Group #13031517 — subgroups

Subgroups — #13031517

Order 1280.

612 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 232 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	#10617206	C₂	1	2	yes	maximal
2	C₈×#713	C₂	1	1	yes	maximal, characteristic
3	#10604020	C₂	1	6	yes	maximal, complemented
4	#10603817	C₂	1	3	yes	maximal, complemented
5	#10603815	C₂	1	3	yes	maximal, complemented
6	C₄×#713	C₂²	1	1	yes	characteristic
7	Q₈×#170	—	2	2	no	—
8	#65705	—	2	6	no	—
9	#65681	C₂²	1	6	yes	—
10	#65680	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	1	yes	characteristic
18	#64883	C₂²	1	3	yes	—
19	#64883	C₂²	1	6	yes	—
20	#64882	C₂²	1	6	yes	—
21	#17835	—	5	1	no	maximal, Sylow
22	C₂×#713	Q₈	1	1	yes	characteristic
23	C₂×#700	—	2	6	no	—
24	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
25	Q₈×Dic₅	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
27	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
28	#4039	D₄	1	3	yes	—
29	#4038	Q₈	1	3	yes	—
30	C₄×D₂₀	C₂×C₄	1	3	yes	—
31	C₄²×D₅	C₂³	1	3	yes	—
32	#4003	—	2	6	no	—
33	C₄⋊#170	—	2	6	no	—
34	C₄⋊#170	C₂³	1	2	yes	—
35	C₄⋊#170	C₂³	1	6	yes	—
36	C₄×#170	—	2	6	no	—
37	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
39	#3952	C₂×C₄	1	6	yes	—
40	C₈×Dic₅	C₂³	1	3	yes	—
41	C₈×Dic₅	C₂³	1	1	yes	characteristic
42	C₄⋊#168	C₂×C₄	1	3	yes	—
43	C₄×#168	C₂×C₄	1	3	yes	—
44	#2991	—	5	2	no	—
45	C₈×#41	—	5	1	no	—
46	#1574	—	5	6	no	—
47	#1566	—	5	3	no	—
48	#1564	—	5	3	no	—
49	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
50	#713	—	2	4	no	—
51	C₂×D₂₀	C₄⋊C₄	1	3	yes	—
52	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
53	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
54	C₂×C₄×D₅	C₂×Q₈	1	3	yes	—
55	C₂×#170	—	2	6	no	—
56	C₂×#170	—	2	2	no	—
57	#700	—	8	6	no	—
58	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
61	C₄²×C₅	C₂²×C₄	1	3	yes	—
62	#685	C₄⋊C₄	1	6	yes	—
63	C₄⋊Dic₅	C₄⋊C₄	1	3	yes	—
64	C₄×Dic₅	C₂×D₄	1	3	yes	—
65	C₄×Dic₅	C₂×Q₈	1	3	yes	—
66	C₄×Dic₅	C₂⁴	1	1	yes	characteristic
67	C₂×#168	C₂²×C₄	1	3	yes	—
68	C₂×#168	C₂²×C₄	1	1	yes	characteristic
69	C₈×D₅	—	2	6	no	—
70	C₈×D₅	—	4	3	no	—
71	C₄×#41	—	5	1	no	—
72	C₈×Q₈	—	5	1	no	—
73	C₈×Q₈	—	10	2	no	—
74	C₈×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
75	#436	—	10	6	no	—
76	C₈×D₄	—	5	3	no	—
77	C₈×D₄	—	5	3	no	—
78	#431	—	5	3	no	—
79	#431	—	5	6	no	—
80	#423	—	5	6	no	—
81	C₂×C₄×C₈	—	5	3	no	—
82	#335	—	5	6	no	—
83	#334	—	5	1	no	—
84	#334	—	5	3	no	—
85	#334	—	5	6	no	—
86	C₂²×D₅	C₂×C₄⋊C₄	1	3	yes	—
87	C₅×Q₈	#108	1	4	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₅	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×Dic₅	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₅	C₂²×Q₈	1	1	yes	characteristic
94	D₂₀	—	2	6	no	—
95	C₄×D₅	—	2	3	no	—
96	C₄×D₅	—	2	3	no	—
97	C₄×D₅	—	2	6	no	—
