	1	yes	characteristic
182	C₂³	#4099	1	1	yes	characteristic
183	Q₈	—	8	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	40	2	no	—
195	C₂×C₄	—	40	1	no	—
196	C₂×C₄	#4078	1	2	yes	—
197	C₂×C₄	C₂×#687	1	2	yes	—
198	C₂×C₄	C₂×#688	1	2	yes	—
199	C₂×C₄	#4088	1	2	yes	—
200	C₂×C₄	#4090	1	1	yes	characteristic
201	C₂×C₄	#4090	1	2	yes	—
202	C₂×C₄	#4092	1	1	yes	characteristic
203	C₂×C₄	#4092	1	2	yes	—
204	C₈	—	10	4	no	—
205	C₈	—	20	4	no	—
206	C₅	#10574	1	1	yes	characteristic, complemented, Sylow
207	C₂²	#65507	1	2	yes	—
208	C₂²	#65516	1	2	yes	—
209	C₂²	#65518	1	1	yes	characteristic
210	C₂²	#65530	1	1	yes	characteristic
211	C₂²	#65535	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	—	80	1	no	—
218	C₄	#65368	1	2	yes	—
219	C₄	#65369	1	2	yes	—
220	C₄	#65373	1	2	yes	—
221	C₄	#65383	1	2	yes	—
222	C₂	#10602576	1	2	yes	—
223	C₂	#10602693	1	1	yes	characteristic
224	C₂	#10602700	1	1	yes	characteristic
225	C₂	#10602710	1	2	yes	—
226	C₂	#10602732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2551

C₂

yes

maximal, characteristic

#10600369

C₂

yes

maximal, characteristic

#10600260

C₂

yes

maximal

#10598125

C₂

yes

maximal, characteristic

#10598110

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

—

C₅×#395

—

C₅×#389

—

C₅×#384

C₂²

yes

characteristic

C₄×C₅×C₄⋊C₄

C₂²

yes

—

C₂×#3940

—

C₂×#3940

—

#65224

C₂²

yes

characteristic

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

—

C₂²×C₅×Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

—

C₂²×#168

—

C₅×#95

—

C₅×#95

—

—

—

—

Q₈

yes

complemented

C₄×#168

—

C₄×#168

Q₈

yes

complemented

#2551

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

SD₁₆

yes

—

C₅×C₄⋊C₄

SD₁₆

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

—

—

—

C₂×D₅

yes

characteristic

C₄×C₄⋊C₄

C₂×D₅

yes

—

#340

—

C₂³×C₅

#120

yes

characteristic

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

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

100

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

101

C₂×C₄×C₅

C₄⋊Q₈

yes

—

102

C₂×C₄×C₅

C₂×SD₁₆

yes

—

103

C₂×C₄×C₅

C₂×SD₁₆

yes

—

104

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

105

C₂×C₄×C₅

#137

yes

—

106

C₂×Dic₅

—

107

C₂×Dic₅

—

108

#168

—

109

#168

—

110

C₂²×Q₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

120

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

121

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂²×D₅

yes

characteristic

124

#106

—

125

#106

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

#95

—

131

#95

—

132

C₂²×C₅

#395

yes

characteristic

133

C₂²×C₅

#448

yes

—

134

C₂²×C₅

#450

yes

—

135

C₂²×C₅

#453

yes

characteristic

136

C₂²×C₅

#456

yes

characteristic

137

C₄×C₅

—

138

C₄×C₅

—

139

C₄×C₅

—

140

C₄×C₅

—

141

C₄×C₅

—

142

C₄×C₅

#475

yes

—

143

C₄×C₅

#476

yes

—

144

C₄×C₅

#498

yes

—

145

C₄×C₅

#501

yes

—

146

Dic₅

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

#687

yes

—

171

C₄⋊C₄

#688

yes

—

172

C₄²

—

173

C₄²

—

174

C₄²

Q₈×D₅

yes

characteristic

175

C₄²

Q₈×D₅

yes

—

176

C₄²

#713

yes

characteristic

177

C₂×C₅

#1623

yes

characteristic

178

C₂×C₅

#1629

yes

characteristic

179

C₂×C₅

#1651

yes

—

180

C₂×C₅

#2058

yes

—

181

C₂×C₅

#2060

yes

characteristic

182

C₂³

#4099

yes

characteristic

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

#4078

yes

—

197

C₂×C₄

C₂×#687

yes

—

198

C₂×C₄

C₂×#688

yes

—

199

C₂×C₄

#4088

yes

—

200

C₂×C₄

#4090

yes

characteristic

201

C₂×C₄

#4090

yes

—

202

C₂×C₄

#4092

yes

characteristic

203

C₂×C₄

#4092

yes

—

204

C₈

—

205

C₈

—

206

C₅

#10574

yes

characteristic, complemented, Sylow

207

C₂²

#65507

yes

—

208

C₂²

#65516

yes

—

209

C₂²

#65518

yes

characteristic

210

C₂²

#65530

yes

characteristic

211

C₂²

#65535

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#65368

yes

—

219

C₄

#65369

yes

—

220

C₄

#65373

yes

—

221

C₄

#65383

yes

—

222

C₂

#10602576

yes

—

223

C₂

#10602693

yes

characteristic

224

C₂

#10602700

yes

characteristic

225

C₂

#10602710

yes

—

226

C₂

#10602732

yes

characteristic

	subgroup	G/H
center	C₂³	#4099
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2551	C₂
socle, prime radical	C₂³×C₅	#120
hypercenter	#2551	D₅
hypocenter	C₅	#10574
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2551

subgroup

G/H

center

C₂³

#4099

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2551

N	H ≅ G/N	type
37. C₄⋊#168	Q₈	semidirect
39. C₄×#168	Q₈	semidirect
206. C₅	#10574	semidirect

H ≅ G/N

type

37. C₄⋊#168

Q₈

semidirect

39. C₄×#168

Q₈

semidirect

206. C₅

#10574

semidirect

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

subgroup	index
C₅×#2551	2
#10600369	2
#10600260	2
#10598125	2
#10598110	2
#10574	5

subgroup

index

Subgroups — #12003739

Conjugacy classes of subgroups

Complements

Maximal subgroups