Group #11936159 — subgroups

Subgroups — C₅×#13439 #11936159

Order 1280.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
4	C₅×#1978	C₂	1	2	yes	maximal, complemented
5	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	C₂²	1	2	yes	—
7	C₂×C₅×D₁₆	—	2	2	no	—
8	C₂×C₅×D₁₆	—	2	4	no	—
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	C₂²	1	2	yes	—
12	C₅×#418	—	2	2	no	—
13	C₂×C₅×#100	—	2	2	no	—
14	C₅×#405	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	2	no	—
16	C₅×#361	—	2	4	no	—
17	#13439	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	—	2	2	no	—
19	C₂×C₅×D₈	—	2	2	no	—
20	C₂×C₅×D₈	—	2	2	no	—
21	C₂×C₅×D₈	—	2	2	no	—
22	C₂×C₅×D₈	—	2	2	no	—
23	C₂×C₅×D₈	—	2	2	no	—
24	C₂×C₅×D₈	D₄	1	4	yes	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×#117	D₄	1	1	yes	characteristic
29	C₅×D₁₆	—	2	8	no	—
30	C₅×M₃₂	—	2	2	no	—
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₂×C₅×C₁₆	—	2	1	no	—
33	C₂×C₅×C₁₆	D₄	1	2	yes	—
34	C₅×#108	—	2	2	no	—
35	C₅×#108	—	2	2	no	—
36	C₅×#102	—	4	4	no	—
37	C₅×#100	—	4	4	no	—
38	C₅×#97	—	2	1	no	—
39	C₄×C₅×C₈	—	2	1	no	—
40	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic
41	#2151	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×#361	C₂×C₅	1	2	yes	complemented
43	#1978	C₂×C₅	1	2	yes	complemented
44	#1381	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂⁴×C₅	—	4	2	no	—
46	C₂×C₅×D₄	—	2	4	no	—
47	C₂×C₅×D₄	—	4	2	no	—
48	C₂×C₅×D₄	—	4	2	no	—
49	C₂×C₅×D₄	—	4	2	no	—
50	C₂×C₅×D₄	—	4	2	no	—
51	C₂×C₅×D₄	—	4	2	no	—
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₅×D₈	—	2	4	no	—
54	C₅×D₈	—	2	4	no	—
55	C₅×D₈	—	4	2	no	—
56	C₅×D₈	—	4	2	no	—
57	C₅×M₁₆	—	2	2	no	—
58	C₅×M₁₆	—	4	2	no	—
59	C₅×M₁₆	—	4	2	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	1	yes	characteristic
65	C₂×C₅×C₈	#41	1	2	yes	—
66	C₅×C₄⋊C₄	D₈	1	2	yes	—
67	C₅×C₂²⋊C₄	—	2	1	no	—
68	C₄²×C₅	—	2	1	no	—
69	C₅×C₁₆	—	2	2	no	—
70	C₅×C₁₆	—	4	1	no	—
71	C₂²×D₈	C₂²×C₅	1	2	yes	—
72	C₂×D₁₆	—	2	4	no	—
73	C₂×D₁₆	—	2	2	no	—
74	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
76	C₂×#108	C₂²×C₅	1	2	yes	—
77	#418	—	2	2	no	—
78	C₂×#100	—	2	2	no	—
79	#405	C₂²×C₅	1	1	yes	characteristic
80	C₄⋊C₁₆	—	2	2	no	—
81	#361	—	2	4	no	—
82	C₂³×C₅	—	4	4	no	—
83	C₂³×C₅	—	8	2	no	—
84	C₂³×C₅	—	8	2	no	—
85	C₂³×C₅	—	16	4	no	—
86	C₂³×C₅	#135	1	1	yes	characteristic
87	C₅×D₄	—	4	4	no	—
88	C₅×D₄	—	4	4	no	—
89	C₅×D₄	—	8	2	no	—
