Group #12003444 — subgroups

Subgroups — #12003444

Order 1280.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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	#10610795	C₂	1	1	yes	maximal, characteristic
2	#10601612	C₂	1	2	yes	maximal, complemented
3	#10601599	C₂	1	2	yes	maximal, complemented
4	#10601310	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1307	C₂	1	1	yes	maximal, characteristic, complemented
6	#65811	C₂²	1	2	yes	—
7	C₅×#432	C₂²	1	2	yes	—
8	C₅×#423	C₂²	1	1	yes	characteristic
9	#65424	—	2	2	no	—
10	#65213	C₂²	1	1	yes	characteristic
11	C₂×#3935	C₂²	1	1	yes	characteristic
12	#65132	—	2	2	no	—
13	#65131	—	2	2	no	—
14	#64778	—	2	2	no	—
15	#64705	—	2	2	no	—
16	#64702	—	2	2	no	—
17	#64684	—	2	4	no	—
18	#10525	—	5	1	no	maximal, Sylow
19	C₂×#709	—	2	2	no	—
20	C₂×C₅×M₁₆	D₄	1	2	yes	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	#4076	—	2	2	no	—
23	#4076	—	2	2	no	—
24	C₂×#681	—	2	1	no	—
25	C₂²×#168	—	2	1	no	—
26	C₂×D₄₀	—	2	2	no	—
27	#4021	—	2	2	no	—
28	#4020	D₄	1	2	yes	—
29	C₄×D₂₀	—	2	2	no	—
30	C₄×D₂₀	—	2	2	no	—
31	#4015	D₄	1	2	yes	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×C₂²⋊C₈	—	2	2	no	—
35	C₅×#97	—	2	2	no	—
36	#3953	—	2	4	no	—
37	#3949	—	2	2	no	—
38	C₄⋊#168	—	2	2	no	—
39	#3935	—	2	1	no	—
40	#3935	D₄	1	2	yes	—
41	#3253	—	5	1	no	—
42	#1838	—	5	1	no	—
43	#1557	—	5	2	no	—
44	#1530	—	5	2	no	—
45	#1307	D₅	1	1	yes	characteristic, complemented
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	#41	1	2	yes	—
48	C₂×C₂²⋊D₅	—	4	2	no	—
49	#709	—	8	4	no	—
50	C₂×D₂₀	—	2	2	no	—
51	C₂×D₂₀	—	2	2	no	—
52	C₂×D₂₀	—	2	2	no	—
53	C₂×C₄×D₅	—	4	2	no	—
54	C₂×Dic₁₀	—	2	2	no	—
55	C₅×M₁₆	—	4	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	4	2	no	—
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅	#41	1	1	yes	characteristic
61	#685	—	4	2	no	—
62	#685	—	4	2	no	—
63	C₄⋊Dic₅	—	2	2	no	—
64	C₄⋊Dic₅	—	2	2	no	—
65	#681	—	4	2	no	—
66	C₂×#168	—	2	2	no	—
67	C₂×#168	—	4	1	no	—
68	C₂×#168	—	4	1	no	—
69	C₂×#168	—	4	1	no	—
70	D₄₀	—	8	2	no	—
71	#532	—	5	2	no	—
72	#476	—	10	2	no	—
73	#469	—	10	2	no	—
74	#459	—	10	2	no	—
75	#432	C₂×D₅	1	2	yes	—
76	#423	—	5	1	no	—
77	#423	C₂×D₅	1	1	yes	characteristic
78	C₂×#97	—	5	1	no	—
79	#341	—	10	2	no	—
80	#332	—	10	2	no	—
81	#331	—	10	2	no	—
82	#329	—	10	4	no	—
83	C₂³×C₅	#124	1	1	yes	characteristic
84	C₂²×D₅	—	4	2	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	4	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	C₂²⋊D₅	—	8	4	no	—
93	C₂×Dic₅	—	4	2	no	—
94	D₂₀	—	4	2	no	—
95	D₂₀	—	4	2	no	—
96	D₂₀	—	8	2	no	—
97	D₂₀	—	8	2	no	—
