Group #12003496 — subgroups

Subgroups — #12003496

Order 1280.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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	#10610800	C₂	1	1	yes	maximal, characteristic
2	C₅×#1848	C₂	1	1	yes	maximal, characteristic
3	#10601304	C₂	1	2	yes	maximal
4	#10600531	C₂	1	2	yes	maximal
5	#10598203	C₂	1	1	yes	maximal, characteristic
6	#65804	C₂²	1	2	yes	—
7	C₂×C₅×#108	—	2	2	no	—
8	C₅×#423	C₂²	1	1	yes	characteristic
9	C₅×#423	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#65426	—	2	2	no	—
12	#65213	C₂²	1	2	yes	—
13	#65140	—	2	2	no	—
14	#65138	—	2	2	no	—
15	#64706	—	2	2	no	—
16	#64702	—	2	2	no	—
17	#64676	—	2	4	no	—
18	#10534	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂×C₅×M₁₆	D₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	#4072	—	2	2	no	—
24	#4070	—	2	2	no	—
25	#4070	—	2	2	no	—
26	C₂×#681	—	2	2	no	—
27	C₂×Dic₂₀	—	2	2	no	—
28	#4016	—	2	2	no	—
29	#4015	D₄	1	2	yes	—
30	#4015	D₄	1	2	yes	—
31	C₄×Dic₁₀	—	2	2	no	—
32	C₄×Dic₁₀	—	2	2	no	—
33	C₅×#108	—	4	4	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	#3949	—	2	2	no	—
39	#3948	—	2	4	no	—
40	C₄⋊#168	—	2	2	no	—
41	C₄⋊#168	—	2	2	no	—
42	#3255	—	5	1	no	—
43	#1848	D₅	1	1	yes	characteristic
44	#1561	—	5	2	no	—
45	#1493	—	5	2	no	—
46	#1309	—	5	1	no	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	2	yes	—
49	C₂×Dic₁₀	—	2	2	no	—
50	C₂×Dic₁₀	—	2	2	no	—
51	C₅×M₁₆	—	4	2	no	—
52	C₅×M₁₆	—	4	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	4	1	no	—
58	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅	C₂×D₄	1	2	yes	—
60	C₄²×C₅	#41	1	1	yes	characteristic
61	C₂²⋊Dic₅	—	4	2	no	—
62	C₄⋊Dic₅	—	2	2	no	—
63	C₄⋊Dic₅	—	2	2	no	—
64	C₄⋊Dic₅	—	2	2	no	—
65	C₄⋊Dic₅	—	2	2	no	—
66	#683	—	4	2	no	—
67	#683	—	4	2	no	—
68	C₄×Dic₅	—	4	2	no	—
69	#681	—	8	2	no	—
70	C₂×#168	—	4	2	no	—
71	Dic₂₀	—	8	2	no	—
72	#533	—	5	2	no	—
73	#475	—	10	2	no	—
74	#470	—	10	2	no	—
75	#462	—	10	2	no	—
76	C₂×#108	—	2	2	no	—
77	#423	—	5	2	no	—
78	#423	C₂×D₅	1	1	yes	characteristic
79	#423	C₂×D₅	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	#332	—	10	2	no	—
82	#332	—	10	2	no	—
83	Q₈⋊C₈	—	10	4	no	—
84	C₂³×C₅	#124	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	#120	1	1	yes	characteristic
90	C₂×C₄×C₅	#121	1	2	yes	—
91	C₂×C₄×C₅	#121	1	1	yes	characteristic
92	C₂×C₄×C₅	#123	1	2	yes	—
93	C₂×Dic₅	—	4	2	no	—
94	C₂×Dic₅	—	4	2	no	—
95	Dic₁₀	—	4	2	no	—
96	Dic₁₀	—	4	2	no	—
97	Dic₁₀	—	4	2	no	—
98	Dic₁₀	—	4	2	no	—
99	C₅×C₈	—	2	4	no	—
100	C₅×C₈	—	4	2	no	—
101	C₅×C₈	—	8	1	no	—
102	#168	—	8	2	no	—
103	C₂×Q₁₆	—	10	2	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	10	2	no	—
106	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
107	C₂²×C₈	—	2	1	no	—
108	C₄⋊Q₈	—	5	2	no	—
109	C₄⋊Q₈	—	5	2	no	—
110	#125	—	10	2	no	—
111	C₂²⋊Q₈	—	10	2	no	—
112	C₂²⋊Q₈	—	10	2	no	—
113	C₄×Q₈	—	10	2	no	—
114	C₄×Q₈	—	10	2	no	—
115	#117	—	10	2	no	—
116	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
117	#108	—	4	4	no	—
118	#106	—	10	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	10	2	no	—
122	C₄⋊C₈	—	10	2	no	—
123	#103	—	10	4	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	D₂₀	1	2	yes	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	2	1	no	—
128	C₂²×C₅	#394	1	1	yes	characteristic
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	#453	1	1	yes	characteristic
132	C₄×C₅	#456	1	1	yes	characteristic
133	C₄×C₅	#471	1	2	yes	—
134	C₄×C₅	#473	1	2	yes	—
135	Dic₅	—	8	2	no	—
136	Dic₅	—	8	2	no	—
137	C₂×Q₈	—	10	2	no	—
138	C₂×Q₈	—	10	2	no	—
139	C₂²×C₄	#713	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
141	Q₁₆	—	40	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	40	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	20	2	no	—
151	C₄⋊C₄	—	10	2	no	—
152	C₄⋊C₄	—	10	2	no	—
153	C₄⋊C₄	—	10	2	no	—
154	C₄⋊C₄	—	10	2	no	—
155	C₄⋊C₄	—	20	2	no	—
156	C₄⋊C₄	—	20	2	no	—
157	C₂²⋊C₄	—	20	2	no	—
158	C₄²	—	20	2	no	—
159	C₄²	C₂×D₂₀	1	1	yes	characteristic
160	C₄²	#709	1	1	yes	characteristic
161	C₄²	D₄×D₅	1	2	yes	—
162	C₂×C₅	—	4	1	no	—
163	C₂×C₅	#2039	1	1	yes	characteristic
164	C₂×C₅	#2052	1	2	yes	—
165	C₂³	#4093	1	1	yes	characteristic
166	Q₈	—	20	2	no	—
167	Q₈	—	20	2	no	—
168	Q₈	—	20	2	no	—
169	Q₈	—	20	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	20	2	no	—
175	C₂×C₄	—	20	2	no	—
176	C₂×C₄	#4040	1	2	yes	—
177	C₂×C₄	#4041	1	2	yes	—
178	C₂×C₄	#4076	1	1	yes	characteristic
179	C₂×C₄	#4083	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	4	2	no	—
182	C₈	—	8	1	no	—
183	C₈	—	40	2	no	—
184	C₅	#10534	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	2	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#65280	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	1	no	—
190	C₄	—	40	2	no	—
191	C₄	—	40	2	no	—
192	C₄	#65185	1	2	yes	—
193	C₄	#65187	1	2	yes	—
194	C₄	#65366	1	1	yes	characteristic
195	C₄	#65384	1	1	yes	characteristic
196	C₂	—	4	1	no	—
197	C₂	#10600386	1	1	yes	characteristic
198	C₂	#10601509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610800