90	C₅×D₄	—	8	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	#121	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
96	C₅×C₈	—	4	1	no	—
97	C₅×C₈	—	8	2	no	—
98	C₅×C₈	#120	1	1	yes	characteristic
99	C₅×C₈	#123	1	2	yes	—
100	C₅×C₈	#124	1	1	yes	characteristic
101	C₂²×D₄	—	2	2	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	2	2	no	—
107	C₂×D₈	C₅×D₄	1	4	yes	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
111	#117	C₅×D₄	1	1	yes	characteristic
112	D₁₆	—	2	8	no	—
113	M₃₂	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	C₅×D₄	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#102	—	4	4	no	—
120	#100	—	4	4	no	—
121	#97	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	8	4	no	—
125	C₂²×C₅	—	16	4	no	—
126	C₂²×C₅	—	16	4	no	—
127	C₂²×C₅	—	16	4	no	—
128	C₂²×C₅	#463	1	2	yes	—
129	C₂²×C₅	#466	1	1	yes	characteristic
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	#394	1	1	yes	characteristic
132	C₄×C₅	#459	1	2	yes	—
133	C₄×C₅	#469	1	1	yes	characteristic
134	C₂⁴	—	4	2	no	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
142	D₈	—	2	4	no	—
143	D₈	—	2	4	no	—
144	D₈	—	4	2	no	—
145	D₈	—	4	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
152	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
153	C₂×C₈	C₅×#41	1	1	yes	characteristic
154	C₂×C₈	C₅×#41	1	2	yes	—
155	C₄⋊C₄	C₅×D₈	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	4	1	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	16	4	no	—
162	C₂×C₅	#2054	1	1	yes	characteristic
163	C₂×C₅	#2221	1	2	yes	—
164	C₂³	—	4	4	no	—
165	C₂³	—	8	2	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	16	4	no	—
168	C₂³	C₅×#135	1	1	yes	characteristic
169	D₄	—	4	4	no	—
170	D₄	—	4	4	no	—
171	D₄	—	8	2	no	—
172	D₄	—	8	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₅×#121	1	2	yes	—
175	C₂×C₄	C₅×#121	1	1	yes	characteristic
176	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
177	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
178	C₈	—	4	1	no	—
179	C₈	—	8	2	no	—
180	C₈	C₅×#120	1	1	yes	characteristic
181	C₈	C₅×#123	1	2	yes	—
182	C₈	C₅×#124	1	1	yes	characteristic
183	C₅	#13439	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	C₅×#463	1	2	yes	—
190	C₂²	C₅×#466	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	C₅×#394	1	1	yes	characteristic
193	C₄	C₅×#459	1	2	yes	—
194	C₄	C₅×#469	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	C₅×#2054	1	1	yes	characteristic
198	C₂	C₅×#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅×D₁₆