98	C₄×D₅	—	8	4	no	—
99	Dic₁₀	—	8	2	no	—
100	C₅×C₈	—	4	2	no	—
101	C₅×C₈	—	8	2	no	—
102	#168	—	4	2	no	—
103	#168	—	8	1	no	—
104	C₂×#41	—	10	2	no	—
105	C₂×D₈	—	10	2	no	—
106	C₂×M₁₆	—	10	1	no	—
107	C₂×M₁₆	D₂₀	1	2	yes	—
108	C₂²×C₈	—	10	1	no	—
109	C₄⋊Q₈	—	5	2	no	—
110	#127	—	5	2	no	—
111	#124	—	10	2	no	—
112	#121	—	10	2	no	—
113	#121	—	10	2	no	—
114	C₄×D₄	—	10	2	no	—
115	C₄×D₄	—	10	2	no	—
116	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
117	#106	—	10	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	10	2	no	—
121	#102	—	10	4	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	5	2	no	—
125	#97	—	10	1	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	2	1	no	—
128	C₂²×C₅	#394	1	1	yes	characteristic
129	C₂×D₅	—	8	2	no	—
130	C₂×D₅	—	8	2	no	—
131	C₂×D₅	—	8	2	no	—
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	#454	1	1	yes	characteristic
136	C₄×C₅	#455	1	1	yes	characteristic
137	Dic₅	—	8	2	no	—
138	#41	—	40	4	no	—
139	C₂×Q₈	—	10	2	no	—
140	C₂×D₄	—	10	2	no	—
141	C₂×D₄	—	10	2	no	—
142	C₂×D₄	—	10	2	no	—
143	C₂×D₄	—	20	2	no	—
144	C₂²×C₄	—	20	2	no	—
145	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
146	C₂²×C₄	#709	1	2	yes	—
147	D₈	—	40	2	no	—
148	M₁₆	—	4	2	no	—
149	M₁₆	—	20	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	10	2	no	—
153	C₂×C₈	—	20	1	no	—
154	C₂×C₈	—	20	1	no	—
155	C₂×C₈	—	20	1	no	—
156	C₄⋊C₄	—	10	2	no	—
157	C₄⋊C₄	—	10	2	no	—
158	C₂²⋊C₄	—	20	2	no	—
159	C₂²⋊C₄	—	20	2	no	—
160	C₄²	D₄×D₅	1	2	yes	—
161	C₄²	D₄×D₅	1	1	yes	characteristic
162	C₄²	#713	1	1	yes	characteristic
163	C₂×C₅	—	4	1	no	—
164	C₂×C₅	#2040	1	1	yes	characteristic
165	C₂×C₅	#2048	1	2	yes	—
166	D₅	—	16	2	no	—
167	C₂³	—	20	2	no	—
168	C₂³	#4021	1	1	yes	characteristic
169	Q₈	—	40	2	no	—
170	D₄	—	20	2	no	—
171	D₄	—	20	2	no	—
172	D₄	—	40	2	no	—
173	D₄	—	40	2	no	—
174	D₄	—	40	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	20	2	no	—
179	C₂×C₄	—	40	4	no	—
180	C₂×C₄	#4028	1	1	yes	characteristic
181	C₂×C₄	#4030	1	2	yes	—
182	C₂×C₄	#4040	1	2	yes	—
183	C₂×C₄	#4041	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	8	2	no	—
186	C₈	—	20	2	no	—
187	C₈	—	40	1	no	—
188	C₅	#10525	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	40	2	no	—
192	C₂²	—	40	2	no	—
193	C₂²	—	40	2	no	—
194	C₂²	#64961	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	40	2	no	—
199	C₄	#65035	1	1	yes	characteristic
200	C₄	#65042	1	1	yes	characteristic
201	C₂	—	4	1	no	—
202	C₂	—	80	2	no	—
203	C₂	#10599770	1	1	yes	characteristic
204	C₂	#10599811	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610795

