Group #12004046 — subgroups

Subgroups — #12004046

Order 1280.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#10601278	C₂	1	2	yes	maximal
2	#10600581	C₂	1	1	yes	maximal, characteristic
3	#10600555	C₂	1	1	yes	maximal, characteristic
4	C₅×#1277	C₂	1	1	yes	maximal, characteristic
5	#10598193	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3948	—	2	2	no	—
10	#65224	C₂²	1	2	yes	—
11	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
12	#64771	—	2	2	no	—
13	#64771	—	2	2	no	—
14	#64770	—	2	2	no	—
15	#10623	—	5	1	no	maximal, Sylow
16	C₂²×Dic₁₀	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	D₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	—	2	2	no	—
21	C₂×C₄⋊Dic₅	—	2	2	no	—
22	C₂×C₄⋊Dic₅	—	2	2	no	—
23	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
24	C₂×#683	—	2	1	no	—
25	C₂×C₄×Dic₅	—	2	1	no	—
26	C₂²×#168	—	2	1	no	—
27	C₄×Dic₁₀	—	2	2	no	—
28	C₄×Dic₁₀	—	2	2	no	—
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	#3948	—	4	4	no	—
33	C₄⋊#168	—	2	2	no	—
34	#1984	—	5	1	no	—
35	#1834	—	5	1	no	—
36	#1803	—	5	2	no	—
37	#1296	—	5	2	no	—
38	#1277	D₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	#41	1	1	yes	characteristic
42	C₂²×Dic₅	—	2	1	no	—
43	C₂²×Dic₅	—	2	1	no	—
44	C₂²×Dic₅	—	4	2	no	—
45	C₂×Dic₁₀	—	2	2	no	—
46	C₂×Dic₁₀	—	2	2	no	—
47	C₂×Dic₁₀	—	2	1	no	—
48	C₂×Dic₁₀	D₈	1	2	yes	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	—	4	2	no	—
54	C₂×C₅×C₈	—	4	2	no	—
55	C₄²×C₅	#41	1	2	yes	—
56	C₄²×C₅	#41	1	2	yes	—
57	C₄⋊Dic₅	—	2	2	no	—
58	C₄⋊Dic₅	—	4	2	no	—
59	C₄⋊Dic₅	—	4	2	no	—
60	C₄⋊Dic₅	—	4	2	no	—
61	C₄⋊Dic₅	—	4	2	no	—
62	C₄⋊Dic₅	—	8	2	no	—
63	#683	—	4	2	no	—
64	C₄×Dic₅	—	4	1	no	—
65	C₄×Dic₅	—	4	1	no	—
66	C₂×#168	—	4	1	no	—
67	C₂×#168	—	4	2	no	—
68	C₂×C₄×Q₈	—	5	1	no	—
69	C₂×C₄⋊C₈	—	5	1	no	—
70	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
71	C₂×#103	—	10	2	no	—
72	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#384	—	5	2	no	—
74	#340	—	10	2	no	—
75	#340	—	10	2	no	—
76	#336	—	10	2	no	—
77	C₂³×C₅	#121	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	#123	1	2	yes	—
80	C₂×C₄×C₅	#123	1	1	yes	characteristic
81	C₂×C₄×C₅	#124	1	1	yes	characteristic
82	C₂×C₄×C₅	#126	1	2	yes	—
83	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	#135	1	2	yes	—
85	C₂×C₄×C₅	#135	1	1	yes	characteristic
86	C₂×C₄×C₅	#137	1	2	yes	—
87	C₂×C₄×C₅	#137	1	2	yes	—
88	C₂×Dic₅	—	2	2	no	—
89	C₂×Dic₅	—	4	1	no	—
90	C₂×Dic₅	—	4	1	no	—
91	C₂×Dic₅	—	4	1	no	—
92	C₂×Dic₅	—	4	2	no	—
93	C₂×Dic₅	—	8	2	no	—
94	C₂×Dic₅	—	8	2	no	—
95	C₂×Dic₅	—	8	2	no	—
96	Dic₁₀	—	2	4	no	—
97	Dic₁₀	—	4	2	no	—
98	C₅×C₈	—	4	2	no	—
99	C₅×C₈	—	8	2	no	—
100	#168	—	8	1	no	—
101	C₂²×Q₈	—	5	1	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	10	1	no	—
104	C₂²×C₈	D₂₀	1	2	yes	—
105	C₄×Q₈	—	10	2	no	—
106	C₄×Q₈	—	10	2	no	—
