Group #13020841 — subgroups

Subgroups — #13020841

Order 1280.

294 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 102 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	#10604738	C₂	1	2	yes	maximal
2	C₂×#64882	C₂	1	1	yes	maximal, characteristic
3	#10604689	C₂	1	1	yes	maximal, characteristic
4	#10599709	C₂	1	1	yes	maximal, characteristic
5	#10598793	C₂	1	2	yes	maximal
6	#65748	C₂²	1	1	yes	characteristic
7	#65748	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#170	C₂²	1	2	yes	—
9	C₂×#3954	C₄	1	2	yes	—
10	M₁₆×Dic₅	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	#64882	—	2	2	no	—
13	#64882	—	2	2	no	—
14	#64881	—	2	2	no	—
15	#170⋊C₈	—	2	2	no	—
16	C₅×#341	C₄	1	1	yes	characteristic
17	#64772	C₄	1	1	yes	characteristic
18	#9610	—	5	1	no	maximal, Sylow
19	C₂²×#170	—	2	2	no	—
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
23	C₂×#681	C₂×C₄	1	2	yes	—
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊#170	—	2	1	no	—
26	C₂²⋊#170	Q₈	1	2	yes	—
27	C₄⋊#170	—	2	2	no	—
28	C₄⋊#170	—	2	2	no	—
29	C₄⋊#170	—	2	1	no	—
30	C₄⋊#170	D₄	1	1	yes	characteristic
31	C₄⋊#170	D₄	1	1	yes	characteristic
32	C₄×#170	—	2	1	no	—
33	C₄×#170	—	2	1	no	—
34	#3954	—	2	4	no	—
35	#3947	—	2	2	no	—
36	C₈×Dic₅	—	2	2	no	—
37	C₈×Dic₅	D₄	1	2	yes	—
38	C₈×Dic₅	Q₈	1	2	yes	—
39	C₄×#108	—	5	1	no	—
40	#1565	—	5	2	no	—
41	C₂×#335	—	5	1	no	—
42	#1452	—	5	1	no	—
43	#1283	—	5	2	no	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
46	C₂²×Dic₅	#41	1	1	yes	characteristic
47	C₂×#170	—	2	1	no	—
48	C₂×#170	—	2	1	no	—
49	C₂×#170	—	2	2	no	—
50	C₂×#170	—	4	2	no	—
51	C₂×#170	—	8	2	no	—
52	C₅×M₁₆	—	2	2	no	—
53	C₅×M₁₆	M₁₆	1	4	yes	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₅	#41	1	1	yes	characteristic
61	#681	—	2	2	no	—
62	#681	—	2	2	no	—
63	C₂×#168	—	2	2	no	—
64	C₂×#168	—	2	2	no	—
65	C₂×#168	C₄⋊C₄	1	2	yes	—
66	#431	—	5	1	no	—
67	#431	—	5	1	no	—
68	C₂×#108	—	5	2	no	—
69	C₂×C₄⋊C₈	—	5	2	no	—
70	C₄×M₁₆	—	5	2	no	—
71	C₂×C₄×C₈	—	5	1	no	—
72	#341	—	5	1	no	—
73	#341	AGL(1,𝔽₅)	1	1	yes	characteristic
74	#335	—	10	2	no	—
75	#335	—	10	2	no	—
76	#334	—	10	2	no	—
77	#322	—	10	2	no	—
78	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₅	#131	1	2	yes	—
83	C₂×Dic₅	—	2	1	no	—
84	C₂×Dic₅	—	2	1	no	—
85	C₂×Dic₅	—	4	1	no	—
86	C₂×Dic₅	#121	1	1	yes	characteristic
87	C₂×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×Dic₅	#125	1	1	yes	characteristic
89	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
90	#170	—	4	2	no	—
91	#170	—	4	1	no	—
92	#170	—	4	1	no	—
93	#170	—	8	2	no	—
94	C₅×C₈	—	2	2	no	—
95	C₅×C₈	—	2	4	no	—
96	#168	—	2	2	no	—
97	#168	—	4	2	no	—
98	C₂×M₁₆	—	5	2	no	—
99	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₂²×C₈	—	5	1	no	—
101	C₂²×C₈	—	10	2	no	—
102	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₂×C₄²	—	5	1	no	—
104	#108	—	10	4	no	—
105	C₄⋊C₈	—	5	1	no	—
106	C₄⋊C₈	—	5	1	no	—
107	C₄⋊C₈	—	10	1	no	—
108	C₄⋊C₈	—	10	2	no	—
109	C₄⋊C₈	—	10	2	no	—
110	C₂²⋊C₈	—	5	2	no	—
111	C₂²⋊C₈	—	10	1	no	—
112	#97	—	10	2	no	—
113	C₄×C₈	—	5	2	no	—
114	C₄×C₈	—	5	2	no	—
115	C₄×C₈	—	10	2	no	—
116	C₄×C₈	—	10	1	no	—
117	C₄×C₈	—	10	1	no	—
118	C₂²×C₅	—	2	1	no	—
119	C₂²×C₅	—	2	1	no	—
120	C₂²×C₅	#389	1	1	yes	characteristic
121	C₂²×C₅	#446	1	2	yes	—
122	C₄×C₅	#423	1	1	yes	characteristic
123	C₄×C₅	#424	1	1	yes	characteristic
124	C₄×C₅	#435	1	2	yes	—
125	Dic₅	—	2	1	no	—
126	Dic₅	—	2	1	no	—
127	Dic₅	—	2	1	no	—
128	Dic₅	#472	1	1	yes	characteristic
129	Dic₅	#473	1	1	yes	characteristic
130	C₂²×C₄	—	5	1	no	—
131	C₂²×C₄	—	5	1	no	—
132	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	10	2	no	—
135	M₁₆	—	10	2	no	—
136	M₁₆	#700	1	4	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	5	2	no	—
140	C₂×C₈	—	10	2	no	—
141	C₂×C₈	—	10	2	no	—
142	C₂×C₈	—	10	1	no	—
143	C₂×C₈	—	10	1	no	—
144	C₂×C₈	—	10	2	no	—
145	C₂×C₈	—	20	2	no	—
146	C₂×C₈	—	40	2	no	—
147	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
148	C₄²	—	5	1	no	—
149	C₄²	—	5	1	no	—
150	C₄²	—	5	1	no	—
151	C₄²	—	5	1	no	—
152	C₂×C₅	—	2	2	no	—
153	C₂×C₅	#1940	1	1	yes	characteristic
154	C₂×C₅	#1952	1	1	yes	characteristic
155	C₂×C₅	#1953	1	1	yes	characteristic
156	C₂³	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₂×C₄	—	5	1	no	—
158	C₂×C₄	—	5	1	no	—
159	C₂×C₄	—	5	1	no	—
160	C₂×C₄	—	5	1	no	—
161	C₂×C₄	—	10	1	no	—
162	C₂×C₄	—	10	1	no	—
163	C₂×C₄	—	20	1	no	—
164	C₂×C₄	C₂×#700	1	2	yes	—
165	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₄	#4131	1	2	yes	—
167	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	2	2	no	—
170	C₈	—	10	2	no	—
171	C₈	—	20	1	no	—
172	C₈	—	20	1	no	—
173	C₈	—	20	2	no	—
174	C₈	—	20	2	no	—
175	C₈	—	40	2	no	—
176	C₅	#9610	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#65709	1	2	yes	—
180	C₂²	#65712	1	1	yes	characteristic
181	C₄	—	5	1	no	—
182	C₄	—	5	1	no	—
183	C₄	—	10	1	no	—
184	C₄	—	10	1	no	—
185	C₄	—	10	1	no	—
186	C₄	#65681	1	1	yes	characteristic
187	C₄	#65693	1	2	yes	—
188	C₄	#65696	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#10603798	1	1	yes	characteristic
191	C₂	#10604088	1	1	yes	characteristic
192	C₂	#10604092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604738

