Group #13031465 — subgroups

Subgroups — #13031465

Order 1280.

702 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 272 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	#10617185	C₂	1	2	yes	maximal
2	C₈×#713	C₂	1	1	yes	maximal, characteristic
3	#10604015	C₂	1	6	yes	maximal, complemented
4	#10603816	C₂	1	3	yes	maximal, complemented
5	D₅⋊#334	C₂	1	3	yes	maximal, complemented
6	C₂×#4133	—	2	2	no	—
7	C₄×#713	C₂²	1	1	yes	characteristic
8	#65708	—	2	6	no	—
9	#65681	C₂²	1	6	yes	—
10	D₅⋊(C₄⋊C₈)	C₂²	1	6	yes	—
11	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
12	Q₈×#168	C₄	1	1	yes	characteristic
13	#65123	C₄	1	3	yes	—
14	#65122	C₂²	1	3	yes	—
15	C₈×D₂₀	C₄	1	3	yes	—
16	C₄×C₈×D₅	C₂²	1	3	yes	—
17	#64883	C₂²	1	6	yes	—
18	#64881	C₂²	1	1	yes	characteristic
19	#64881	C₂²	1	3	yes	—
20	#64881	C₂²	1	6	yes	—
21	#17826	—	5	1	no	maximal, Sylow
22	C₂×#713	D₄	1	1	yes	characteristic
23	#4133	—	4	8	no	—
24	C₂×#700	—	2	6	no	—
25	C₂×D₅⋊C₈	—	2	6	no	—
26	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
27	Q₈×Dic₅	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
30	#4039	Q₈	1	3	yes	—
31	#4038	D₄	1	3	yes	—
32	C₄×D₂₀	C₂×C₄	1	3	yes	—
33	C₄²×D₅	C₂³	1	3	yes	—
34	#4004	—	2	6	no	—
35	C₄⋊#170	—	2	6	no	—
36	C₄⋊#170	C₂³	1	6	yes	—
37	C₄⋊#170	C₂³	1	2	yes	—
38	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
40	#3952	C₂×C₄	1	6	yes	—
41	C₈×Dic₅	C₂³	1	3	yes	—
42	C₈×Dic₅	C₂³	1	1	yes	characteristic
43	C₄⋊#168	C₂×C₄	1	3	yes	—
44	C₄×#168	C₂×C₄	1	3	yes	—
45	C₈×#41	—	5	1	no	—
46	#2906	—	5	2	no	—
47	#1570	—	5	6	no	—
48	#1566	—	5	3	no	—
49	C₂×#334	—	5	3	no	—
50	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
51	#713	SD₁₆	1	8	yes	—
52	C₂×D₂₀	C₄⋊C₄	1	3	yes	—
53	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
54	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
55	C₂×C₄×D₅	C₂×Q₈	1	3	yes	—
56	C₂×#170	—	2	2	no	—
57	C₂×#170	—	2	6	no	—
58	#700	—	4	12	no	—
59	D₅⋊C₈	—	4	12	no	—
60	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₄²×C₅	C₂²×C₄	1	3	yes	—
64	#685	C₄⋊C₄	1	6	yes	—
65	C₄⋊Dic₅	C₄⋊C₄	1	3	yes	—
66	C₄×Dic₅	C₂×D₄	1	3	yes	—
67	C₄×Dic₅	C₂×Q₈	1	3	yes	—
68	C₄×Dic₅	C₂⁴	1	1	yes	characteristic
69	C₂×#168	C₂²×C₄	1	1	yes	characteristic
70	C₂×#168	C₂²×C₄	1	3	yes	—
71	C₈×D₅	—	2	6	no	—
72	C₈×D₅	—	4	3	no	—
73	C₂×#131	—	10	2	no	—
74	C₄×#41	—	5	1	no	—
75	C₈×Q₈	—	5	1	no	—
76	C₈×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₈×D₄	—	5	3	no	—
78	C₈×D₄	—	5	3	no	—
79	#431	—	5	3	no	—
80	#424	—	10	6	no	—
81	#423	—	5	6	no	—
82	C₂×C₄⋊C₈	—	5	6	no	—
83	C₂×C₄×C₈	—	5	3	no	—
84	#334	—	5	1	no	—
85	#334	—	5	3	no	—
86	#334	—	5	6	no	—
87	#334	—	5	6	no	—
88	C₂²×D₅	C₂×C₄⋊C₄	1	3	yes	—
89	C₅×Q₈	#106	1	4	yes	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂×Dic₅	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×Dic₅	C₂²×D₄	1	1	yes	characteristic
95	C₂×Dic₅	C₂²×Q₈	1	1	yes	characteristic
96	D₂₀	#106	1	12	yes	—
97	C₄×D₅	—	2	3	no	—
98	C₄×D₅	—	2	3	no	—
99	C₄×D₅	—	4	3	no	—