C₂

yes

maximal, characteristic

C₅×#1848

C₂

yes

maximal, characteristic

#10601304

C₂

yes

maximal

#10600531

C₂

yes

maximal

#10598203

C₂

yes

maximal, characteristic

#65804

C₂²

yes

—

C₂×C₅×#108

—

C₅×#423

C₂²

yes

characteristic

C₅×#423

C₂²

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65426

—

#65213

C₂²

yes

—

—

—

—

—

—

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

D₄

yes

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

—

C₂×Dic₂₀

—

#4016

—

#4015

D₄

yes

—

#4015

D₄

yes

—

C₄×Dic₁₀

—

C₄×Dic₁₀

—

C₅×#108

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

D₅

yes

—

—

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

#681

—

C₂×#168

—

Dic₂₀

—

—

—

—

—

—

—

C₂×D₅

yes

characteristic

#423

C₂×D₅

yes

characteristic

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#332

—

#332

—

Q₈⋊C₈

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#120

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

100

C₅×C₈

—

101

C₅×C₈

—

102

#168

—

103

C₂×Q₁₆

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂×M₁₆

C₂²⋊D₅

yes

—

107

C₂²×C₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

112

C₂²⋊Q₈

—

113

C₄×Q₈

—

114

C₄×Q₈

—

115

#117

—

116

C₂×C₄²

C₂²×D₅

yes

characteristic

117

#108

—

118

#106

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#103

—

124

C₄×C₈

—

125

C₄×C₈

D₂₀

yes

—

126

C₂²×C₅

—

127

C₂²×C₅

—

128

C₂²×C₅

#394

yes

characteristic

129

C₄×C₅

—

130

C₄×C₅

—

131

C₄×C₅

#453

yes

characteristic

132

C₄×C₅

#456

yes

characteristic

133

C₄×C₅

#471

yes

—

134

C₄×C₅

#473

yes

—

135

Dic₅

—

136

Dic₅

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

#713

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

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

C₂²⋊C₄

—

158

C₄²

—

159

C₄²

C₂×D₂₀

yes

characteristic

160

C₄²

#709

yes

characteristic

161

C₄²

D₄×D₅

yes

—

162

C₂×C₅

—

163

C₂×C₅

#2039

yes

characteristic

164

C₂×C₅

#2052

yes

—

165

C₂³

#4093

yes

characteristic

166

Q₈

—

167

Q₈

—

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

#4040

yes

—

177

C₂×C₄

#4041

yes

—

178

C₂×C₄

#4076

yes

characteristic

179

C₂×C₄

#4083

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₅

#10534

yes

characteristic, complemented, Sylow

185

C₂²

—

186

C₂²

—

187

C₂²

#65280

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#65185

yes

—

193

C₄

#65187

yes

—

194

C₄

#65366

yes

characteristic

195

C₄

#65384

yes

characteristic

196

C₂

—

197

C₂

#10600386

yes

characteristic

198

C₂

#10601509

yes

—

	subgroup	G/H
center	C₂²	#65280
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1848	C₂
socle	C₂²×C₅	#394
prime radical	C₂³×C₅	#124
hypercenter	#1848	D₅
hypocenter	C₅	#10534
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#65280

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
184. C₅	#10534	semidirect

H ≅ G/N

type

184. C₅

#10534

semidirect

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

subgroup	index
#10610800	2
C₅×#1848	2
#10601304	2
#10600531	2
#10598203	2
#10534	5

subgroup

index

Subgroups — #12003496

Conjugacy classes of subgroups

Complements

Maximal subgroups