107	C₂×C₄⋊C₄	—	5	1	no	—
108	C₂×C₄⋊C₄	—	10	1	no	—
109	C₂×C₄⋊C₄	—	10	2	no	—
110	C₂×C₄⋊C₄	—	10	2	no	—
111	C₂×C₄⋊C₄	—	10	2	no	—
112	C₂×C₄²	—	10	1	no	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	10	2	no	—
117	#103	—	20	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₂²×C₅	#397	1	1	yes	characteristic
120	C₂²×C₅	#464	1	2	yes	—
121	C₂²×C₅	#466	1	1	yes	characteristic
122	C₂²×C₅	#470	1	1	yes	characteristic
123	C₂²×C₅	#471	1	2	yes	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	#480	1	2	yes	—
126	C₄×C₅	#485	1	2	yes	—
127	C₄×C₅	#485	1	2	yes	—
128	C₄×C₅	#488	1	1	yes	characteristic
129	C₄×C₅	#491	1	1	yes	characteristic
130	Dic₅	—	4	2	no	—
131	Dic₅	—	8	1	no	—
132	Dic₅	—	16	2	no	—
133	C₂×Q₈	—	5	2	no	—
134	C₂×Q₈	—	10	1	no	—
135	C₂×Q₈	—	10	2	no	—
136	C₂×Q₈	—	10	2	no	—
137	C₂²×C₄	—	10	1	no	—
138	C₂²×C₄	—	10	1	no	—
139	C₂²×C₄	—	20	2	no	—
140	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
141	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
142	C₂²×C₄	#713	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	20	1	no	—
150	C₂×C₈	—	20	2	no	—
151	C₄⋊C₄	—	10	2	no	—
152	C₄⋊C₄	—	20	2	no	—
153	C₄⋊C₄	—	20	2	no	—
154	C₄⋊C₄	—	20	2	no	—
155	C₄⋊C₄	—	20	2	no	—
156	C₄⋊C₄	—	20	2	no	—
157	C₄⋊C₄	—	40	2	no	—
158	C₄²	—	20	1	no	—
159	C₄²	—	20	1	no	—
160	C₄²	#709	1	2	yes	—
161	C₄²	#711	1	2	yes	—
162	C₂×C₅	#1682	1	2	yes	—
163	C₂×C₅	#1704	1	2	yes	—
164	C₂×C₅	#2074	1	1	yes	characteristic
165	C₂×C₅	#2075	1	1	yes	characteristic
166	C₂×C₅	#2079	1	1	yes	characteristic
167	C₂³	#4041	1	1	yes	characteristic
168	Q₈	—	10	4	no	—
169	Q₈	—	20	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	10	2	no	—
172	C₂×C₄	—	20	1	no	—
173	C₂×C₄	—	20	1	no	—
174	C₂×C₄	—	20	2	no	—
175	C₂×C₄	—	20	1	no	—
176	C₂×C₄	—	40	2	no	—
177	C₂×C₄	—	40	2	no	—
178	C₂×C₄	—	40	2	no	—
179	C₂×C₄	#4021	1	1	yes	characteristic
180	C₂×C₄	#4029	1	2	yes	—
181	C₂×C₄	#4032	1	1	yes	characteristic
182	C₂×C₄	#4044	1	2	yes	—
183	C₂×C₄	#4050	1	2	yes	—
184	C₂×C₄	#4055	1	2	yes	—
185	C₂×C₄	D₅×D₈	1	1	yes	characteristic
186	C₂×C₄	#4061	1	2	yes	—
187	C₂×C₄	#4065	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	40	1	no	—
191	C₅	#10623	1	1	yes	characteristic, complemented, Sylow
192	C₂²	#64963	1	1	yes	characteristic
193	C₂²	#65134	1	2	yes	—
194	C₂²	#65157	1	1	yes	characteristic
195	C₂²	#65172	1	1	yes	characteristic
196	C₂²	#65187	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	20	2	no	—
199	C₄	—	40	1	no	—
200	C₄	—	80	2	no	—
201	C₄	#64986	1	1	yes	characteristic
202	C₄	#64992	1	1	yes	characteristic
203	C₄	#65012	1	2	yes	—
204	C₄	#65063	1	2	yes	—
205	C₄	#65103	1	2	yes	—
206	C₂	#10599348	1	2	yes	—
207	C₂	#10599383	1	2	yes	—
208	C₂	#10599431	1	1	yes	characteristic
209	C₂	#10599623	1	1	yes	characteristic
210	C₂	#10599715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10601278

