Group #12004337 — subgroups

Subgroups — #12004337

Order 1280.

494 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 147 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×#1957	C₂	1	1	yes	maximal, characteristic
2	#10602557	C₂	1	2	yes	maximal
3	#10600558	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600367	C₂	1	2	yes	maximal, complemented
5	#10598144	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#127	C₂²	1	1	yes	characteristic
7	C₂×C₅×#102	—	2	2	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#384	C₂²	1	1	yes	characteristic
10	C₂×#3964	—	2	4	no	—
11	C₂×#3950	—	2	2	no	—
12	#65224	C₂²	1	2	yes	—
13	C₂×C₄×#168	C₂²	1	2	yes	—
14	#64749	—	2	4	no	—
15	#10671	—	5	1	no	maximal, Sylow
16	C₂²×C₅×D₄	—	2	1	no	—
17	C₂²×C₅×D₄	—	2	2	no	—
18	C₂²×C₅×C₈	D₄	1	2	yes	—
19	C₅×#127	—	2	1	no	—
20	C₅×#127	—	2	2	no	—
21	C₅×#127	Q₈	1	2	yes	—
22	C₂×C₅×C₄⋊C₄	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂²×#168	D₄	1	4	yes	—
29	C₅×#102	—	4	4	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	#3964	—	4	8	no	—
32	#3950	—	4	4	no	—
33	C₄×#168	—	2	2	no	—
34	C₄×#168	—	2	2	no	—
35	#1957	—	5	2	no	—
36	#1957	D₅	1	1	yes	characteristic
37	#1845	—	5	1	no	—
38	#1834	—	5	2	no	—
39	#1273	—	5	1	no	—
40	C₂⁴×C₅	—	4	1	no	—
41	C₂×C₅×D₄	—	2	4	no	—
42	C₂×C₅×D₄	—	2	2	no	—
43	C₂×C₅×D₄	—	4	2	no	—
44	C₂×C₅×D₄	—	4	1	no	—
45	C₂×C₅×D₄	—	4	2	no	—
46	C₂×C₅×D₄	—	4	4	no	—
47	C₂×C₅×D₄	—	4	4	no	—
48	C₂×C₅×D₄	—	4	2	no	—
49	C₂²×C₄×C₅	—	4	1	no	—
50	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅	—	4	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	D₈	1	4	yes	—
56	C₅×C₄⋊C₄	—	4	2	no	—
57	C₅×C₄⋊C₄	—	4	4	no	—
58	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅	#41	1	1	yes	characteristic
60	C₄²×C₅	#41	1	2	yes	—
61	C₄⋊Dic₅	—	4	4	no	—
62	C₄⋊Dic₅	—	4	4	no	—
63	C₄⋊Dic₅	—	4	4	no	—
64	C₂×#168	—	2	4	no	—
65	C₂×#168	—	2	4	no	—
66	C₂×#168	—	2	4	no	—
67	C₂×#127	C₂×D₅	1	1	yes	characteristic
68	C₂×#107	—	10	2	no	—
69	C₂×#102	—	2	2	no	—
70	C₂×#102	—	10	4	no	—
71	C₂×C₄×C₈	—	5	2	no	—
72	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#384	—	5	2	no	—
74	#384	C₂×D₅	1	1	yes	characteristic
75	#340	—	10	4	no	—
76	C₂³×C₅	—	4	2	no	—
77	C₂³×C₅	—	8	1	no	—
78	C₂³×C₅	—	8	2	no	—
79	C₂³×C₅	—	8	2	no	—
80	C₂³×C₅	—	8	1	no	—
81	C₂³×C₅	#127	1	1	yes	characteristic
82	C₅×D₄	—	4	4	no	—
83	C₅×D₄	—	4	8	no	—
84	C₅×D₄	—	8	2	no	—
85	C₅×D₄	—	8	4	no	—
86	C₂×C₄×C₅	—	8	1	no	—
87	C₂×C₄×C₅	—	8	2	no	—
88	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	#123	1	2	yes	—
91	C₂×C₄×C₅	#123	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₅	#135	1	4	yes	—
94	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
95	C₂×C₄×C₅	#137	1	2	yes	—
96	C₂×Dic₅	—	8	2	no	—
97	C₂×Dic₅	—	8	2	no	—
98	C₂×Dic₅	—	8	2	no	—
99	C₅×C₈	—	2	4	no	—
100	#168	—	4	4	no	—
101	C₂²×D₄	—	2	2	no	—
102	C₂²×D₄	—	2	1	no	—
103	C₂²×C₈	—	5	4	no	—
104	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