98	C₄×D₅	—	2	6	no	—
99	#170	—	4	6	no	—
100	#170	—	4	2	no	—
101	C₅×C₈	—	2	3	no	—
102	C₅×C₈	—	2	1	no	—
103	#168	—	2	3	no	—
104	#168	#131	1	2	yes	—
105	C₂×#41	—	5	1	no	—
106	C₂×M₁₆	—	10	6	no	—
107	C₂²×C₈	—	5	3	no	—
108	C₂²×C₈	—	5	3	no	—
109	C₄×Q₈	—	5	1	no	—
110	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₄×D₄	—	5	3	no	—
112	C₄×D₄	—	5	3	no	—
113	#117	—	5	3	no	—
114	C₂×C₄²	—	5	3	no	—
115	C₄⋊C₈	—	5	6	no	—
116	C₄⋊C₈	—	5	2	no	—
117	C₄⋊C₈	—	5	3	no	—
118	C₄⋊C₈	—	10	6	no	—
119	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
120	C₂²⋊C₈	—	5	6	no	—
121	C₂²⋊C₈	—	10	6	no	—
122	C₄×C₈	—	5	1	no	—
123	C₄×C₈	—	5	3	no	—
124	C₄×C₈	—	5	3	no	—
125	C₄×C₈	—	10	6	no	—
126	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
127	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂×D₅	—	2	3	no	—
129	C₂×D₅	—	2	3	no	—
130	C₂×D₅	—	2	3	no	—
131	C₄×C₅	—	2	3	no	—
132	C₄×C₅	C₂×#108	1	6	yes	—
133	C₄×C₅	C₂×#131	1	1	yes	characteristic
134	C₄×C₅	#568	1	1	yes	characteristic
135	Dic₅	—	2	3	no	—
136	Dic₅	—	2	1	no	—
137	Dic₅	#577	1	2	yes	—
138	#41	—	10	4	no	—
139	C₂×Q₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×D₄	—	5	3	no	—
141	C₂²×C₄	—	5	3	no	—
142	C₂²×C₄	—	5	3	no	—
143	C₂²×C₄	—	5	3	no	—
144	M₁₆	—	40	6	no	—
145	C₂×C₈	—	5	1	no	—
146	C₂×C₈	—	5	3	no	—
147	C₂×C₈	—	10	6	no	—
148	C₂×C₈	—	10	2	no	—
149	C₂×C₈	—	10	6	no	—
150	C₂×C₈	—	20	3	no	—
151	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
153	C₄⋊C₄	—	5	3	no	—
154	C₄⋊C₄	C₄⋊AGL(1,𝔽₅)	1	3	yes	—
155	C₂²⋊C₄	—	5	6	no	—
156	C₄²	—	5	3	no	—
157	C₄²	—	5	3	no	—
158	C₄²	—	5	1	no	—
159	C₄²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
160	C₂×C₅	#2906	1	1	yes	characteristic
161	C₂×C₅	#2912	1	1	yes	characteristic
162	C₂×C₅	C₂²×#108	1	1	yes	characteristic
163	D₅	—	4	3	no	—
164	C₂³	—	5	3	no	—
165	Q₈	#3996	1	4	yes	—
166	D₄	—	10	6	no	—
167	C₂×C₄	—	5	1	no	—
168	C₂×C₄	—	5	1	no	—
169	C₂×C₄	—	5	3	no	—
170	C₂×C₄	—	10	6	no	—
171	C₂×C₄	—	10	6	no	—
172	C₂×C₄	—	10	3	no	—
173	C₂×C₄	—	10	3	no	—
174	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
175	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
176	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₈	—	2	1	no	—
178	C₈	—	2	3	no	—
179	C₈	—	5	2	no	—
180	C₈	—	10	3	no	—
181	C₈	—	20	6	no	—
182	C₈	—	20	2	no	—
183	C₅	#17835	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	10	3	no	—
185	C₂²	—	10	3	no	—
186	C₂²	—	10	3	no	—
187	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	5	2	no	—
190	C₄	—	10	1	no	—
191	C₄	—	10	3	no	—
192	C₄	C₂×#3996	1	6	yes	—
193	C₄	#66260	1	1	yes	characteristic
194	C₄	D₅⋊#131	1	1	yes	characteristic
195	C₂	—	20	3	no	—
196	C₂	#10617186	1	1	yes	characteristic
197	C₂	C₂²×#3996	1	1	yes	characteristic
198	C₂	#10617416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10617206

