Group #11935976 — subgroups

Subgroups — C₅×#13256 #11935976

Order 1280.

462 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#362	C₂	1	4	yes	maximal
4	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×Q₁₆	—	2	1	no	—
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#494	—	2	2	no	—
10	C₅×#494	C₄	1	4	yes	—
11	C₂×C₅×#108	C₂²	1	2	yes	—
12	C₄×C₅×M₁₆	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	—	2	2	no	—
15	C₅×#362	—	2	8	no	—
16	C₅×#341	—	2	1	no	—
17	#13256	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	4	no	—
20	C₂×C₅×Q₁₆	—	2	4	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	4	4	no	—
24	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	—	2	4	no	—
29	C₅×C₄⋊Q₈	—	4	1	no	—
30	C₅×C₄⋊Q₈	—	4	1	no	—
31	C₂×C₅×C₄⋊C₄	—	4	2	no	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₅×M₃₂	—	2	4	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₅×#108	—	2	2	no	—
36	C₅×#108	D₄	1	4	yes	—
37	C₅×#97	—	4	1	no	—
38	C₄×C₅×C₈	—	4	1	no	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₂×#494	C₂×C₅	1	1	yes	characteristic
42	#2150	C₂×C₅	1	1	yes	characteristic
43	C₂×#362	C₂×C₅	1	4	yes	—
44	C₄×#108	C₂×C₅	1	1	yes	characteristic
45	C₂×C₅×Q₈	—	2	4	no	—
46	C₂×C₅×Q₈	—	4	2	no	—
47	C₂×C₅×Q₈	—	4	4	no	—
48	C₂×C₅×Q₈	—	4	4	no	—
49	C₂²×C₄×C₅	—	4	2	no	—
50	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₅×Q₁₆	—	2	8	no	—
53	C₅×Q₁₆	—	4	8	no	—
54	C₅×Q₁₆	—	4	2	no	—
55	C₅×Q₁₆	—	4	2	no	—
56	C₅×M₁₆	—	2	4	no	—
57	C₅×M₁₆	—	4	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
65	C₅×C₄⋊C₄	—	4	4	no	—
66	C₅×C₄⋊C₄	—	4	4	no	—
67	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅	D₈	1	1	yes	characteristic
69	C₄²×C₅	D₈	1	1	yes	characteristic
70	C₅×C₁₆	—	4	2	no	—
71	C₂²×Q₁₆	—	2	1	no	—
72	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
73	C₂×C₄⋊Q₈	—	2	1	no	—
74	C₂×M₃₂	C₂²×C₅	1	2	yes	—
75	#494	—	2	2	no	—
76	#494	C₄×C₅	1	4	yes	—
77	C₂×#108	C₂²×C₅	1	2	yes	—
78	C₄×M₁₆	—	2	1	no	—
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₄⋊C₁₆	—	2	2	no	—
81	#362	—	2	8	no	—
82	#341	—	2	1	no	—
83	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
84	C₅×Q₈	—	4	4	no	—
85	C₅×Q₈	—	4	4	no	—
86	C₅×Q₈	—	8	4	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	8	4	no	—
90	C₂×C₄×C₅	#102	1	2	yes	—
91	C₂×C₄×C₅	#102	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	#121	1	2	yes	—
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	—	4	4	no	—
98	C₅×C₈	C₄×D₄	1	2	yes	—
99	C₅×C₈	#124	1	1	yes	characteristic
