Group #13103779 — subgroups

Subgroups — #13103779

Order 1600.

447 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂²×#6036	C₂	1	1	yes	maximal, characteristic
2	C₂×#87853	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#704	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716754	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅⋊D₅	C₄	1	1	yes	characteristic
6	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
7	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
8	C₂×#6036	—	2	2	no	—
9	C₂×#6036	—	2	2	no	—
10	C₂×#6036	C₄	1	2	yes	—
11	C₂×#6036	C₂²	1	2	yes	—
12	#87853	—	2	2	no	—
13	C₂×#6032	C₂²	1	2	yes	complemented
14	C₅×#704	—	2	2	no	—
15	C₂×C₅×#170	C₂²	1	2	yes	complemented
16	#65755	—	5	1	no	maximal
17	#65415	—	5	1	no	maximal
18	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅⋊D₅	—	2	2	no	—
20	C₂²×C₅⋊D₅	—	2	4	no	—
21	C₂²×C₅⋊D₅	C₂×C₄	1	2	yes	—
22	C₂×C₅×Dic₅	—	2	2	no	—
23	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×Dic₅	D₄	1	2	yes	—
26	C₂×C₅×Dic₅	D₄	1	2	yes	—
27	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
28	#6036	—	2	4	no	—
29	#6036	—	2	2	no	—
30	#6036	—	2	2	no	—
31	#6032	—	2	2	no	—
32	C₅×#170	—	2	2	no	—
33	C₂²×C₄×D₅	—	5	1	no	—
34	C₂²×C₄×D₅	—	5	1	no	—
35	C₂×#704	—	5	1	no	—
36	C₂×#704	D₅	1	1	yes	characteristic, complemented
37	C₂×C₅×M₁₆	—	5	1	no	—
38	C₂×#681	—	5	1	no	—
39	#4003	—	5	4	no	—
40	#3952	—	5	4	no	—
41	C₂²×C₅²	—	2	2	no	—
42	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₅⋊D₅	—	2	8	no	—
45	C₂×C₅⋊D₅	—	2	2	no	—
46	C₂×C₅⋊D₅	—	2	2	no	—
47	C₂×C₅⋊D₅	M₁₆	1	4	yes	complemented
48	C₅×Dic₅	—	2	2	no	—
49	C₅×Dic₅	C₂²⋊C₄	1	2	yes	—
50	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
51	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
52	C₂³×D₅	—	5	1	no	—
53	C₂³×D₅	—	5	1	no	—
54	C₂³×D₅	—	20	1	no	—
55	C₂²×C₄×C₅	—	5	1	no	—
56	C₂²×C₄×C₅	—	5	1	no	—
57	C₂²×Dic₅	AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
59	C₂×C₄×D₅	—	5	2	no	—
60	C₂×C₄×D₅	—	5	2	no	—
61	C₂×C₄×D₅	—	5	2	no	—
62	C₂×C₄×D₅	—	5	2	no	—
63	C₂×C₄×D₅	—	10	2	no	—
64	C₂×C₄×D₅	—	10	2	no	—
65	C₂×C₄×D₅	—	10	2	no	—
66	C₂×C₄×D₅	—	10	2	no	—
67	#704	—	2	2	no	—
68	#704	—	10	2	no	—
69	C₂×#170	—	5	2	no	—
70	C₂×#170	C₂×D₅	1	2	yes	complemented
71	C₅×M₁₆	—	10	2	no	—
72	C₂×C₅×C₈	—	5	2	no	—
73	#681	—	10	2	no	—
74	C₂×#168	—	5	2	no	—
75	#407	—	25	1	no	Sylow
76	C₂×C₅²	—	2	2	no	—
77	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅²	C₂×M₁₆	1	2	yes	—
79	C₅⋊D₅	—	2	4	no	—
80	C₂³×C₅	—	4	1	no	—
81	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
82	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂²×D₅	—	5	2	no	—
84	C₂²×D₅	—	5	2	no	—
85	C₂²×D₅	—	10	4	no	—
86	C₂²×D₅	—	10	4	no	—
87	C₂²×D₅	—	10	2	no	—
88	C₂²×D₅	—	10	2	no	—
89	C₂²×D₅	—	20	2	no	—
90	C₂²×D₅	—	20	8	no	—
91	C₂²×D₅	—	20	4	no	—
92	C₂×C₄×C₅	—	5	2	no	—
93	C₂×C₄×C₅	—	5	2	no	—
94	C₂×C₄×C₅	—	5	1	no	—
95	C₂×C₄×C₅	—	5	1	no	—
96	C₂×C₄×C₅	—	5	2	no	—
97	C₂×C₄×C₅	—	10	2	no	—