C₂

yes

maximal

#10600581

C₂

yes

maximal, characteristic

#10600555

C₂

yes

maximal, characteristic

C₅×#1277

C₂

yes

maximal, characteristic

#10598193

C₂

yes

maximal

C₂×C₄×Dic₁₀

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3948

—

#65224

C₂²

yes

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

C₂²×Dic₁₀

D₄

yes

characteristic

C₂²×C₅×C₈

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

D₄

yes

characteristic

C₂×#683

—

C₂×C₄×Dic₅

—

C₂²×#168

—

C₄×Dic₁₀

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

D₈

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#121

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

#135

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

100

#168

—

101

C₂²×Q₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

D₂₀

yes

—

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

C₂²×D₅

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#103

—

118

C₄×C₈

—

119

C₂²×C₅

#397

yes

characteristic

120

C₂²×C₅

#464

yes

—

121

C₂²×C₅

#466

yes

characteristic

122

C₂²×C₅

#470

yes

characteristic

123

C₂²×C₅

#471

yes

—

124

C₄×C₅

—

125

C₄×C₅

#480

yes

—

126

C₄×C₅

#485

yes

—

127

C₄×C₅

#485

yes

—

128

C₄×C₅

#488

yes

characteristic

129

C₄×C₅

#491

yes

characteristic

130

Dic₅

—

131

Dic₅

—

132

Dic₅

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×D₂₀

yes

characteristic

141

C₂²×C₄

D₄×D₅

yes

characteristic

142

C₂²×C₄

#713

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₄²

#709

yes

—

161

C₄²

#711

yes

—

162

C₂×C₅

#1682

yes

—

163

C₂×C₅

#1704

yes

—

164

C₂×C₅

#2074

yes

characteristic

165

C₂×C₅

#2075

yes

characteristic

166

C₂×C₅

#2079

yes

characteristic

167

C₂³

#4041

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

#4021

yes

characteristic

180

C₂×C₄

#4029

yes

—

181

C₂×C₄

#4032

yes

characteristic

182

C₂×C₄

#4044

yes

—

183

C₂×C₄

#4050

yes

—

184

C₂×C₄

#4055

yes

—

185

C₂×C₄

D₅×D₈

yes

characteristic

186

C₂×C₄

#4061

yes

—

187

C₂×C₄

#4065

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

191

C₅

#10623

yes

characteristic, complemented, Sylow

192

C₂²

#64963

yes

characteristic

193

C₂²

#65134

yes

—

194

C₂²

#65157

yes

characteristic

195

C₂²

#65172

yes

characteristic

196

C₂²

#65187

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#64986

yes

characteristic

202

C₄

#64992

yes

characteristic

203

C₄

#65012

yes

—

204

C₄

#65063

yes

—

205

C₄

#65103

yes

—

206

C₂

#10599348

yes

—

207

C₂

#10599383

yes

—

208

C₂

#10599431

yes

characteristic

209

C₂

#10599623

yes

characteristic

210

C₂

#10599715

yes

characteristic

	subgroup	G/H
center	C₂³	#4041
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1277	C₂
socle, prime radical	C₂³×C₅	#121
hypercenter	#1277	D₅
hypocenter	C₅	#10623
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#4041

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 ◁ #1277

N	H ≅ G/N	type
191. C₅	#10623	semidirect

H ≅ G/N

type

191. C₅

#10623

semidirect

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

subgroup	index
#10601278	2
#10600581	2
#10600555	2
C₅×#1277	2
#10598193	2
#10623	5

subgroup

index

Subgroups — #12004046

Conjugacy classes of subgroups

Complements

Maximal subgroups