100	C₅×C₈	#127	1	1	yes	characteristic
101	C₂²×Q₈	—	2	2	no	—
102	C₂×Q₁₆	—	2	4	no	—
103	C₂×Q₁₆	—	2	4	no	—
104	C₂×Q₁₆	—	2	2	no	—
105	C₂×Q₁₆	—	2	4	no	—
106	C₂×Q₁₆	—	4	4	no	—
107	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	—	4	1	no	—
113	C₄⋊Q₈	—	4	1	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
116	M₃₂	—	2	4	no	—
117	C₂×C₁₆	—	2	2	no	—
118	#108	—	2	2	no	—
119	#108	C₅×D₄	1	4	yes	—
120	#97	—	4	1	no	—
121	C₄×C₈	—	4	1	no	—
122	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
123	C₄×C₈	C₅×D₄	1	2	yes	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	C₂×#102	1	1	yes	characteristic
126	C₂²×C₅	#460	1	2	yes	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	—	8	4	no	—
129	C₄×C₅	#362	1	4	yes	complemented
130	C₄×C₅	#390	1	1	yes	characteristic
131	C₄×C₅	#418	1	1	yes	characteristic
132	C₄×C₅	#459	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	4	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
139	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	Q₁₆	—	2	8	no	—
141	Q₁₆	—	4	8	no	—
142	Q₁₆	—	4	2	no	—
143	Q₁₆	—	4	2	no	—
144	M₁₆	—	2	4	no	—
145	M₁₆	—	4	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
151	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₈	C₅×#41	1	2	yes	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
156	C₄²	C₅×D₈	1	1	yes	characteristic
157	C₄²	C₅×D₈	1	1	yes	characteristic
158	C₁₆	—	4	2	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	#1982	1	1	yes	characteristic
161	C₂×C₅	C₂×#362	1	2	yes	—
162	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	Q₈	—	4	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	C₅×#102	1	2	yes	—
170	C₂×C₄	C₅×#102	1	2	yes	—
171	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₅×#121	1	2	yes	—
173	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
174	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
175	C₈	—	2	2	no	—
176	C₈	—	4	4	no	—
177	C₈	C₄×C₅×D₄	1	2	yes	—
178	C₈	C₅×#124	1	1	yes	characteristic
179	C₈	C₅×#127	1	1	yes	characteristic
180	C₅	#13256	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	C₂×C₅×#102	1	1	yes	characteristic
183	C₂²	C₅×#460	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	C₅×#362	1	4	yes	complemented
187	C₄	C₅×#390	1	1	yes	characteristic
188	C₄	C₅×#418	1	1	yes	characteristic
189	C₄	C₅×#459	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	C₅×#1982	1	1	yes	characteristic
192	C₂	C₂×C₅×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#494

