Group #12003501 — subgroups

Subgroups — #12003501

Order 1280.

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

normal subgroups only

This group has 342 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×#1837	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	C₅×#423	C₂²	1	1	yes	characteristic
8	C₅×#423	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	#65426	—	2	2	no	—
11	#65213	C₂²	1	2	yes	—
12	#65140	—	2	2	no	—
13	#65138	—	2	2	no	—
14	C₅×#341	—	2	2	no	—
15	#64706	—	2	2	no	—
16	#64702	—	2	2	no	—
17	#64676	—	2	4	no	—
18	#10535	—	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×C₄⋊C₈	—	2	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	#3949	—	2	2	no	—
38	#3948	—	2	4	no	—
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	—	2	2	no	—
41	#3255	—	5	1	no	—
42	#1837	D₅	1	1	yes	characteristic
43	#1561	—	5	2	no	—
44	#1493	—	5	2	no	—
45	#1309	—	5	1	no	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	#41	1	2	yes	—
48	C₂×Dic₁₀	—	2	2	no	—
49	C₂×Dic₁₀	—	2	2	no	—
50	C₅×M₁₆	—	4	2	no	—
51	C₅×M₁₆	—	4	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	4	1	no	—
55	C₂×C₅×C₈	—	4	1	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄²×C₅	#41	1	1	yes	characteristic
60	C₂²⋊Dic₅	—	4	2	no	—
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	#683	—	4	2	no	—
66	#683	—	4	2	no	—
67	C₄×Dic₅	—	4	2	no	—
68	#681	—	8	2	no	—
69	C₂×#168	—	4	2	no	—
70	Dic₂₀	—	8	2	no	—
71	#533	—	5	2	no	—
72	#475	—	10	2	no	—
73	#470	—	10	2	no	—
74	#462	—	10	2	no	—
75	#423	—	5	2	no	—
76	#423	C₂×D₅	1	1	yes	characteristic
77	#423	C₂×D₅	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
79	#341	—	2	2	no	—
80	#332	—	10	2	no	—
81	#332	—	10	2	no	—
82	Q₈⋊C₈	—	10	4	no	—
83	C₂³×C₅	#124	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	#120	1	1	yes	characteristic
89	C₂×C₄×C₅	#121	1	2	yes	—
90	C₂×C₄×C₅	#121	1	1	yes	characteristic
91	C₂×C₄×C₅	#123	1	2	yes	—
92	C₂×Dic₅	—	4	2	no	—
93	C₂×Dic₅	—	4	2	no	—
94	Dic₁₀	—	4	2	no	—
95	Dic₁₀	—	4	2	no	—
96	Dic₁₀	—	4	2	no	—
97	Dic₁₀	—	4	2	no	—
98	C₅×C₈	—	4	2	no	—
99	C₅×C₈	—	4	2	no	—
100	C₅×C₈	—	8	1	no	—
101	#168	—	8	2	no	—
102	C₂×Q₁₆	—	10	2	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	10	2	no	—
105	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
106	C₂²×C₈	—	2	1	no	—
107	C₄⋊Q₈	—	5	2	no	—
108	C₄⋊Q₈	—	5	2	no	—
109	#125	—	10	2	no	—
110	C₂²⋊Q₈	—	10	2	no	—
111	C₂²⋊Q₈	—	10	2	no	—
112	C₄×Q₈	—	10	2	no	—
113	C₄×Q₈	—	10	2	no	—
114	#117	—	10	2	no	—
115	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
116	#106	—	10	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	10	2	no	—
120	C₄⋊C₈	—	10	2	no	—
121	#103	—	10	4	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	D₂₀	1	2	yes	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	—	2	1	no	—
126	C₂²×C₅	#394	1	1	yes	characteristic
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	—	2	1	no	—
129	C₄×C₅	#454	1	1	yes	characteristic
130	C₄×C₅	#455	1	1	yes	characteristic
131	C₄×C₅	#471	1	2	yes	—
132	C₄×C₅	#473	1	2	yes	—
133	Dic₅	—	8	2	no	—
134	Dic₅	—	8	2	no	—
135	C₂×Q₈	—	10	2	no	—
136	C₂×Q₈	—	10	2	no	—
137	C₂²×C₄	#713	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
139	Q₁₆	—	40	2	no	—
140	M₁₆	—	4	2	no	—
141	M₁₆	—	4	2	no	—
142	M₁₆	—	40	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	20	2	no	—
149	C₄⋊C₄	—	10	2	no	—
150	C₄⋊C₄	—	10	2	no	—
151	C₄⋊C₄	—	10	2	no	—
152	C₄⋊C₄	—	10	2	no	—
153	C₄⋊C₄	—	20	2	no	—
154	C₄⋊C₄	—	20	2	no	—
155	C₂²⋊C₄	—	20	2	no	—
156	C₄²	—	20	2	no	—
157	C₄²	C₂×D₂₀	1	1	yes	characteristic
158	C₄²	#709	1	1	yes	characteristic
159	C₄²	D₄×D₅	1	2	yes	—
160	C₂×C₅	—	4	1	no	—
161	C₂×C₅	#2040	1	1	yes	characteristic
162	C₂×C₅	#2052	1	2	yes	—
163	C₂³	#4093	1	1	yes	characteristic
164	Q₈	—	20	2	no	—
165	Q₈	—	20	2	no	—
166	Q₈	—	20	2	no	—
167	Q₈	—	20	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	20	2	no	—
173	C₂×C₄	—	20	2	no	—
174	C₂×C₄	#4040	1	2	yes	—
175	C₂×C₄	#4041	1	2	yes	—
176	C₂×C₄	#4076	1	1	yes	characteristic
177	C₂×C₄	#4083	1	1	yes	characteristic
178	C₈	—	4	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	8	1	no	—
181	C₈	—	40	2	no	—
182	C₅	#10535	1	1	yes	characteristic, complemented, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	#65280	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	40	2	no	—
189	C₄	—	40	2	no	—
190	C₄	#65185	1	2	yes	—
191	C₄	#65187	1	2	yes	—
192	C₄	#65350	1	1	yes	characteristic
193	C₄	#65351	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	#10600356	1	1	yes	characteristic
196	C₂	#10601509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610800

