Group #11933344 — subgroups

Subgroups — C₅×#10624 #11933344

Order 1280.

350 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 134 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	C₅×#1984	C₂	1	1	yes	maximal, characteristic
2	C₅×#1834	C₂	1	1	yes	maximal, characteristic
3	C₅×#1803	C₂	1	2	yes	maximal
4	C₅×#1296	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#103	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#384	C₂²	1	2	yes	—
12	C₅×#340	—	2	2	no	—
13	C₅×#340	—	2	2	no	—
14	C₅×#336	—	2	2	no	—
15	#10624	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	1	no	—
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	D₄	1	2	yes	—
20	C₄×C₅×Q₈	—	2	2	no	—
21	C₄×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	—	2	1	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₅×#103	—	4	4	no	—
33	C₄×C₅×C₈	—	2	2	no	—
34	#1984	C₂×C₅	1	1	yes	characteristic
35	#1834	C₂×C₅	1	1	yes	characteristic
36	#1803	C₂×C₅	1	2	yes	—
37	#1296	C₂×C₅	1	2	yes	—
38	#1277	C₂×C₅	1	1	yes	characteristic
39	C₂×C₅×Q₈	—	2	1	no	—
40	C₂×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₈	—	2	2	no	—
42	C₂×C₅×Q₈	Q₁₆	1	2	yes	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	—	2	1	no	—
45	C₂²×C₄×C₅	—	4	2	no	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	1	yes	characteristic
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	1	no	—
54	C₂×C₅×C₈	—	4	2	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	2	no	—
57	C₅×C₄⋊C₄	—	2	2	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	4	2	no	—
62	C₅×C₄⋊C₄	—	4	2	no	—
63	C₅×C₄⋊C₄	—	8	2	no	—
64	C₄²×C₅	—	4	1	no	—
65	C₄²×C₅	—	4	1	no	—
66	C₄²×C₅	#41	1	2	yes	—
67	C₄²×C₅	#41	1	2	yes	—
68	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×#103	—	2	2	no	—
72	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	#384	C₂²×C₅	1	2	yes	—
74	#340	—	2	2	no	—
75	#340	—	2	2	no	—
76	#336	—	2	2	no	—
77	C₂³×C₅	#121	1	1	yes	characteristic
78	C₅×Q₈	—	2	4	no	—
79	C₅×Q₈	—	4	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	4	1	no	—
83	C₂×C₄×C₅	—	4	1	no	—
84	C₂×C₄×C₅	—	4	1	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	8	2	no	—
87	C₂×C₄×C₅	—	8	2	no	—
88	C₂×C₄×C₅	—	8	2	no	—
89	C₂×C₄×C₅	#123	1	2	yes	—
90	C₂×C₄×C₅	#123	1	1	yes	characteristic
91	C₂×C₄×C₅	#124	1	1	yes	characteristic
92	C₂×C₄×C₅	#126	1	2	yes	—
93	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	#135	1	1	yes	characteristic
95	C₂×C₄×C₅	#135	1	2	yes	—
96	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₅	#137	1	2	yes	—
98	C₅×C₈	—	4	2	no	—
99	C₅×C₈	—	8	2	no	—
100	C₅×C₈	—	8	1	no	—
101	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	C₅×D₄	1	2	yes	—
105	C₄×Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#103	—	4	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	#467	1	1	yes	characteristic
122	C₂²×C₅	#469	1	1	yes	characteristic
123	C₂²×C₅	#471	1	2	yes	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	4	2	no	—
126	C₄×C₅	—	8	1	no	—
127	C₄×C₅	—	16	2	no	—
128	C₄×C₅	#482	1	2	yes	—
129	C₄×C₅	#484	1	2	yes	—
130	C₄×C₅	#485	1	2	yes	—
131	C₄×C₅	#488	1	1	yes	characteristic
132	C₄×C₅	#491	1	1	yes	characteristic
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	C₅×Q₁₆	1	2	yes	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
142	C₂²×C₄	C₅×#41	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	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	8	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₅×#41	1	2	yes	—
161	C₄²	C₅×#41	1	2	yes	—
162	C₂×C₅	#1684	1	2	yes	—
163	C₂×C₅	#1702	1	2	yes	—
164	C₂×C₅	#2074	1	1	yes	characteristic
165	C₂×C₅	#2076	1	1	yes	characteristic
166	C₂×C₅	#2079	1	1	yes	characteristic
167	C₂³	C₅×#121	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	Q₈	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	C₅×#123	1	2	yes	—
180	C₂×C₄	C₅×#123	1	1	yes	characteristic
181	C₂×C₄	C₅×#124	1	1	yes	characteristic
182	C₂×C₄	C₅×#126	1	2	yes	—
183	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
184	C₂×C₄	C₅×#135	1	1	yes	characteristic
185	C₂×C₄	C₅×#135	1	2	yes	—
186	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
187	C₂×C₄	C₅×#137	1	2	yes	—
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	8	1	no	—
191	C₅	#10624	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₅×#397	1	1	yes	characteristic
193	C₂²	C₅×#464	1	2	yes	—
194	C₂²	C₅×#467	1	1	yes	characteristic
195	C₂²	C₅×#469	1	1	yes	characteristic
196	C₂²	C₅×#471	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	16	2	no	—
201	C₄	C₅×#482	1	2	yes	—
202	C₄	C₅×#484	1	2	yes	—
203	C₄	C₅×#485	1	2	yes	—
204	C₄	C₅×#488	1	1	yes	characteristic
205	C₄	C₅×#491	1	1	yes	characteristic
206	C₂	C₅×#1684	1	2	yes	—
207	C₂	C₅×#1702	1	2	yes	—
208	C₂	C₅×#2074	1	1	yes	characteristic
209	C₂	C₅×#2076	1	1	yes	characteristic
210	C₂	C₅×#2079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1984