98	C₂×Dic₅	—	2	2	no	—
99	C₂×Dic₅	C₄×D₅	1	1	yes	characteristic
100	C₂×Dic₅	D₂₀	1	2	yes	—
101	C₂×Dic₅	C₂²⋊D₅	1	2	yes	—
102	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
103	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
104	C₄×D₅	—	10	4	no	—
105	C₄×D₅	—	10	2	no	—
106	C₄×D₅	—	10	2	no	—
107	C₄×D₅	—	10	4	no	—
108	C₄×D₅	—	10	2	no	—
109	C₄×D₅	—	10	2	no	—
110	#170	—	2	2	no	—
111	#170	—	10	2	no	—
112	C₅×C₈	—	10	2	no	—
113	#168	—	10	2	no	—
114	C₂³×C₄	—	25	1	no	—
115	C₂×M₁₆	—	5	1	no	—
116	C₂×M₁₆	—	25	1	no	—
117	C₂²⋊C₈	—	25	4	no	—
118	C₅²	#407	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	—	2	2	no	—
121	C₂²×C₅	—	4	4	no	—
122	C₂²×C₅	—	4	2	no	—
123	C₂²×C₅	—	4	1	no	—
124	C₂²×C₅	#685	1	2	yes	—
125	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
126	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
127	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₂×D₅	—	5	4	no	—
129	C₂×D₅	—	5	4	no	—
130	C₂×D₅	—	10	8	no	—
131	C₂×D₅	—	10	8	no	—
132	C₂×D₅	—	10	2	no	—
133	C₂×D₅	—	10	2	no	—
134	C₂×D₅	—	10	2	no	—
135	C₂×D₅	—	10	2	no	—
136	C₂×D₅	—	20	4	no	—
137	C₂×D₅	—	20	16	no	—
138	C₂×D₅	—	20	4	no	—
139	C₂×D₅	—	20	4	no	—
140	C₄×C₅	—	5	2	no	—
141	C₄×C₅	—	5	1	no	—
142	C₄×C₅	—	5	1	no	—
143	C₄×C₅	—	10	2	no	—
144	Dic₅	—	2	2	no	—
145	Dic₅	#685	1	2	yes	—
146	Dic₅	C₂×D₂₀	1	1	yes	characteristic
147	Dic₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
148	C₂⁴	—	25	1	no	—
149	C₂²×C₄	—	5	1	no	—
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	25	2	no	—
152	C₂²×C₄	—	25	2	no	—
153	C₂²×C₄	—	50	2	no	—
154	C₂²×C₄	—	50	2	no	—
155	M₁₆	—	10	2	no	—
156	M₁₆	—	50	2	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	25	2	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	4	4	no	—
162	C₂×C₅	—	4	1	no	—
163	C₂×C₅	—	4	2	no	—
164	C₂×C₅	D₅×M₁₆	1	2	yes	—
165	C₂×C₅	C₂×#685	1	1	yes	characteristic
166	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
167	C₂×C₅	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
168	D₅	—	10	4	no	—
169	D₅	—	10	4	no	—
170	D₅	—	20	8	no	—
171	C₂³	—	25	2	no	—
172	C₂³	—	50	4	no	—
173	C₂³	—	50	2	no	—
174	C₂³	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₂×C₄	—	5	2	no	—
176	C₂×C₄	—	5	2	no	—
177	C₂×C₄	—	5	1	no	—
178	C₂×C₄	—	5	1	no	—
179	C₂×C₄	—	5	2	no	—
180	C₂×C₄	—	10	2	no	—
181	C₂×C₄	—	50	2	no	—
182	C₂×C₄	—	50	2	no	—
183	C₂×C₄	—	50	4	no	—
184	C₈	—	10	2	no	—
185	C₈	—	50	2	no	—
186	C₅	—	4	1	no	—
187	C₅	#65415	1	1	yes	characteristic, complemented
188	C₅	#65755	1	1	yes	characteristic, complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	25	4	no	—
191	C₂²	—	50	8	no	—
192	C₂²	—	50	2	no	—
193	C₂²	—	50	2	no	—
194	C₂²	(C₂×AGL(1,𝔽₅))⋊D₅	1	2	yes	—
195	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₄	—	5	1	no	—
197	C₄	—	5	1	no	—
198	C₄	—	5	2	no	—
199	C₄	—	10	2	no	—
200	C₂	—	2	2	no	—
201	C₂	—	50	4	no	—
202	C₂	C₂×(C₂×AGL(1,𝔽₅))⋊D₅	1	1	yes	characteristic
203	C₂	#11717347	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6036