C₂

yes

maximal

C₈×#713

C₂

yes

maximal, characteristic

#10604020

C₂

yes

maximal, complemented

#10603817

C₂

yes

maximal, complemented

#10603815

C₂

yes

maximal, complemented

C₄×#713

C₂²

yes

—

—

C₂²

yes

—

#65680

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

characteristic

#64883

C₂²

yes

—

#64883

C₂²

yes

—

#64882

C₂²

yes

—

#17835

—

maximal, Sylow

C₂×#713

Q₈

yes

characteristic

C₂×#700

—

C₄×C₅×Q₈

C₂×C₄

yes

characteristic

Q₈×Dic₅

D₄

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

#4039

D₄

yes

—

#4038

Q₈

yes

—

C₄×D₂₀

C₂×C₄

yes

—

C₄²×D₅

C₂³

yes

—

#4003

—

C₄⋊#170

—

C₄⋊#170

C₂³

yes

—

C₄⋊#170

C₂³

yes

—

C₄×#170

—

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

—

#2991

—

C₈×#41

—

#1574

—

#1566

—

#1564

—

C₂×C₅×Q₈

C₄⋊C₄

yes

characteristic

#713

—

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

—

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

—

C₂×#168

C₂²×C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

AGL(1,𝔽₅)

yes

characteristic

#436

—

C₈×D₄

—

C₈×D₄

—

#431

—

#431

—

#423

—

C₂×C₄×C₈

—

—

—

—

—

C₂²×D₅

C₂×C₄⋊C₄

yes

—

C₅×Q₈

#108

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

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

100

#170

—

101

C₅×C₈

—

102

C₅×C₈

—

103

#168

—

104

#168

#131

yes

—

105

C₂×#41

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄×Q₈

—

110

C₄×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

111

C₄×D₄

—

112

C₄×D₄

—

113

#117

—

114

C₂×C₄²

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

120

C₂²⋊C₈

—

121

C₂²⋊C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

127

C₂²×C₅

C₂²×C₄⋊C₄

yes

characteristic

128

C₂×D₅

—

129

C₂×D₅

—

130

C₂×D₅

—

131

C₄×C₅

—

132

C₄×C₅

C₂×#108

yes

—

133

C₄×C₅

C₂×#131

yes

characteristic

134

C₄×C₅

#568

yes

characteristic

135

Dic₅

—

136

Dic₅

—

137

Dic₅

#577

yes

—

138

#41

—

139

C₂×Q₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

140

C₂×D₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

152

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊AGL(1,𝔽₅)

yes

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

C₂²×AGL(1,𝔽₅)

yes

—

160

C₂×C₅

#2906

yes

characteristic

161

C₂×C₅

#2912

yes

characteristic

162

C₂×C₅

C₂²×#108

yes

characteristic

163

D₅

—

164

C₂³

—

165

Q₈

#3996

yes

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

175

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

176

C₂×C₄

C₂³×AGL(1,𝔽₅)

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₅

#17835

yes

characteristic, complemented, Sylow

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₂²×C₄⋊AGL(1,𝔽₅)

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

C₂×#3996

yes

—

193

C₄

#66260

yes

characteristic

194

C₄

D₅⋊#131

yes

characteristic

195

C₂

—

196

C₂

#10617186

yes

characteristic

197

C₂

C₂²×#3996

yes

characteristic

198

C₂

#10617416

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	Q₈
cosocle	C₄×Dic₅	C₂⁴
hypercenter	C₈×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#17835
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

Q₈

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. #10604020	C₂	semidirect
4. #10603817	C₂	semidirect
5. #10603815	C₂	semidirect
183. C₅	#17835	semidirect

H ≅ G/N

type

3. #10604020

C₂

semidirect

4. #10603817

C₂

semidirect

5. #10603815

C₂

semidirect

183. C₅

#17835

semidirect

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

subgroup	index
#10617206	2
C₈×#713	2
#10604020	2
#10603817	2
#10603815	2
#17835	5

subgroup

index

Subgroups — #13031517

Conjugacy classes of subgroups

Complements

Maximal subgroups