C₂

yes

maximal, characteristic

C₅×#1834

C₂

yes

maximal, characteristic

C₅×#1803

C₂

yes

maximal

C₅×#1296

C₂

yes

maximal

C₅×#1277

C₂

yes

maximal, characteristic

C₂×C₄×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₅×#103

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

—

—

—

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

D₄

yes

characteristic

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

D₄

yes

—

C₄×C₅×Q₈

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#103

—

C₄×C₅×C₈

—

#1984

C₂×C₅

yes

characteristic

#1834

C₂×C₅

yes

characteristic

#1803

C₂×C₅

yes

—

#1296

C₂×C₅

yes

—

#1277

C₂×C₅

yes

characteristic

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

Q₁₆

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

—

C₂×C₄×Q₈

C₂²×C₅

yes

characteristic

C₂×C₄⋊C₈

C₂²×C₅

yes

—

C₂×C₄⋊C₈

C₂²×C₅

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²×C₅

yes

characteristic

#384

C₂²×C₅

yes

—

#340

—

#340

—

#336

—

C₂³×C₅

#121

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

C₅×C₈

—

100

C₅×C₈

—

101

C₂²×Q₈

C₅×D₄

yes

characteristic

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

C₅×D₄

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

C₅×D₄

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

C₂³×C₅

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

#467

yes

characteristic

122

C₂²×C₅

#469

yes

characteristic

123

C₂²×C₅

#471

yes

—

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

—

127

C₄×C₅

—

128

C₄×C₅

#482

yes

—

129

C₄×C₅

#484

yes

—

130

C₄×C₅

#485

yes

—

131

C₄×C₅

#488

yes

characteristic

132

C₄×C₅

#491

yes

characteristic

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

C₅×Q₁₆

yes

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

141

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

142

C₂²×C₄

C₅×#41

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

C₅×#41

yes

—

161

C₄²

C₅×#41

yes

—

162

C₂×C₅

#1684

yes

—

163

C₂×C₅

#1702

yes

—

164

C₂×C₅

#2074

yes

characteristic

165

C₂×C₅

#2076

yes

characteristic

166

C₂×C₅

#2079

yes

characteristic

167

C₂³

C₅×#121

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

C₅×#123

yes

—

180

C₂×C₄

C₅×#123

yes

characteristic

181

C₂×C₄

C₅×#124

yes

characteristic

182

C₂×C₄

C₅×#126

yes

—

183

C₂×C₄

C₂×C₅×Q₁₆

yes

characteristic

184

C₂×C₄

C₅×#135

yes

characteristic

185

C₂×C₄

C₅×#135

yes

—

186

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

187

C₂×C₄

C₅×#137

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₅

#10624

yes

characteristic, direct product factor, Sylow

192

C₂²

C₅×#397

yes

characteristic

193

C₂²

C₅×#464

yes

—

194

C₂²

C₅×#467

yes

characteristic

195

C₂²

C₅×#469

yes

characteristic

196

C₂²

C₅×#471

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

C₅×#482

yes

—

202

C₄

C₅×#484

yes

—

203

C₄

C₅×#485

yes

—

204

C₄

C₅×#488

yes

characteristic

205

C₄

C₅×#491

yes

characteristic

206

C₂

C₅×#1684

yes

—

207

C₂

C₅×#1702

yes

—

208

C₂

C₅×#2074

yes

characteristic

209

C₂

C₅×#2076

yes

characteristic

210

C₂

C₅×#2079

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅

#121

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

N	H ≅ G/N	type
15. #10624	C₅	direct (N × H)
191. C₅	#10624	direct (N × H)

H ≅ G/N

type

15. #10624

C₅

direct (N × H)

191. C₅

#10624

direct (N × H)

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

subgroup	index
C₅×#1984	2
C₅×#1834	2
C₅×#1803	2
C₅×#1296	2
C₅×#1277	2
#10624	5

subgroup

index

Subgroups — C₅×#10624 #11933344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#10624 #11933344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624 #11933344