Group #13026032 — subgroups

Subgroups — #13026032

Order 1280.

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

normal subgroups only

This group has 318 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	#10604107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	#10604089	C₂	1	2	yes	maximal
4	D₅×#368	C₂	1	1	yes	maximal, characteristic
5	#10598626	C₂	1	2	yes	maximal
6	#65728	—	2	2	no	—
7	#65721	C₂²	1	2	yes	—
8	C₂×C₈×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
11	D₅×#108	C₂²	1	2	yes	—
12	#64900	—	2	2	no	—
13	#64850	C₄	1	2	yes	complemented
14	#64849	C₄	1	2	yes	complemented
15	C₁₆×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
16	C₁₆×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
17	C₅×#368	C₄	1	1	yes	characteristic, complemented
18	#64726	C₄	1	1	yes	characteristic, complemented
19	#64713	C₄	1	2	yes	complemented
20	#12670	—	5	1	no	maximal, Sylow
21	#4130	—	2	2	no	—
22	C₂×C₄×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
23	C₂×D₅⋊C₈	D₄	1	1	yes	characteristic
24	D₅×M₁₆	—	2	2	no	—
25	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
26	C₂×#674	—	2	1	no	—
27	#3994	C₂×C₄	1	4	yes	—
28	#3993	—	2	2	no	—
29	#3992	—	4	2	no	—
30	C₈×AGL(1,𝔽₅)	—	4	2	no	—
31	C₈×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
32	C₈×AGL(1,𝔽₅)	Q₈	1	2	yes	—
33	D₅⋊C₁₆	—	2	1	no	—
34	D₅⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₅×#108	C₂×C₄	1	2	yes	—
37	#3954	—	2	2	no	—
38	#3951	C₂×C₄	1	2	yes	—
39	C₂×#672	C₂×C₄	1	1	yes	characteristic
40	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
41	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
42	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
43	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×#368	—	5	1	no	—
45	#2158	—	5	1	no	—
46	C₂×C₄×C₁₆	—	5	1	no	—
47	#1772	—	5	2	no	—
48	#1382	—	5	2	no	—
49	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
50	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊AGL(1,𝔽₅)	—	4	2	no	—
52	C₂×#170	Q₁₆	1	2	yes	—
53	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
54	C₄×AGL(1,𝔽₅)	—	2	1	no	—
55	C₄×AGL(1,𝔽₅)	—	4	2	no	—
56	C₄×AGL(1,𝔽₅)	C₄⋊C₄	1	2	yes	—
57	D₅⋊C₈	C₄⋊C₄	1	2	yes	—
58	D₅⋊C₈	D₈	1	1	yes	characteristic
59	D₅⋊C₈	D₈	1	1	yes	characteristic
60	C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	#681	—	2	2	no	—
63	C₂×#168	#41	1	1	yes	characteristic
64	#676	—	4	2	no	—
65	C₈×D₅	—	4	2	no	—
66	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
67	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
68	C₈×D₅	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₅	#41	1	1	yes	characteristic
70	#674	—	2	2	no	—
71	C₅×C₁₆	C₄²	1	1	yes	characteristic
72	C₅×C₁₆	C₄²	1	1	yes	characteristic
73	#672	C₄²	1	1	yes	characteristic
74	#672	C₄²	1	1	yes	characteristic
75	C₂²×C₁₆	—	5	1	no	—
76	C₂²×C₁₆	—	5	1	no	—
77	C₂×#108	—	5	2	no	—
78	#427	—	5	2	no	—
79	#405	—	10	2	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	#368	—	5	1	no	—
82	#368	—	5	2	no	—
83	#368	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
84	#367	—	5	2	no	—
85	#366	—	5	2	no	—
86	C₄×C₁₆	—	5	2	no	—
87	C₄×C₁₆	—	5	2	no	—
88	#339	—	10	2	no	—
89	C₂²×D₅	#135	1	1	yes	characteristic
90	C₂×AGL(1,𝔽₅)	—	2	4	no	—