C₂

yes

maximal, characteristic

C₂×#87853

C₂

yes

maximal, characteristic, complemented

C₂×C₅×#704

C₂

yes

maximal, characteristic, complemented

#11716754

C₂

yes

maximal, complemented

C₂³×C₅⋊D₅

C₄

yes

characteristic

C₂²×C₅×Dic₅

C₄

yes

characteristic

C₂²×C₅×Dic₅

C₂²

yes

—

—

C₄

yes

—

C₂×#6036

C₂²

yes

—

#87853

—

C₂×#6032

C₂²

yes

complemented

C₅×#704

—

C₂×C₅×#170

C₂²

yes

—

—

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

D₄

yes

—

C₂×C₅×Dic₅

D₄

yes

—

C₂×C₅×Dic₅

C₂³

yes

—

—

—

—

—

C₂²×C₄×D₅

—

C₂²×C₄×D₅

—

C₂×#704

—

C₂×#704

D₅

yes

characteristic, complemented

C₂×C₅×M₁₆

—

C₂×#681

—

#4003

—

#3952

—

C₂²×C₅²

—

C₂²×C₅²

C₂²⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

M₁₆

yes

complemented

C₅×Dic₅

—

C₅×Dic₅

C₂²⋊C₄

yes

—

C₅×Dic₅

C₂×D₄

yes

characteristic

C₅×Dic₅

C₂×D₄

yes

characteristic

C₂³×D₅

—

C₂³×D₅

—

C₂³×D₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×Dic₅

AGL(1,𝔽₅)

yes

characteristic

C₂²×Dic₅

C₂×D₅

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

—

—

—

C₂×D₅

yes

complemented

C₅×M₁₆

—

C₂×C₅×C₈

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

—

C₅⋊D₅

—

C₂³×C₅

—

C₂³×C₅

C₄×D₅

yes

characteristic

C₂³×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

C₄×D₅

yes

characteristic

100

C₂×Dic₅

D₂₀

yes

—

101

C₂×Dic₅

C₂²⋊D₅

yes

—

102

C₂×Dic₅

C₂×AGL(1,𝔽₅)

