Group #12003752 — subgroups

Subgroups — #12003752

Order 1280.

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

normal subgroups only

This group has 406 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	#10606073	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic
3	#10600361	C₂	1	2	yes	maximal
4	#10598146	C₂	1	2	yes	maximal
5	#10598125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#107	—	2	2	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	C₂²	1	1	yes	characteristic
9	C₅×#384	C₂²	1	1	yes	characteristic
10	#65275	—	2	2	no	—
11	#65274	—	2	2	no	—
12	#65266	—	2	2	no	—
13	C₂×#3940	—	2	2	no	—
14	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	#64749	—	2	2	no	—
17	#10576	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	—	2	1	no	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	1	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
26	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
27	C₂×C₄×Dic₅	—	2	2	no	—
28	C₂²×#168	—	2	2	no	—
29	C₅×#107	—	4	4	no	—
30	C₄×C₅×C₈	—	2	1	no	—
31	C₄×C₅×C₈	Q₈	1	2	yes	—
32	#3963	—	2	2	no	—
33	#3963	—	2	4	no	—
34	#3963	—	2	2	no	—
35	#3940	—	4	4	no	—
36	C₄⋊#168	—	2	2	no	—
37	C₄⋊#168	Q₈	1	4	yes	—
38	#2571	—	5	1	no	—
39	#1948	—	5	2	no	—
40	#1845	D₅	1	1	yes	characteristic
41	#1296	—	5	1	no	—
42	#1279	—	5	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	—	4	1	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	2	2	no	—
48	C₂²×Dic₅	—	2	2	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₅×C₄⋊C₄	—	4	2	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅	C₂×Q₈	1	2	yes	—
60	C₄²×C₅	#41	1	1	yes	characteristic
61	C₄⋊Dic₅	—	2	2	no	—
62	C₄⋊Dic₅	—	2	2	no	—
63	C₄⋊Dic₅	—	2	2	no	—
64	C₄⋊Dic₅	—	2	2	no	—
65	C₄⋊Dic₅	—	8	2	no	—
66	C₄×Dic₅	—	4	2	no	—
67	C₄×Dic₅	—	4	2	no	—
68	C₂×#168	—	2	4	no	—
69	C₂×#168	—	4	2	no	—
70	C₂×#168	—	4	2	no	—
71	C₂×#107	—	2	2	no	—
72	C₂×#106	—	10	2	no	—
73	C₂×C₄⋊C₈	—	5	2	no	—
74	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
75	#400	—	10	2	no	—
76	#389	—	10	2	no	—
77	#384	C₂×D₅	1	1	yes	characteristic
78	#384	C₂×D₅	1	1	yes	characteristic
79	#382	—	10	2	no	—
80	C₄×C₄⋊C₄	—	5	2	no	—
81	#340	—	10	2	no	—
82	C₂³×C₅	#120	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	8	2	no	—
88	C₂×C₄×C₅	—	8	1	no	—
89	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅	#135	1	2	yes	—
94	C₂×C₄×C₅	#135	1	2	yes	—
95	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₅	#137	1	2	yes	—
97	C₂×Dic₅	—	4	2	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₂×Dic₅	—	4	2	no	—
100	C₂×Dic₅	—	4	2	no	—
101	C₂×Dic₅	—	4	2	no	—
102	C₂×Dic₅	—	4	2	no	—
103	C₅×C₈	—	2	4	no	—
104	#168	—	4	4	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	10	2	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	5	2	no	—
111	C₂×C₄⋊C₄	—	5	2	no	—
112	C₂×C₄⋊C₄	—	10	2	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₅	1	2	yes	—
114	C₂×C₄²	—	10	2	no	—
115	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
116	#107	—	4	4	no	—
117	#106	—	20	4	no	—
118	C₄⋊C₈	—	5	4	no	—
119	C₄⋊C₈	—	10	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	Dic₁₀	1	2	yes	—
122	#95	—	10	2	no	—
123	#95	—	10	4	no	—
124	#95	—	10	2	no	—
125	C₂²×C₅	#395	1	1	yes	characteristic
126	C₂²×C₅	#449	1	2	yes	—
127	C₂²×C₅	#452	1	2	yes	—
128	C₂²×C₅	#453	1	1	yes	characteristic
129	C₂²×C₅	#456	1	1	yes	characteristic
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	—	2	1	no	—
132	C₄×C₅	—	8	2	no	—
133	C₄×C₅	—	16	1	no	—
134	C₄×C₅	#478	1	2	yes	—
135	C₄×C₅	#479	1	2	yes	—
136	C₄×C₅	#499	1	2	yes	—
137	C₄×C₅	#501	1	2	yes	—
138	Dic₅	—	8	2	no	—
139	Dic₅	—	8	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	10	2	no	—
143	C₂²×C₄	—	10	2	no	—
144	C₂²×C₄	D₄×D₅	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	10	4	no	—
150	C₂×C₈	—	20	2	no	—
151	C₂×C₈	—	20	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	10	2	no	—
159	C₄⋊C₄	—	10	2	no	—
160	C₄⋊C₄	—	10	2	no	—
161	C₄⋊C₄	—	10	2	no	—
162	C₄⋊C₄	—	40	2	no	—
163	C₄²	—	20	2	no	—
164	C₄²	—	20	2	no	—
165	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
166	C₄²	#709	1	1	yes	characteristic
167	C₄²	Q₈×D₅	1	2	yes	—
168	C₂×C₅	#1620	1	1	yes	characteristic
169	C₂×C₅	#1634	1	1	yes	characteristic
170	C₂×C₅	#1650	1	2	yes	—
171	C₂×C₅	#2059	1	2	yes	—
172	C₂×C₅	#2060	1	1	yes	characteristic
173	C₂³	#4083	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	20	2	no	—
181	C₂×C₄	—	20	2	no	—
182	C₂×C₄	—	20	2	no	—
183	C₂×C₄	—	20	2	no	—
184	C₂×C₄	—	20	2	no	—
185	C₂×C₄	—	20	2	no	—
186	C₂×C₄	#4034	1	2	yes	—
187	C₂×C₄	#4042	1	2	yes	—
188	C₂×C₄	#4057	1	2	yes	—
189	C₂×C₄	#4058	1	2	yes	—
190	C₂×C₄	#4064	1	2	yes	—
191	C₂×C₄	#4065	1	2	yes	—
192	C₂×C₄	#4070	1	1	yes	characteristic
193	C₂×C₄	#4090	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₈	—	20	4	no	—
196	C₅	#10576	1	1	yes	characteristic, complemented, Sylow
197	C₂²	#65268	1	1	yes	characteristic
198	C₂²	#65366	1	1	yes	characteristic
199	C₂²	#65384	1	1	yes	characteristic
200	C₂²	#65447	1	2	yes	—
201	C₂²	#65476	1	2	yes	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	—	40	2	no	—
207	C₄	—	40	2	no	—
208	C₄	#65165	1	2	yes	—
209	C₄	#65166	1	2	yes	—
210	C₄	#65167	1	2	yes	—
211	C₄	#65180	1	2	yes	—
212	C₂	#10600366	1	1	yes	characteristic
213	C₂	#10601411	1	2	yes	—
214	C₂	#10602077	1	1	yes	characteristic
215	C₂	#10602097	1	1	yes	characteristic
216	C₂	#10602163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10606073