100	C₄×D₅	C₂×SD₁₆	1	12	yes	—
101	#170	—	4	2	no	—
102	#170	—	4	6	no	—
103	C₅×C₈	—	2	3	no	—
104	C₅×C₈	#131	1	2	yes	complemented
105	#168	—	2	3	no	—
106	#168	—	2	1	no	—
107	C₂×#41	—	5	1	no	—
108	#131	—	20	8	no	—
109	C₂×M₁₆	—	10	6	no	—
110	C₂²×C₈	—	5	3	no	—
111	C₂²×C₈	—	5	3	no	—
112	C₂²×C₈	—	10	6	no	—
113	C₄×Q₈	—	5	1	no	—
114	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₄×D₄	—	5	3	no	—
116	C₄×D₄	—	5	3	no	—
117	#117	—	5	3	no	—
118	C₂×C₄²	—	5	3	no	—
119	C₄⋊C₈	—	5	2	no	—
120	C₄⋊C₈	—	5	6	no	—
121	C₄⋊C₈	—	5	3	no	—
122	C₄⋊C₈	—	10	6	no	—
123	C₄⋊C₈	—	10	6	no	—
124	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
125	C₂²⋊C₈	—	5	6	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	—	5	3	no	—
128	C₄×C₈	—	5	3	no	—
129	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
130	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂×D₅	—	2	3	no	—
132	C₂×D₅	—	2	3	no	—
133	C₂×D₅	C₂×#106	1	6	yes	—
134	C₄×C₅	—	2	3	no	—
135	C₄×C₅	C₂×#106	1	6	yes	—
136	C₄×C₅	C₂×#131	1	1	yes	characteristic
137	C₄×C₅	#568	1	1	yes	characteristic
138	Dic₅	—	2	3	no	—
139	Dic₅	—	2	1	no	—
140	Dic₅	C₂²×SD₁₆	1	2	yes	—
141	#41	—	5	8	no	—
142	C₂×Q₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₂×D₄	—	5	3	no	—
144	C₂²×C₄	—	5	3	no	—
145	C₂²×C₄	—	5	3	no	—
146	C₂²×C₄	—	5	3	no	—
147	M₁₆	—	20	12	no	—
148	C₂×C₈	—	5	1	no	—
149	C₂×C₈	—	5	3	no	—
150	C₂×C₈	—	10	6	no	—
151	C₂×C₈	—	10	2	no	—
152	C₂×C₈	—	10	6	no	—
153	C₂×C₈	—	20	3	no	—
154	C₂×C₈	—	20	12	no	—
155	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
156	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₄⋊C₄	—	5	3	no	—
158	C₄⋊C₄	C₄⋊AGL(1,𝔽₅)	1	3	yes	—
159	C₂²⋊C₄	—	5	6	no	—
160	C₄²	—	5	3	no	—
161	C₄²	—	5	3	no	—
162	C₄²	—	5	1	no	—
163	C₄²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
164	C₂×C₅	#2906	1	1	yes	characteristic
165	C₂×C₅	C₂²×#106	1	1	yes	characteristic
166	C₂×C₅	#2916	1	1	yes	characteristic
167	D₅	—	2	6	no	—
168	C₂³	—	5	3	no	—
169	Q₈	#3993	1	4	yes	—
170	D₄	—	5	12	no	—
171	C₂×C₄	—	5	1	no	—
172	C₂×C₄	—	5	12	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	5	3	no	—
175	C₂×C₄	—	10	3	no	—
176	C₂×C₄	—	10	3	no	—
177	C₂×C₄	—	20	3	no	—
178	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
179	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
180	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	10	3	no	—
183	C₈	—	10	1	no	—
184	C₈	—	20	2	no	—
185	C₈	—	20	6	no	—
186	C₈	#4133	1	2	yes	complemented
187	C₅	#17826	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	5	6	no	—
189	C₂²	—	10	3	no	—
190	C₂²	—	10	3	no	—
191	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	5	2	no	—
194	C₄	—	10	1	no	—
195	C₄	—	10	3	no	—
196	C₄	C₂×#3993	1	6	yes	—
197	C₄	C₂×#4133	1	1	yes	characteristic
198	C₄	#66264	1	1	yes	characteristic
199	C₂	—	10	6	no	—
200	C₂	#10617185	1	1	yes	characteristic
201	C₂	C₂²×#3993	1	1	yes	characteristic
202	C₂	#10617418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10617185