105	#127	—	2	1	no	—
106	#127	—	2	2	no	—
107	#127	Dic₁₀	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	10	2	no	—
111	C₂×C₄⋊C₄	—	10	2	no	—
112	C₂×C₄⋊C₄	—	10	2	no	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	#107	—	20	4	no	—
115	#102	—	4	4	no	—
116	#102	—	20	8	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	10	2	no	—
119	C₄×C₈	—	10	2	no	—
120	C₂²×C₅	—	8	4	no	—
121	C₂²×C₅	—	8	2	no	—
122	C₂²×C₅	—	16	1	no	—
123	C₂²×C₅	—	16	2	no	—
124	C₂²×C₅	—	16	1	no	—
125	C₂²×C₅	#399	1	1	yes	characteristic
126	C₂²×C₅	#493	1	1	yes	characteristic
127	C₂²×C₅	#495	1	2	yes	—
128	C₂²×C₅	#496	1	2	yes	—
129	C₂²×C₅	#497	1	1	yes	characteristic
130	C₄×C₅	—	16	1	no	—
131	C₄×C₅	#474	1	2	yes	—
132	C₄×C₅	#478	1	4	yes	—
133	C₄×C₅	#480	1	2	yes	—
134	C₄×C₅	#482	1	4	yes	—
135	Dic₅	—	16	2	no	—
136	C₂⁴	—	4	1	no	—
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂×D₄	—	4	4	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	20	2	no	—
147	C₂²×C₄	D₄×D₅	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	10	4	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	10	4	no	—
154	C₂×C₈	#686	1	4	yes	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	20	4	no	—
158	C₄⋊C₄	—	20	4	no	—
159	C₄⋊C₄	—	20	4	no	—
160	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
161	C₄²	#709	1	1	yes	characteristic
162	C₄²	#711	1	2	yes	—
163	C₂×C₅	—	16	2	no	—
164	C₂×C₅	#1709	1	1	yes	characteristic
165	C₂×C₅	#1719	1	2	yes	—
166	C₂×C₅	#1721	1	1	yes	characteristic
167	C₂×C₅	#2092	1	2	yes	—
168	C₂×C₅	#2093	1	1	yes	characteristic
169	C₂³	—	4	2	no	—
170	C₂³	—	8	1	no	—
171	C₂³	—	8	1	no	—
172	C₂³	—	8	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	#4086	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	4	8	no	—
177	D₄	—	8	2	no	—
178	D₄	—	8	4	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	40	2	no	—
182	C₂×C₄	—	40	2	no	—
183	C₂×C₄	—	40	2	no	—
184	C₂×C₄	#4024	1	2	yes	—
185	C₂×C₄	#4029	1	2	yes	—
186	C₂×C₄	#4058	1	4	yes	—
187	C₂×C₄	#4060	1	4	yes	—
188	C₂×C₄	#4070	1	1	yes	characteristic
189	C₂×C₄	C₂×#686	1	2	yes	—
190	C₂×C₄	#4081	1	1	yes	characteristic
191	C₂×C₄	#4097	1	2	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	20	4	no	—
194	C₅	#10671	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	8	2	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	16	1	no	—
198	C₂²	—	16	1	no	—
199	C₂²	—	16	2	no	—
200	C₂²	#65238	1	1	yes	characteristic
201	C₂²	#65349	1	1	yes	characteristic
202	C₂²	#65362	1	1	yes	characteristic
203	C₂²	#65444	1	2	yes	—
204	C₂²	#65465	1	2	yes	—
205	C₄	—	16	1	no	—
206	C₄	—	80	2	no	—
207	C₄	#65055	1	4	yes	—
208	C₄	#65073	1	4	yes	—
209	C₄	#65296	1	2	yes	—
210	C₄	#65322	1	2	yes	—
211	C₂	—	16	2	no	—
212	C₂	#10600448	1	1	yes	characteristic
213	C₂	#10600974	1	2	yes	—
214	C₂	#10601983	1	2	yes	—
215	C₂	#10602055	1	1	yes	characteristic
216	C₂	#10602068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1957