C₂

yes

maximal, characteristic

#10601612

C₂

yes

maximal, complemented

#10601599

C₂

yes

maximal, complemented

#10601310

C₂

yes

maximal, characteristic, complemented

C₅×#1307

C₂

yes

maximal, characteristic, complemented

#65811

C₂²

yes

—

C₅×#432

C₂²

yes

—

C₅×#423

C₂²

yes

characteristic

#65424

—

#65213

C₂²

yes

characteristic

C₂×#3935

C₂²

yes

—

—

—

—

—

—

—

—

C₂×C₅×M₁₆

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₂²×#168

—

C₂×D₄₀

—

#4021

—

#4020

D₄

yes

—

C₄×D₂₀

—

C₄×D₂₀

—

#4015

D₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

—

—

—

—

D₄

yes

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×C₂²⋊D₅

—

#709

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

C₄²×C₅

yes

—

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

D₄₀

—

—

—

—

—

C₂×D₅

yes

—

#423

—

#423

C₂×D₅

yes

characteristic

C₂×#97

—

—

—

—

—

C₂³×C₅

#124

yes

characteristic

C₂²×D₅

—

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

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

100

C₅×C₈

—

101

C₅×C₈

—

102

#168

—

103

#168

—

104

C₂×#41

—

105

C₂×D₈

—

106

C₂×M₁₆

—

107

C₂×M₁₆

D₂₀

yes

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#127

—

111

#124

—

112

#121

—

113

#121

—

114

C₄×D₄

—

115

C₄×D₄

—

116

C₂×C₄²

C₂²×D₅

yes

characteristic

117

#106

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#102

—

122

C₂²⋊C₈

—

123

#97

—

124

#97

—

125

#97

—

126

C₂²×C₅

—

127

C₂²×C₅

—

128

C₂²×C₅

#394

yes

characteristic

129

C₂×D₅

—

130

C₂×D₅

—

131

C₂×D₅

—

132

C₄×C₅

—

133

C₄×C₅

—

134

C₄×C₅

—

135

C₄×C₅

#454

yes

characteristic

136

C₄×C₅

#455

yes

characteristic

137

Dic₅

—

138

#41

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₂₀

yes

characteristic

146

C₂²×C₄

#709

yes

—

147

D₈

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

D₄×D₅

yes

—

161

C₄²

D₄×D₅

yes

characteristic

162

C₄²

#713

yes

characteristic

163

C₂×C₅

—

164

C₂×C₅

#2040

yes

characteristic

165

C₂×C₅

#2048

yes

—

166

D₅

—

167

C₂³

—

168

C₂³

#4021

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#4028

yes

characteristic

181

C₂×C₄

#4030

yes

—

182

C₂×C₄

#4040

yes

—

183

C₂×C₄

#4041

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₅

#10525

yes

characteristic, complemented, Sylow

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#64961

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#65035

yes

characteristic

200

C₄

#65042

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#10599770

yes

characteristic

204

C₂

#10599811

yes

—

	subgroup	G/H
center	C₂²	#64961
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1307	C₂
socle	C₂²×C₅	#394
prime radical	#10610795	C₂
hypercenter	#1307	D₅
hypocenter	C₅	#10525
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#64961

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
2. #10601612	C₂	semidirect
3. #10601599	C₂	semidirect
4. #10601310	C₂	semidirect
5. C₅×#1307	C₂	semidirect
45. #1307	D₅	semidirect
188. C₅	#10525	semidirect

H ≅ G/N

type

2. #10601612

C₂

semidirect

3. #10601599

C₂

semidirect

4. #10601310

C₂

semidirect

5. C₅×#1307

C₂

semidirect

45. #1307

D₅

semidirect

188. C₅

#10525

semidirect

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

subgroup	index
#10610795	2
#10601612	2
#10601599	2
#10601310	2
C₅×#1307	2
#10525	5

subgroup

index

Subgroups — #12003444

Conjugacy classes of subgroups

Complements

Maximal subgroups