C₂

yes

maximal

C₂×#64882

C₂

yes

maximal, characteristic

#10604689

C₂

yes

maximal, characteristic

#10599709

C₂

yes

maximal, characteristic

#10598793

C₂

yes

maximal

#65748

C₂²

yes

characteristic

#65748

C₂²

yes

characteristic

C₂×C₄⋊#170

C₂²

yes

—

C₂×#3954

C₄

yes

—

M₁₆×Dic₅

C₂²

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#64772

C₄

yes

characteristic

#9610

—

maximal, Sylow

C₂²×#170

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂×#681

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²⋊#170

—

C₂²⋊#170

Q₈

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#170

D₄

yes

—

—

—

—

C₈×Dic₅

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₂×Q₈

yes

C₂²×Dic₅

yes

—

—

—

—

—

C₅×M₁₆

—

C₅×M₁₆

M₁₆

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

C₄×Dic₅

yes

—

—

—

—

C₄⋊C₄

yes

—

#431

—

#431

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

yes

—

—

—

—

C₂³×C₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#121

yes

characteristic

C₂×Dic₅

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅

#125

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

—

—

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

C₂×AGL(1,𝔽₅)

yes

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

103

C₂×C₄²

—

104

#108

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₂²×C₅

—

119

C₂²×C₅

—

120

C₂²×C₅

#389

yes

characteristic

121

C₂²×C₅

#446

yes

—

122

C₄×C₅

#423

yes

characteristic

123

C₄×C₅

#424

yes

characteristic

124

C₄×C₅

#435

yes

—

125

Dic₅

—

126

Dic₅

—

127

Dic₅

—

128

Dic₅

#472

yes

characteristic

129

Dic₅

#473

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

136

M₁₆

#700

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₂×C₅

—

153

C₂×C₅

#1940

yes

characteristic

154

C₂×C₅

#1952

yes

characteristic

155

C₂×C₅

#1953

yes

characteristic

156

C₂³

Q₈×AGL(1,𝔽₅)

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×#700

yes

—

165

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

166

C₂×C₄

#4131

yes

—

167

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₅

#9610

yes

characteristic, complemented, Sylow

177

C₂²

—

178

C₂²

—

179

C₂²

#65709

yes

—

180

C₂²

#65712

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#65681

yes

characteristic

187

C₄

#65693

yes

—

188

C₄

#65696

yes

characteristic

189

C₂

—

190

C₂

#10603798

yes

characteristic

191

C₂

#10604088

yes

characteristic

192

C₂

#10604092

yes

characteristic

	subgroup	G/H
center	C₂²	#65712
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#341	C₄
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	#341	AGL(1,𝔽₅)
hypocenter	C₅	#9610
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#65712

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂³×C₅

C₄×Q₈

cosocle

C₂×C₄×Dic₅

C₂³

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
176. C₅	#9610	semidirect

H ≅ G/N

type

176. C₅

#9610

semidirect

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

subgroup	index
#10604738	2
C₂×#64882	2
#10604689	2
#10599709	2
#10598793	2
#9610	5

subgroup

index

Subgroups — #13020841

Conjugacy classes of subgroups

Complements

Maximal subgroups