91	C₂×AGL(1,𝔽₅)	—	2	2	no	—
92	C₂×AGL(1,𝔽₅)	—	4	4	no	—
93	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
95	C₄×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₅	C₄×D₄	1	1	yes	characteristic
97	C₄×D₅	C₂×D₈	1	1	yes	characteristic
98	C₄×D₅	#135	1	1	yes	characteristic
99	#170	#107	1	2	yes	—
100	#170	#107	1	2	yes	—
101	C₅×C₈	—	4	2	no	—
102	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
103	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
104	#168	—	4	2	no	—
105	#168	#117	1	1	yes	characteristic
106	#168	C₄×Q₈	1	1	yes	characteristic
107	C₂×M₁₆	—	10	2	no	—
108	C₂²×C₈	—	5	1	no	—
109	C₂²×C₈	—	5	1	no	—
110	#117	—	10	2	no	—
111	C₂×C₄²	—	5	1	no	—
112	C₂×C₁₆	—	5	1	no	—
113	C₂×C₁₆	—	5	1	no	—
114	C₂×C₁₆	—	5	1	no	—
115	C₂×C₁₆	—	5	1	no	—
116	C₂×C₁₆	—	5	1	no	—
117	C₂×C₁₆	—	5	2	no	—
118	C₂×C₁₆	—	10	1	no	—
119	C₂×C₁₆	—	10	1	no	—
120	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	#108	—	5	2	no	—
122	#108	—	10	2	no	—
123	#108	C₂×AGL(1,𝔽₅)	1	2	yes	—
124	#107	—	5	4	no	—
125	#106	—	10	2	no	—
126	#97	—	20	2	no	—
127	C₄×C₈	—	5	2	no	—
128	C₄×C₈	—	5	2	no	—
129	C₄×C₈	—	20	2	no	—
130	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
131	C₂×D₅	—	2	2	no	—
132	C₂×D₅	#427	1	1	yes	characteristic
133	C₂×D₅	C₄×D₈	1	1	yes	characteristic
134	AGL(1,𝔽₅)	—	2	4	no	—
135	AGL(1,𝔽₅)	—	8	4	no	—
136	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₅	C₄×D₈	1	1	yes	characteristic
138	Dic₅	C₂×#107	1	1	yes	characteristic
139	Dic₅	C₄×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄	—	5	1	no	—
141	C₂²×C₄	—	10	1	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	—	10	2	no	—
144	M₁₆	—	20	2	no	—
145	C₂×C₈	—	5	2	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	1	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	5	1	no	—
152	C₂×C₈	—	5	1	no	—
153	C₂×C₈	—	5	1	no	—
154	C₂×C₈	—	20	2	no	—
155	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₄⋊C₄	—	10	4	no	—
157	C₂²⋊C₄	—	20	2	no	—
158	C₄²	—	5	2	no	—
159	C₄²	—	10	1	no	—
160	C₄²	—	20	2	no	—
161	C₁₆	—	5	1	no	—
162	C₁₆	—	5	1	no	—
163	C₁₆	—	10	2	no	—
164	C₁₆	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₁₆	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₅	—	2	1	no	—
167	C₂×C₅	C₄×#107	1	1	yes	characteristic
168	D₅	—	2	1	no	—
169	D₅	#2178	1	2	yes	—
170	C₂³	—	5	1	no	—
171	C₂×C₄	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	5	1	no	—
175	C₂×C₄	—	5	1	no	—
176	C₂×C₄	—	10	2	no	—
177	C₂×C₄	—	10	4	no	—
178	C₂×C₄	—	20	4	no	—
179	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₈	—	4	2	no	—
181	C₈	—	5	2	no	—
182	C₈	—	5	2	no	—
183	C₈	—	5	1	no	—
184	C₈	—	5	1	no	—
185	C₈	—	20	2	no	—
186	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₈	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₅	#12670	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	5	1	no	—
190	C₂²	—	5	1	no	—
191	C₂²	—	10	2	no	—
192	C₂²	Q₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₄	—	5	1	no	—
194	C₄	—	5	1	no	—
195	C₄	—	10	4	no	—
196	C₄	—	40	4	no	—
197	C₄	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₄	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	5	2	no	—
201	C₂	—	10	1	no	—
202	C₂	AGL(1,𝔽₅)×#107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604107