C₂

yes

maximal

C₈×#713

C₂

yes

maximal, characteristic

#10604015

C₂

yes

maximal, complemented

#10603816

C₂

yes

maximal, complemented

D₅⋊#334

C₂

yes

maximal, complemented

C₂×#4133

—

C₄×#713

C₂²

yes

characteristic

#65708

—

#65681

C₂²

yes

—

D₅⋊(C₄⋊C₈)

C₂²

yes

—

C₅×C₈×Q₈

C₄

yes

characteristic

Q₈×#168

C₄

yes

characteristic

#65123

C₄

yes

—

#65122

C₂²

yes

—

C₈×D₂₀

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

—

#64883

C₂²

yes

—

#64881

C₂²

yes

characteristic

#64881

C₂²

yes

—

#64881

C₂²

yes

—

#17826

—

maximal, Sylow

C₂×#713

D₄

yes

characteristic

#4133

—

C₂×#700

—

C₂×D₅⋊C₈

—

C₄×C₅×Q₈

C₂×C₄

yes

characteristic

Q₈×Dic₅

Q₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

#4039

Q₈

yes

—

#4038

D₄

yes

—

C₄×D₂₀

C₂×C₄

yes

—

C₄²×D₅

C₂³

yes

—

#4004

—

C₄⋊#170

—

C₄⋊#170

C₂³

yes

—

C₄⋊#170

C₂³

yes

—

C₅×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

#3952

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₄⋊#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₈×#41

—

—

—

—

—

C₂×C₅×Q₈

C₄⋊C₄

yes

characteristic

#713

SD₁₆

yes

—

C₂×D₂₀

C₄⋊C₄

yes

—

C₂×C₄×D₅

C₂²×C₄

yes

—

C₂×C₄×D₅

C₂×D₄

yes

—

C₂×C₄×D₅

C₂×Q₈

yes

—

C₂×#170

—

C₂×#170

—

#700

—

D₅⋊C₈

—

C₂×C₅×C₈

C₂²×C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

—

#685

C₄⋊C₄

yes

—

C₄⋊Dic₅

C₄⋊C₄

yes

—

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

—

C₄×Dic₅

C₂⁴

yes

characteristic

C₂×#168

C₂²×C₄

yes

characteristic

C₂×#168

C₂²×C₄

yes

—

C₈×D₅

—

C₈×D₅

—

C₂×#131

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₈×D₄

—

C₈×D₄

—

#431

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

—

—

—

—

C₂²×D₅

C₂×C₄⋊C₄

yes

—

C₅×Q₈

#106

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂³×C₄

yes

characteristic

C₂×Dic₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

C₂²×D₄

yes

characteristic

C₂×Dic₅

C₂²×Q₈

yes

characteristic

D₂₀

#106

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

100

C₄×D₅

C₂×SD₁₆

yes

—

101

#170

—

102

#170

—

103

C₅×C₈

—

104

C₅×C₈

#131

yes

complemented

105

#168

—

106

#168

—

107

C₂×#41

—

108

#131

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄×Q₈

—

114

C₄×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

115

C₄×D₄

—

116

C₄×D₄

—

117

#117

—

118

C₂×C₄²

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

130

C₂²×C₅

C₂²×C₄⋊C₄

yes

characteristic

131

C₂×D₅

—

132

C₂×D₅

—

133

C₂×D₅

C₂×#106

yes

—

134

C₄×C₅

—

135

C₄×C₅

C₂×#106

yes

—

136

C₄×C₅

C₂×#131

yes

characteristic

137

C₄×C₅

#568

yes

characteristic

138

Dic₅

—

139

Dic₅

—

140

Dic₅

C₂²×SD₁₆

yes

—

141

#41

—

142

C₂×Q₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

143

C₂×D₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

—

156

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

C₄⋊AGL(1,𝔽₅)

yes

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

C₂²×AGL(1,𝔽₅)

yes

—

164

C₂×C₅

#2906

yes

characteristic

165

C₂×C₅

C₂²×#106

yes

characteristic

166

C₂×C₅

#2916

yes

characteristic

167

D₅

—

168

C₂³

—

169

Q₈

#3993

yes

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

179

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

180

C₂×C₄

C₂³×AGL(1,𝔽₅)

yes

characteristic

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

#4133

yes

complemented

187

C₅

#17826

yes

characteristic, complemented, Sylow

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂²×C₄⋊AGL(1,𝔽₅)

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

C₂×#3993

yes

—

197

C₄

C₂×#4133

yes

characteristic

198

C₄

#66264

yes

characteristic

199

C₂

—

200

C₂

#10617185

yes

characteristic

201

C₂

C₂²×#3993

yes

characteristic

202

C₂

#10617418

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×C₈×Q₈	C₄
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#713	D₄
cosocle	C₄×Dic₅	C₂⁴
hypercenter	C₈×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#17826
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂³×C₄

Frattini

C₂×C₄

C₂³×AGL(1,𝔽₅)

Fitting

C₅×C₈×Q₈

C₄

socle

C₂²×C₅

C₂²×C₄⋊C₄

prime radical

C₂×#713

D₄

cosocle

C₄×Dic₅

C₂⁴

C₈×Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
3. #10604015	C₂	semidirect
4. #10603816	C₂	semidirect
5. D₅⋊#334	C₂	semidirect
104. C₅×C₈	#131	semidirect
186. C₈	#4133	semidirect
187. C₅	#17826	semidirect

H ≅ G/N

type

3. #10604015

C₂

semidirect

4. #10603816

C₂

semidirect

5. D₅⋊#334

C₂

semidirect

104. C₅×C₈

#131

semidirect

186. C₈

#4133

semidirect

187. C₅

#17826

semidirect

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

subgroup	index
#10617185	2
C₈×#713	2
#10604015	2
#10603816	2
D₅⋊#334	2
#17826	5

subgroup

index

Subgroups — #13031465

Conjugacy classes of subgroups

Complements

Maximal subgroups