Group #11931352 — subgroups

Subgroups — C₅×#8632 #11931352

Order 1280.

388 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	2	yes	maximal, complemented
3	C₅×#1536	C₂	1	2	yes	maximal
4	C₅×#1307	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#131	C₄	1	2	yes	—
6	C₅×#532	C₂²	1	1	yes	characteristic
7	C₅×#436	—	2	2	no	—
8	C₅×C₈×D₄	—	2	2	no	—
9	C₅×#432	C₂²	1	4	yes	—
10	C₅×#423	C₂²	1	1	yes	characteristic
11	C₅×#408	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#335	—	2	2	no	—
14	C₅×#334	—	2	2	no	—
15	C₅×#329	—	2	4	no	—
16	#8632	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×#41	C₂×C₄	1	2	yes	—
18	C₅×#131	—	4	4	no	—
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
25	C₅×#127	Q₈	1	1	yes	characteristic
26	C₅×#124	D₄	1	2	yes	—
27	C₅×#121	—	2	2	no	—
28	C₄×C₅×D₄	—	2	2	no	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	4	no	—
32	C₅×C₂²⋊C₈	—	2	4	no	—
33	C₅×C₂²⋊C₈	—	2	4	no	—
34	C₅×#97	—	2	4	no	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	#2931	C₂×C₅	1	1	yes	characteristic
38	#1568	C₂×C₅	1	2	yes	complemented
39	#1536	C₂×C₅	1	2	yes	—
40	#1307	C₂×C₅	1	2	yes	complemented
41	C₅×#41	—	4	2	no	—
42	C₅×#41	—	4	2	no	—
43	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₅×D₄	—	2	2	no	—
45	C₂×C₅×D₄	—	2	2	no	—
46	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	#41	1	2	yes	—
50	C₅×M₁₆	—	4	4	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	4	no	—
56	C₂×C₅×C₈	—	4	4	no	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₅×C₂²⋊C₄	—	2	4	no	—
62	C₄²×C₅	C₂×D₄	1	2	yes	—
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂×#131	C₄×C₅	1	2	yes	—
66	#532	C₂²×C₅	1	1	yes	characteristic
67	#436	—	2	2	no	—
68	C₈×D₄	—	2	2	no	—
69	#432	C₂²×C₅	1	4	yes	—
70	#423	C₂²×C₅	1	1	yes	characteristic
71	#408	C₄×C₅	1	2	yes	—
72	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	#335	—	2	2	no	—
74	#334	—	2	2	no	—
75	#329	—	2	4	no	—
76	C₂³×C₅	—	2	2	no	—
77	C₂³×C₅	#117	1	1	yes	characteristic
78	C₅×Q₈	—	4	2	no	—
79	C₅×D₄	—	4	2	no	—
80	C₅×D₄	—	4	4	no	—
81	C₅×D₄	—	4	4	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	#121	1	2	yes	—
91	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
92	C₅×C₈	—	2	1	no	—
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	—	4	2	no	—
95	C₅×C₈	—	8	4	no	—
96	C₅×C₈	#100	1	2	yes	—
97	C₂×#41	C₂×C₄×C₅	1	2	yes	—
98	#131	—	4	4	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
104	C₄⋊Q₈	C₅×Q₈	1	1	yes	characteristic
105	#127	C₅×Q₈	1	1	yes	characteristic
106	#124	C₅×D₄	1	2	yes	—
107	#121	—	2	2	no	—
108	C₄×D₄	—	2	2	no	—
109	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₂²⋊C₈	—	2	4	no	—
113	C₂²⋊C₈	—	2	4	no	—
114	#97	—	2	4	no	—
115	C₄×C₈	C₅×D₄	1	2	yes	—
116	C₄×C₈	C₅×D₄	1	2	yes	—
117	C₂²×C₅	—	2	1	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	4	2	no	—
120	C₂²×C₅	—	4	4	no	—
121	C₂²×C₅	#380	1	1	yes	characteristic
122	C₄×C₅	—	2	1	no	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	4	2	no	—
126	C₄×C₅	C₂×#100	1	1	yes	characteristic
127	C₄×C₅	#413	1	1	yes	characteristic
128	#41	—	4	2	no	—
129	#41	—	4	2	no	—
130	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₄	C₅×#41	1	2	yes	—
137	M₁₆	—	4	4	no	—
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
145	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	C₂×C₅×D₄	1	2	yes	—
150	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
151	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
152	C₂×C₅	—	4	1	no	—
153	C₂×C₅	—	8	2	no	—
154	C₂×C₅	#1800	1	1	yes	characteristic
155	C₂×C₅	#1814	1	2	yes	—
156	C₂³	—	2	2	no	—
157	C₂³	C₅×#117	1	1	yes	characteristic
158	Q₈	—	4	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	4	no	—
161	D₄	—	4	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₅×#121	1	2	yes	—
171	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
172	C₈	—	2	1	no	—
173	C₈	—	2	2	no	—
174	C₈	—	4	2	no	—
175	C₈	—	8	4	no	—
176	C₈	C₅×#100	1	2	yes	—
177	C₅	#8632	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₅×#380	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×C₅×#100	1	1	yes	characteristic
188	C₄	C₅×#413	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	—	8	2	no	—
191	C₂	C₅×#1800	1	1	yes	characteristic
192	C₂	C₅×#1814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2931