C₂

yes

maximal, characteristic

#10602557

C₂

yes

maximal

#10600558

C₂

yes

maximal, characteristic, complemented

#10600367

C₂

yes

maximal, complemented

#10598144

C₂

yes

maximal, characteristic, complemented

C₂×C₅×#127

C₂²

yes

characteristic

C₂×C₅×#102

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

#64749

—

#10671

—

maximal, Sylow

C₂²×C₅×D₄

—

C₂²×C₅×D₄

—

C₂²×C₅×C₈

D₄

yes

—

C₅×#127

—

C₅×#127

—

C₅×#127

Q₈

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

D₄

yes

—

C₅×#102

—

C₄×C₅×C₈

—

—

—

—

—

—

D₅

yes

—

—

—

C₂⁴×C₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₈

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Q₈

yes

C₄²×C₅

yes

C₄²×C₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

C₂×D₅

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#384

—

#384

C₂×D₅

yes

characteristic

#340

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

#127

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#123

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

100

#168

—

101

C₂²×D₄

—

102

C₂²×D₄

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂²⋊D₅

yes

—

105

#127

—

106

#127

—

107

#127

Dic₁₀

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

C₂²×D₅

yes

characteristic

114

#107

—

115

#102

—

116

#102

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₅

—

121

C₂²×C₅

—

122

C₂²×C₅

—

123

C₂²×C₅

—

124

C₂²×C₅

—

125

C₂²×C₅

#399

yes

characteristic

126

C₂²×C₅

#493

yes

characteristic

127

C₂²×C₅

#495

yes

—

128

C₂²×C₅

#496

yes

—

129

C₂²×C₅

#497

yes

characteristic

130

C₄×C₅

—

131

C₄×C₅

#474

yes

—

132

C₄×C₅

#478

yes

—

133

C₄×C₅

#480

yes

—

134

C₄×C₅

#482

yes

—

135

Dic₅

—

136

C₂⁴

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

D₄×D₅

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#686

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄²

C₂×Dic₁₀

yes

characteristic

161

C₄²

#709

yes

characteristic

162

C₄²

#711

yes

—

163

C₂×C₅

—

164

C₂×C₅

#1709

yes

characteristic

165

C₂×C₅

#1719

yes

—

166

C₂×C₅

#1721

yes

characteristic

167

C₂×C₅

#2092

yes

—

168

C₂×C₅

#2093

yes

characteristic

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

#4086

yes

characteristic

175

D₄

—

176

D₄

—

177

D₄

—

178

D₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

#4024

yes

—

185

C₂×C₄

#4029

yes

—

186

C₂×C₄

#4058

yes

—

187

C₂×C₄

#4060

yes

—

188

C₂×C₄

#4070

yes

characteristic

189

C₂×C₄

C₂×#686

yes

—

190

C₂×C₄

#4081

yes

characteristic

191

C₂×C₄

#4097

yes

—

192

C₈

—

193

C₈

—

194

C₅

#10671

yes

characteristic, complemented, Sylow

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#65238

yes

characteristic

201

C₂²

#65349

yes

characteristic

202

C₂²

#65362

yes

characteristic

203

C₂²

#65444

yes

—

204

C₂²

#65465

yes

—

205

C₄

—

206

C₄

—

207

C₄

#65055

yes

—

208

C₄

#65073

yes

—

209

C₄

#65296

yes

—

210

C₄

#65322

yes

—

211

C₂

—

212

C₂

#10600448

yes

characteristic

213

C₂

#10600974

yes

—

214

C₂

#10601983

yes

—

215

C₂

#10602055

yes

characteristic

216

C₂

#10602068

yes

characteristic

	subgroup	G/H
center	C₂³	#4086
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1957	C₂
socle	C₂³×C₅	#127
prime radical	C₂×C₅×#127	C₂²
hypercenter	#1957	D₅
hypocenter	C₅	#10671
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#4086

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

C₂×C₅×#127

C₂²

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
3. #10600558	C₂	semidirect
4. #10600367	C₂	semidirect
5. #10598144	C₂	semidirect
194. C₅	#10671	semidirect

H ≅ G/N

type

3. #10600558

C₂

semidirect

4. #10600367

C₂

semidirect

5. #10598144

C₂

semidirect

194. C₅

#10671

semidirect

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

subgroup	index
C₅×#1957	2
#10602557	2
#10600558	2
#10600367	2
#10598144	2
#10671	5

subgroup

index

Subgroups — #12004337

Conjugacy classes of subgroups

Complements

Maximal subgroups