C₂

yes

maximal, characteristic

C₅×#2150

C₂

yes

maximal, characteristic

C₂×C₅×#362

C₂

yes

maximal

C₄×C₅×#108

C₂

yes

maximal, characteristic

C₂²×C₅×Q₁₆

—

C₂²×C₅×Q₁₆

C₂²

yes

—

C₂×C₅×C₄⋊Q₈

—

C₂×C₅×M₃₂

C₂²

yes

—

C₅×#494

—

C₅×#494

C₄

yes

—

C₂×C₅×#108

C₂²

yes

—

C₄×C₅×M₁₆

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×C₄⋊C₁₆

—

C₅×#362

—

C₅×#341

—

#13256

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

C₂×C₄

yes

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₅×#108

—

C₅×#108

D₄

yes

—

C₅×#97

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂×#494

C₂×C₅

yes

characteristic

#2150

C₂×C₅

yes

characteristic

C₂×#362

C₂×C₅

yes

—

C₄×#108

C₂×C₅

yes

characteristic

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

SD₁₆

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

D₈

yes

characteristic

C₄²×C₅

D₈

yes

characteristic

C₅×C₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²×C₅

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²×C₅

yes

—

#494

—

#494

C₄×C₅

yes

—

C₂×#108

C₂²×C₅

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂²×C₅

yes

characteristic

C₄⋊C₁₆

—

#362

—

#341

—

C₂³×C₅

C₂×SD₁₆

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#102

yes

—

C₂×C₄×C₅

#102

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₄×D₄

yes

—

C₅×C₈

#124

yes

characteristic

100

C₅×C₈

#127

yes

characteristic

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

C₂×C₄×C₅

yes

—

108

C₂×M₁₆

—

109

C₂²×C₈

C₅×D₄

yes

characteristic

110

C₂²×C₈

C₂³×C₅

yes

characteristic

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₅×D₄

yes

characteristic

116

M₃₂

—

117

C₂×C₁₆

—

118

#108

—

119

#108

C₅×D₄

yes

—

120

#97

—

121

C₄×C₈

—

122

C₄×C₈

C₂×C₄×C₅

yes

—

123

C₄×C₈

C₅×D₄

yes

—

124

C₂²×C₅

—

125

C₂²×C₅

C₂×#102

yes

characteristic

126

C₂²×C₅

#460

yes

—

127

C₄×C₅

—

128

C₄×C₅

—

129

C₄×C₅

#362

yes

complemented

130

C₄×C₅

#390

yes

characteristic

131

C₄×C₅

#418

yes

characteristic

132

C₄×C₅

#459

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

—

138

C₂²×C₄

C₅×SD₁₆

yes

—

139

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

140

Q₁₆

—

141

Q₁₆

—

142

Q₁₆

—

143

Q₁₆

—

144

M₁₆

—

145

M₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

C₅×C₂²⋊C₄

yes

—

149

C₂×C₈

C₂²×C₄×C₅

yes

characteristic

150

C₂×C₈

C₂×C₅×D₄

yes

—

151

C₂×C₈

C₂×C₅×D₄

yes

characteristic

152

C₂×C₈

C₅×#41

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₅×C₂²⋊C₄

yes

—

156

C₄²

C₅×D₈

yes

characteristic

157

C₄²

C₅×D₈

yes

characteristic

158

C₁₆

—

159

C₂×C₅

—

160

C₂×C₅

#1982

yes

characteristic

161

C₂×C₅

C₂×#362

yes

—

162

C₂³

C₂×C₅×SD₁₆

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₅×#102

yes

—

170

C₂×C₄

C₅×#102

yes

—

171

C₂×C₄

C₂×C₅×C₂²⋊C₄

yes

characteristic

172

C₂×C₄

C₅×#121

yes

—

173

C₂×C₄

C₂×C₅×D₈

yes

characteristic

174

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

175

C₈

—

176

C₈

—

177

C₈

C₄×C₅×D₄

yes

—

178

C₈

C₅×#124

yes

characteristic

179

C₈

C₅×#127

yes

characteristic

180

C₅

#13256

yes

characteristic, direct product factor, Sylow

181

C₂²

—

182

C₂²

C₂×C₅×#102

yes

characteristic

183

C₂²

C₅×#460

yes

—

184

C₄

—

185

C₄

—

186

C₄

C₅×#362

yes

complemented

187

C₄

C₅×#390

yes

characteristic

188

C₄

C₅×#418

yes

characteristic

189

C₄

C₅×#459

yes

—

190

C₂

—

191

C₂

C₅×#1982

yes

characteristic

192

C₂

C₂×C₅×#362

yes

—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

subgroup

G/H

center, socle

C₂²×C₅

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₅

Frattini, cosocle

C₂²×C₈

C₂³×C₅

prime radical

C₂³×C₅

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

N	H ≅ G/N	type
17. #13256	C₅	direct (N × H)
129. C₄×C₅	#362	semidirect
180. C₅	#13256	direct (N × H)
186. C₄	C₅×#362	semidirect

H ≅ G/N

type

17. #13256

C₅

direct (N × H)

129. C₄×C₅

#362

semidirect

180. C₅

#13256

direct (N × H)

186. C₄

C₅×#362

semidirect

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

subgroup	index
C₂×C₅×#494	2
C₅×#2150	2
C₂×C₅×#362	2
C₄×C₅×#108	2
#13256	5

subgroup

index

C₂×C₅×#494

C₅×#2150

C₂×C₅×#362

C₄×C₅×#108

#13256

Subgroups — C₅×#13256 #11935976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#13256 #11935976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13256 #11935976