C₂

yes

maximal, characteristic

C₅×#2151

C₂

yes

maximal, characteristic, complemented

C₂×C₅×#361

C₂

yes

maximal, complemented

C₅×#1978

C₂

yes

maximal, complemented

C₅×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×D₈

C₂²

yes

—

C₂×C₅×D₁₆

—

C₂×C₅×D₁₆

—

C₂×C₅×M₃₂

C₂²

yes

characteristic

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

C₅×#418

—

C₂×C₅×#100

—

C₅×#405

C₂²

yes

characteristic

C₅×C₄⋊C₁₆

—

C₅×#361

—

#13439

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×D₄

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

D₄

yes

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×#117

D₄

yes

characteristic

C₅×D₁₆

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

C₅×#97

—

C₄×C₅×C₈

—

C₂²×D₁₆

C₂×C₅

yes

characteristic

#2151

C₂×C₅

yes

characteristic, complemented

C₂×#361

C₂×C₅

yes

complemented

#1978

C₂×C₅

yes

complemented

#1381

C₂×C₅

yes

characteristic, complemented

C₂⁴×C₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×D₈

—

C₅×D₈

—

C₅×D₈

—

C₅×D₈

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

C₂×C₅×C₈

yes

C₂×C₅×C₈

yes

—

C₅×C₄⋊C₄

D₈

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₅×C₁₆

—

C₅×C₁₆

—

C₂²×D₈

C₂²×C₅

yes

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

C₂²×C₅

yes

characteristic

C₂²×C₁₆

C₂²×C₅

yes

characteristic

C₂×#108

C₂²×C₅

yes

—

#418

—

C₂×#100

—

#405

C₂²×C₅

yes

characteristic

C₄⋊C₁₆

—

#361

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

#135

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#120

yes

characteristic

C₅×C₈

#123

yes

—

100

C₅×C₈

#124

yes

characteristic

101

C₂²×D₄

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

C₅×D₄

yes

—

108

C₂×M₁₆

—

109

C₂×M₁₆

C₅×D₄

yes

characteristic

110

C₂²×C₈

C₂³×C₅

yes

characteristic

111

#117

C₅×D₄

yes

characteristic

112

D₁₆

—

113

M₃₂

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

C₅×D₄

yes

—

117

#108

—

118

#108

—

119

#102

—

120

#100

—

121

#97

—

122

C₄×C₈

—

123

C₂²×C₅

—

124

C₂²×C₅

—

125

C₂²×C₅

—

126

C₂²×C₅

—

127

C₂²×C₅

—

128

C₂²×C₅

#463

yes

—

129

C₂²×C₅

#466

yes

characteristic

130

C₄×C₅

—

131

C₄×C₅

#394

yes

characteristic

132

C₄×C₅

#459

yes

—

133

C₄×C₅

#469

yes

characteristic

134

C₂⁴

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

142

D₈

—

143

D₈

—

144

D₈

—

145

D₈

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂×C₅×D₄

yes

—

152

C₂×C₈

C₂×C₅×D₄

yes

characteristic

153

C₂×C₈

C₅×#41

yes

characteristic

154

C₂×C₈

C₅×#41

yes

—

155

C₄⋊C₄

C₅×D₈

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂×C₅

—

161

C₂×C₅

—

162

C₂×C₅

#2054

yes

characteristic

163

C₂×C₅

#2221

yes

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

C₅×#135

yes

characteristic

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

C₅×#121

yes

—

175

C₂×C₄

C₅×#121

yes

characteristic

176

C₂×C₄

C₂×C₅×D₈

yes

characteristic

177

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

178

C₈

—

179

C₈

—

180

C₈

C₅×#120

yes

characteristic

181

C₈

C₅×#123

yes

—

182

C₈

C₅×#124

yes

characteristic

183

C₅

#13439

yes

characteristic, direct product factor, Sylow

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₅×#463

yes

—

190

C₂²

C₅×#466

yes

characteristic

191

C₄

—

192

C₄

C₅×#394

yes

characteristic

193

C₄

C₅×#459

yes

—

194

C₄

C₅×#469

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₅×#2054

yes

characteristic

198

C₂

C₅×#2221

yes

—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₁₆	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₅

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅

prime radical

C₂²×C₅×D₁₆

C₂

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

N	H ≅ G/N	type
2. C₅×#2151	C₂	semidirect
3. C₂×C₅×#361	C₂	semidirect
4. C₅×#1978	C₂	semidirect
5. C₅×#1381	C₂	semidirect
17. #13439	C₅	direct (N × H)
41. #2151	C₂×C₅	semidirect
42. C₂×#361	C₂×C₅	semidirect
43. #1978	C₂×C₅	semidirect
44. #1381	C₂×C₅	semidirect
183. C₅	#13439	direct (N × H)

H ≅ G/N

type

2. C₅×#2151

C₂

semidirect

3. C₂×C₅×#361

C₂

semidirect

4. C₅×#1978

C₂

semidirect

5. C₅×#1381

C₂

semidirect

17. #13439

C₅

direct (N × H)

41. #2151

C₂×C₅

semidirect

42. C₂×#361

C₂×C₅

semidirect

43. #1978

C₂×C₅

semidirect

44. #1381

C₂×C₅

semidirect

183. C₅

#13439

direct (N × H)

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

subgroup	index
C₂²×C₅×D₁₆	2
C₅×#2151	2
C₂×C₅×#361	2
C₅×#1978	2
C₅×#1381	2
#13439	5

subgroup

index

C₂²×C₅×D₁₆

C₂×C₅×#361

Subgroups — C₅×#13439 #11936159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#13439 #11936159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13439 #11936159