C₂

yes

maximal, characteristic

C₅×#1837

C₂

yes

maximal, characteristic

#10601304

C₂

yes

maximal

#10600531

C₂

yes

maximal

#10598203

C₂

yes

maximal, characteristic

#65804

C₂²

yes

—

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×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

—

—

—

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

—

C₅×C₈

—

100

C₅×C₈

—

101

#168

—

102

C₂×Q₁₆

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂×M₁₆

C₂²⋊D₅

yes

—

106

C₂²×C₈

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

#125

—

110

C₂²⋊Q₈

—

111

C₂²⋊Q₈

—

112

C₄×Q₈

—

113

C₄×Q₈

—

114

#117

—

115

C₂×C₄²

C₂²×D₅

yes

characteristic

116

#106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

122

C₄×C₈

—

123

C₄×C₈

D₂₀

yes

—

124

C₂²×C₅

—

125

C₂²×C₅

—

126

C₂²×C₅

#394

yes

characteristic

127

C₄×C₅

—

128

C₄×C₅

—

129

C₄×C₅

#454

yes

characteristic

130

C₄×C₅

#455

yes

characteristic

131

C₄×C₅

#471

yes

—

132

C₄×C₅

#473

yes

—

133

Dic₅

—

134

Dic₅

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

#713

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

139

Q₁₆

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

C₂×D₂₀

yes

characteristic

158

C₄²

#709

yes

characteristic

159

C₄²

D₄×D₅

yes

—

160

C₂×C₅

—

161

C₂×C₅

#2040

yes

characteristic

162

C₂×C₅

#2052

yes

—

163

C₂³

#4093

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

#4040

yes

—

175

C₂×C₄

#4041

yes

—

176

C₂×C₄

#4076

yes

characteristic

177

C₂×C₄

#4083

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₅

#10535

yes

characteristic, complemented, Sylow

183

C₂²

—

184

C₂²

—

185

C₂²

#65280

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

#65185

yes

—

191

C₄

#65187

yes

—

192

C₄

#65350

yes

characteristic

193

C₄

#65351

yes

characteristic

194

C₂

—

195

C₂

#10600356

yes

characteristic

196

C₂

#10601509

yes

—

	subgroup	G/H
center	C₂²	#65280
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1837	C₂
socle	C₂²×C₅	#394
prime radical	C₂³×C₅	#124
hypercenter	#1837	D₅
hypocenter	C₅	#10535
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

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 ◁ #1837

N	H ≅ G/N	type
182. C₅	#10535	semidirect

H ≅ G/N

type

182. C₅

#10535

semidirect

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

subgroup	index
#10610800	2
C₅×#1837	2
#10601304	2
#10600531	2
#10598203	2
#10535	5

subgroup

index

Subgroups — #12003501

Conjugacy classes of subgroups

Complements

Maximal subgroups