yes

—

103

C₂×Dic₅

C₂²×D₅

yes

characteristic

104

C₄×D₅

—

105

C₄×D₅

—

106

C₄×D₅

—

107

C₄×D₅

—

108

C₄×D₅

—

109

C₄×D₅

—

110

#170

—

111

#170

—

112

C₅×C₈

—

113

#168

—

114

C₂³×C₄

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂²⋊C₈

—

118

C₅²

#407

yes

characteristic, complemented, Sylow

119

C₂²×C₅

—

120

C₂²×C₅

—

121

C₂²×C₅

—

122

C₂²×C₅

—

123

C₂²×C₅

—

124

C₂²×C₅

#685

yes

—

125

C₂²×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

126

C₂²×C₅

C₂×C₄×D₅

yes

characteristic

127

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

128

C₂×D₅

—

129

C₂×D₅

—

130

C₂×D₅

—

131

C₂×D₅

—

132

C₂×D₅

—

133

C₂×D₅

—

134

C₂×D₅

—

135

C₂×D₅

—

136

C₂×D₅

—

137

C₂×D₅

—

138

C₂×D₅

—

139

C₂×D₅

—

140

C₄×C₅

—

141

C₄×C₅

—

142

C₄×C₅

—

143

C₄×C₅

—

144

Dic₅

—

145

Dic₅

#685

yes

—

146

Dic₅

C₂×D₂₀

yes

characteristic

147

Dic₅

C₂×C₂²⋊D₅

yes

characteristic

148

C₂⁴

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

M₁₆

—

156

M₁₆

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₅

—

160

C₂×C₅

—

161

C₂×C₅

—

162

C₂×C₅

—

163

C₂×C₅

—

164

C₂×C₅

D₅×M₁₆

yes

—

165

C₂×C₅

C₂×#685

yes

characteristic

166

C₂×C₅

D₅⋊M₁₆

yes

—

167

C₂×C₅

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

168

D₅

—

169

D₅

—

170

D₅

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

D₅×AGL(1,𝔽₅)

yes

characteristic

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

C₈

—

185

C₈

—

186

C₅

—

187

C₅

#65415

yes

characteristic, complemented

188

C₅

#65755

yes

characteristic, complemented

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

(C₂×AGL(1,𝔽₅))⋊D₅

yes

—

195

C₂²

C₂×D₅×AGL(1,𝔽₅)

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₂

—

201

C₂

—

202

C₂

C₂×(C₂×AGL(1,𝔽₅))⋊D₅

yes

characteristic

203

C₂

#11717347

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂³×C₅²	C₂×C₄
prime radical	C₂³×C₅⋊D₅	C₄
cosocle	C₂×C₅×Dic₅	C₂³
hypercenter	C₂³	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#407
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂³

subgroup

G/H

center, Frattini

C₂²

C₂×D₅×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂³×C₅²

C₂×C₄

prime radical

C₂³×C₅⋊D₅

C₄

cosocle

C₂×C₅×Dic₅

C₂³

C₂³

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂³

N	H ≅ G/N	type
2. C₂×#87853	C₂	semidirect
3. C₂×C₅×#704	C₂	semidirect
4. #11716754	C₂	semidirect
13. C₂×#6032	C₂²	semidirect
15. C₂×C₅×#170	C₂²	semidirect
36. C₂×#704	D₅	semidirect
47. C₂×C₅⋊D₅	M₁₆	semidirect
70. C₂×#170	C₂×D₅	semidirect
118. C₅²	#407	semidirect
187. C₅	#65415	semidirect
188. C₅	#65755	semidirect

H ≅ G/N

type

2. C₂×#87853

C₂

semidirect

3. C₂×C₅×#704

C₂

semidirect

4. #11716754

C₂

semidirect

13. C₂×#6032

C₂²

semidirect

15. C₂×C₅×#170

C₂²

semidirect

36. C₂×#704

D₅

semidirect

47. C₂×C₅⋊D₅

M₁₆

semidirect

70. C₂×#170

C₂×D₅

semidirect

118. C₅²

#407

semidirect

187. C₅

#65415

semidirect

188. C₅

#65755

semidirect

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

subgroup	index
C₂²×#6036	2
C₂×#87853	2
C₂×C₅×#704	2
#11716754	2
#65755	5
#65415	5

subgroup

index

C₂²×#6036

C₂×C₅×#704

Subgroups — #13103779

Conjugacy classes of subgroups

Complements

Maximal subgroups