C₂

yes

maximal, characteristic

C₅×#1845

C₂

yes

maximal, characteristic

#10600361

C₂

yes

maximal

#10598146

C₂

yes

maximal

#10598125

C₂

yes

maximal, characteristic

C₂×C₅×#107

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

—

—

—

C₄×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊#168

C₂²

yes

—

#64749

—

#10576

—

maximal, Sylow

C₂²×C₅×C₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

D₄

yes

—

C₂×C₄⋊Dic₅

D₄

yes

—

C₂×C₄×Dic₅

—

C₂²×#168

—

C₅×#107

—

C₄×C₅×C₈

—

C₄×C₅×C₈

Q₈

yes

—

—

—

—

—

—

Q₈

yes

—

#2571

—

#1948

—

#1845

D₅

yes

characteristic

#1296

—

#1279

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

C₂×D₅

yes

characteristic

#384

C₂×D₅

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

#340

—

C₂³×C₅

#120

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

#135

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

—

100

C₂×Dic₅

—

101

C₂×Dic₅

—

102

C₂×Dic₅

—

103

C₅×C₈

—

104

#168

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂²×D₅

yes

characteristic

116

#107

—

117

#106

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₁₀

yes

—

122

#95

—

123

#95

—

124

#95

—

125

C₂²×C₅

#395

yes

characteristic

126

C₂²×C₅

#449

yes

—

127

C₂²×C₅

#452

yes

—

128

C₂²×C₅

#453

yes

characteristic

129

C₂²×C₅

#456

yes

characteristic

130

C₄×C₅

—

131

C₄×C₅

—

132

C₄×C₅

—

133

C₄×C₅

—

134

C₄×C₅

#478

yes

—

135

C₄×C₅

#479

yes

—

136

C₄×C₅

#499

yes

—

137

C₄×C₅

#501

yes

—

138

Dic₅

—

139

Dic₅

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

D₄×D₅

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

C₂×Dic₁₀

yes

characteristic

166

C₄²

#709

yes

characteristic

167

C₄²

Q₈×D₅

yes

—

168

C₂×C₅

#1620

yes

characteristic

169

C₂×C₅

#1634

yes

characteristic

170

C₂×C₅

#1650

yes

—

171

C₂×C₅

#2059

yes

—

172

C₂×C₅

#2060

yes

characteristic

173

C₂³

#4083

yes

characteristic

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

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

#4034

yes

—

187

C₂×C₄

#4042

yes

—

188

C₂×C₄

#4057

yes

—

189

C₂×C₄

#4058

yes

—

190

C₂×C₄

#4064

yes

—

191

C₂×C₄

#4065

yes

—

192

C₂×C₄

#4070

yes

characteristic

193

C₂×C₄

#4090

yes

characteristic

194

C₈

—

195

C₈

—

196

C₅

#10576

yes

characteristic, complemented, Sylow

197

C₂²

#65268

yes

characteristic

198

C₂²

#65366

yes

characteristic

199

C₂²

#65384

yes

characteristic

200

C₂²

#65447

yes

—

201

C₂²

#65476

yes

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

#65165

yes

—

209

C₄

#65166

yes

—

210

C₄

#65167

yes

—

211

C₄

#65180

yes

—

212

C₂

#10600366

yes

characteristic

213

C₂

#10601411

yes

—

214

C₂

#10602077

yes

characteristic

215

C₂

#10602097

yes

characteristic

216

C₂

#10602163

yes

—

	subgroup	G/H
center	C₂³	#4083
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1845	C₂
socle, prime radical	C₂³×C₅	#120
hypercenter	#1845	D₅
hypocenter	C₅	#10576
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#4083

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 ◁ #1845

N	H ≅ G/N	type
196. C₅	#10576	semidirect

H ≅ G/N

type

196. C₅

#10576

semidirect

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

subgroup	index
#10606073	2
C₅×#1845	2
#10600361	2
#10598146	2
#10598125	2
#10576	5

subgroup

index

Subgroups — #12003752

Conjugacy classes of subgroups

Complements

Maximal subgroups