C₂

yes

maximal, characteristic

C₅×#1568

C₂

yes

maximal, complemented

C₅×#1536

C₂

yes

maximal

C₅×#1307

C₂

yes

maximal, complemented

C₂×C₅×#131

C₄

yes

—

C₅×#532

C₂²

yes

characteristic

C₅×#436

—

C₅×C₈×D₄

—

C₅×#432

C₂²

yes

—

C₅×#423

C₂²

yes

characteristic

C₅×#408

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

—

—

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅×#41

C₂×C₄

yes

—

C₅×#131

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₅×C₄⋊Q₈

Q₈

yes

characteristic

C₅×#127

Q₈

yes

characteristic

C₅×#124

D₄

yes

—

C₅×#121

—

C₄×C₅×D₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

#2931

C₂×C₅

yes

characteristic

#1568

C₂×C₅

yes

complemented

#1536

C₂×C₅

yes

—

#1307

C₂×C₅

yes

complemented

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

C₄⋊C₄

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₄⋊C₄

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

characteristic

C₂×#131

C₄×C₅

yes

—

#532

C₂²×C₅

yes

characteristic

#436

—

C₈×D₄

—

#432

C₂²×C₅

yes

—

#423

C₂²×C₅

yes

characteristic

#408

C₄×C₅

yes

—

C₂×C₄×C₈

C₂²×C₅

yes

—

—

—

C₂³×C₅

—

C₂³×C₅

#117

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#100

yes

—

C₂×#41

C₂×C₄×C₅

yes

—

#131

—

C₂×M₁₆

—

100

C₂×M₁₆

C₂×C₄×C₅

yes

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×C₄×C₅

yes

characteristic

103

C₂²×C₈

C₂×C₄×C₅

yes

characteristic

104

C₄⋊Q₈

C₅×Q₈

yes

characteristic

105

#127

C₅×Q₈

yes

characteristic

106

#124

C₅×D₄

yes

—

107

#121

—

108

C₄×D₄

—

109

C₂×C₄²

C₂³×C₅

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

C₄×C₈

C₅×D₄

yes

—

116

C₄×C₈

C₅×D₄

yes

—

117

C₂²×C₅

—

118

C₂²×C₅

—

119

C₂²×C₅

—

120

C₂²×C₅

—

121

C₂²×C₅

#380

yes

characteristic

122

C₄×C₅

—

123

C₄×C₅

—

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

C₂×#100

yes

characteristic

127

C₄×C₅

#413

yes

characteristic

128

#41

—

129

#41

—

130

C₂×Q₈

C₅×C₄⋊C₄

yes

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

C₅×C₄⋊C₄

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

136

C₂²×C₄

C₅×#41

yes

—

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

C₅×C₂²⋊C₄

yes

—

145

C₂×C₈

C₅×C₂²⋊C₄

yes

characteristic

146

C₂×C₈

C₅×C₂²⋊C₄

yes

characteristic

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

C₂×C₅×D₄

yes

—

150

C₄²

C₂×C₅×D₄

yes

characteristic

151

C₄²

C₂×C₅×Q₈

yes

characteristic

152

C₂×C₅

—

153

C₂×C₅

—

154

C₂×C₅

#1800

yes

characteristic

155

C₂×C₅

#1814

yes

—

156

C₂³

—

157

C₂³

C₅×#117

yes

characteristic

158

Q₈

—

159

D₄

—

160

D₄

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₂×C₅×C₂²⋊C₄

yes

characteristic

169

C₂×C₄

C₂×C₅×C₄⋊C₄

yes

characteristic

170

C₂×C₄

C₅×#121

yes

—

171

C₂×C₄

C₅×C₂²⋊Q₈

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

C₅×#100

yes

—

177

C₅

#8632

yes

characteristic, direct product factor, Sylow

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₅×#380

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×C₅×#100

yes

characteristic

188

C₄

C₅×#413

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₅×#1800

yes

characteristic

192

C₂

C₅×#1814

yes

—

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

subgroup

G/H

center, socle

C₂²×C₅

#380

commutator

C₂²×C₄

C₂²×C₄×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

prime radical

C₅×#532

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

N	H ≅ G/N	type
2. C₅×#1568	C₂	semidirect
4. C₅×#1307	C₂	semidirect
16. #8632	C₅	direct (N × H)
38. #1568	C₂×C₅	semidirect
40. #1307	C₂×C₅	semidirect
177. C₅	#8632	direct (N × H)

H ≅ G/N

type

2. C₅×#1568

C₂

semidirect

4. C₅×#1307

C₂

semidirect

16. #8632

C₅

direct (N × H)

38. #1568

C₂×C₅

semidirect

40. #1307

C₂×C₅

semidirect

177. C₅

#8632

direct (N × H)

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

subgroup	index
C₅×#2931	2
C₅×#1568	2
C₅×#1536	2
C₅×#1307	2
#8632	5

subgroup

index

Subgroups — C₅×#8632 #11931352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#8632 #11931352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8632 #11931352