C₂

yes

maximal, characteristic

C₂×C₁₆×AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

#10604089

C₂

yes

maximal

D₅×#368

C₂

yes

maximal, characteristic

#10598626

C₂

yes

maximal

#65728

—

#65721

C₂²

yes

—

C₂×C₈×AGL(1,𝔽₅)

C₂²

yes

characteristic

C₂×D₅⋊C₁₆

C₂²

yes

characteristic

C₂×D₅×C₁₆

C₂²

yes

characteristic

D₅×#108

C₂²

yes

—

#64900

—

#64850

C₄

yes

complemented

#64849

C₄

yes

complemented

C₁₆×AGL(1,𝔽₅)

C₄

yes

complemented

C₁₆×AGL(1,𝔽₅)

C₄

yes

complemented

C₅×#368

C₄

yes

characteristic, complemented

#64726

C₄

yes

characteristic, complemented

#64713

C₄

yes

—

—

C₂×C₄×AGL(1,𝔽₅)

D₄

yes

characteristic

C₂×D₅⋊C₈

D₄

yes

characteristic

D₅×M₁₆

—

C₂×C₈×D₅

C₂³

yes

characteristic

C₂×#674

—

#3994

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₈×AGL(1,𝔽₅)

Q₈

yes

—

D₅⋊C₁₆

—

D₅⋊C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#108

C₂×C₄

yes

—

#3954

—

#3951

C₂×C₄

yes

—

C₂×#672

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

C₂×#368

—

#2158

—

C₂×C₄×C₁₆

—

#1772

—

#1382

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

—

C₂×#170

Q₁₆

yes

—

—

—

—

C₄⋊C₄

yes

—

D₅⋊C₈

C₄⋊C₄

yes

—

D₅⋊C₈

D₈

yes

characteristic

D₅⋊C₈

D₈

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

C₂²×C₄

yes

—

yes

—

C₈×D₅

—

C₈×D₅

C₂²×C₄

yes

characteristic

C₈×D₅

C₂²×C₄

yes

characteristic

C₈×D₅

C₂×Q₈

yes

C₈×D₅

yes

—

C₅×C₁₆

C₄²

yes

characteristic

C₅×C₁₆

C₄²

yes

characteristic

#672

C₄²

yes

characteristic

#672

C₄²

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

—

—

yes

characteristic, complemented

#367

—

#366

—

C₄×C₁₆

—

C₄×C₁₆

—

#339

—

C₂²×D₅

yes

—

—

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×Dic₅

C₂×Q₁₆

yes

characteristic

C₄×D₅

C₂×C₄⋊C₄

yes

characteristic

C₄×D₅

C₄×D₄

yes

characteristic

C₄×D₅

C₂×D₈

yes

C₄×D₅

yes

yes

—

100

#170

#107

yes

—

101

C₅×C₈

—

102

C₅×C₈

C₂×C₄²

yes

characteristic

103

C₅×C₈

C₄×Q₈

yes

characteristic

104

#168

—

105

yes

106

C₄×Q₈

yes

characteristic

107

C₂×M₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

#117

—

111

C₂×C₄²

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

C₂×AGL(1,𝔽₅)

yes

characteristic

121

#108

—

122

#108

—

123

#108

C₂×AGL(1,𝔽₅)

yes

—

124

#107

—

125

#106

—

126

#97

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂²×C₅

C₄×Q₁₆

yes

characteristic

131

C₂×D₅

—

132

C₂×D₅

#427

yes

characteristic

133

C₂×D₅

C₄×D₈

yes

characteristic

134

AGL(1,𝔽₅)

—

135

AGL(1,𝔽₅)

—

136

C₄×C₅

C₄×C₄⋊C₄

yes

characteristic

137

C₄×C₅

C₄×D₈

yes

characteristic

138

Dic₅

C₂×#107

yes

characteristic

139

Dic₅

C₄×Q₁₆

yes

characteristic

140

C₂²×C₄

—

141

C₂²×C₄

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

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

characteristic

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₁₆

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

C₄×AGL(1,𝔽₅)

yes

characteristic

165

C₁₆

C₄×AGL(1,𝔽₅)

yes

characteristic

166

C₂×C₅

—

167

C₂×C₅

C₄×#107

yes

characteristic

168

D₅

—

169

D₅

#2178

yes

—

170

C₂³

—

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

D₄×AGL(1,𝔽₅)

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

187

C₈

Q₈×AGL(1,𝔽₅)

yes

characteristic

188

C₅

#12670

yes

characteristic, complemented, Sylow

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

Q₁₆×AGL(1,𝔽₅)

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₄⋊C₄×AGL(1,𝔽₅)

yes

characteristic

198

C₄

D₈×AGL(1,𝔽₅)

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

AGL(1,𝔽₅)×#107

yes

characteristic

	subgroup	G/H
center	C₄	D₈×AGL(1,𝔽₅)
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#368	C₄
socle	C₂×C₅	C₄×#107
prime radical	C₂²×D₅	#135
cosocle	C₂×C₈×D₅	C₂³
hypercenter	#368	AGL(1,𝔽₅)
hypocenter	C₅	#12670
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×AGL(1,𝔽₅)

commutator

C₅×C₈

C₂×C₄²

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₂²×D₅

C₂×C₈×D₅

C₂³

C₅

G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
13. #64850	C₄	semidirect
14. #64849	C₄	semidirect
15. C₁₆×AGL(1,𝔽₅)	C₄	semidirect
16. C₁₆×AGL(1,𝔽₅)	C₄	semidirect
17. C₅×#368	C₄	semidirect
18. #64726	C₄	semidirect
19. #64713	C₄	semidirect
83. #368	AGL(1,𝔽₅)	semidirect
188. C₅	#12670	semidirect

H ≅ G/N

type

13. #64850

C₄

semidirect

14. #64849

C₄

semidirect

15. C₁₆×AGL(1,𝔽₅)

C₄

semidirect

16. C₁₆×AGL(1,𝔽₅)

C₄

semidirect

17. C₅×#368

C₄

semidirect

18. #64726

C₄

semidirect

19. #64713

C₄

semidirect

83. #368

AGL(1,𝔽₅)

semidirect

188. C₅

#12670

semidirect

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

subgroup	index
#10604107	2
C₂×C₁₆×AGL(1,𝔽₅)	2
#10604089	2
D₅×#368	2
#10598626	2
#12670	5

subgroup

index

C₂×C₁₆×AGL(1,𝔽₅)

Subgroups — #13026032

Conjugacy classes of subgroups

Complements

Maximal subgroups