Group #13027190 — subgroups

Subgroups — #13027190

Order 1280.

290 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 116 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	#10604746	C₂	1	2	yes	maximal
2	C₂×#64889	C₂	1	1	yes	maximal, characteristic
3	C₂×#64886	C₂	1	1	yes	maximal, characteristic
4	#10601495	C₂	1	1	yes	maximal, characteristic
5	#10598796	C₂	1	2	yes	maximal
6	C₂×#3998	C₂²	1	2	yes	—
7	C₂²×#674	C₂²	1	1	yes	characteristic
8	C₂²×#674	C₂²	1	1	yes	characteristic
9	C₂×C₅×#108	C₄	1	1	yes	characteristic
10	#65411	C₂²	1	2	yes	—
11	#65395	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	#64889	—	2	2	no	—
14	#64888	—	2	4	no	—
15	#64886	—	2	2	no	—
16	#64775	C₄	1	2	yes	—
17	#13162	—	5	1	no	maximal, Sylow
18	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	#4072	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×#168	C₂³	1	1	yes	characteristic
23	#3998	—	2	2	no	—
24	#3998	—	2	2	no	—
25	C₂×#674	—	2	1	no	—
26	C₂×#674	—	2	1	no	—
27	C₂×#674	—	2	1	no	—
28	C₂×#674	—	2	1	no	—
29	C₂×#674	—	2	2	no	—
30	C₂×#674	D₄	1	2	yes	—
31	C₂×#674	Q₈	1	2	yes	—
32	C₅×#108	—	2	2	no	—
33	#3950	C₈	1	2	yes	—
34	#3949	C₈	1	2	yes	—
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	C₂×#364	—	5	1	no	—
40	#2151	—	5	2	no	—
41	C₂×#346	—	5	1	no	—
42	#1772	—	5	1	no	—
43	#1379	—	5	2	no	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	—	2	1	no	—
46	C₅×M₁₆	—	2	2	no	—
47	C₅×M₁₆	—	2	2	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₅×C₈	M₁₆	1	2	yes	—
53	C₂²⋊Dic₅	—	2	2	no	—
54	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
55	C₄×Dic₅	—	2	2	no	—
56	C₄×Dic₅	—	2	1	no	—
57	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
58	C₂×#168	C₂×D₄	1	1	yes	characteristic
59	C₂×#168	C₂×D₄	1	1	yes	characteristic
60	C₂×#168	C₂×Q₈	1	1	yes	characteristic
61	C₂×#168	C₂×Q₈	1	1	yes	characteristic
62	C₂×#168	#41	1	1	yes	characteristic
63	C₂×#168	#41	1	1	yes	characteristic
64	#674	—	2	2	no	—
65	#674	—	2	2	no	—
66	#674	—	4	2	no	—
67	C₂×M₃₂	—	5	2	no	—
68	C₂²×C₁₆	—	5	1	no	—
69	C₂²×C₁₆	—	5	1	no	—
70	C₂×#108	AGL(1,𝔽₅)	1	1	yes	characteristic
71	#427	—	5	1	no	—
72	#405	—	5	2	no	—
73	C₂×C₄×C₈	—	5	1	no	—
74	#364	—	10	2	no	—
75	C₄⋊C₁₆	—	10	4	no	—
76	#346	—	10	2	no	—
77	#339	—	5	2	no	—
78	C₂³×C₅	#131	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	C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	#131	1	1	yes	characteristic
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	—	2	4	no	—
87	C₂×Dic₅	—	4	2	no	—
88	C₅×C₈	—	4	2	no	—
89	C₅×C₈	C₄⋊C₈	1	2	yes	—
90	C₅×C₈	C₄⋊C₈	1	2	yes	—
91	#168	#121	1	1	yes	characteristic
92	#168	C₂²⋊Q₈	1	1	yes	characteristic
93	#168	#125	1	1	yes	characteristic
94	#168	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
96	C₂²×C₈	—	5	1	no	—
97	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	#117	—	5	2	no	—
99	C₂×C₄²	—	5	1	no	—
100	M₃₂	—	10	2	no	—
101	M₃₂	—	10	2	no	—
102	C₂×C₁₆	—	5	2	no	—
103	C₂×C₁₆	—	5	2	no	—
104	C₂×C₁₆	—	10	1	no	—
105	C₂×C₁₆	—	10	1	no	—
106	C₂×C₁₆	—	10	1	no	—
107	C₂×C₁₆	—	10	1	no	—
108	C₂×C₁₆	—	10	2	no	—
109	#108	—	2	2	no	—
110	#107	—	5	2	no	—
111	#106	—	5	2	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	C₂²×C₅	—	2	2	no	—
117	C₂²×C₅	#424	1	1	yes	characteristic
118	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₅	#446	1	1	yes	characteristic
120	C₄×C₅	#389	1	1	yes	characteristic
121	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₄×C₅	#436	1	1	yes	characteristic
123	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
124	Dic₅	—	4	2	no	—
125	Dic₅	—	4	4	no	—
126	C₂²×C₄	—	10	1	no	—
127	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	5	1	no	—
132	C₂×C₈	—	5	1	no	—
133	C₂×C₈	—	5	1	no	—
134	C₂×C₈	—	5	1	no	—
135	C₂×C₈	—	5	1	no	—
136	C₂×C₈	—	5	1	no	—
137	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
138	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
139	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
140	C₂×C₈	#704	1	2	yes	—
141	C₄⋊C₄	—	5	4	no	—
142	C₂²⋊C₄	—	10	2	no	—
143	C₄²	—	5	2	no	—
144	C₄²	—	10	1	no	—
145	C₄²	—	10	2	no	—
146	C₁₆	—	10	2	no	—
147	C₁₆	—	10	2	no	—
148	C₁₆	—	20	2	no	—
149	C₂×C₅	—	2	1	no	—
150	C₂×C₅	—	2	1	no	—
151	C₂×C₅	#1939	1	1	yes	characteristic
152	C₂×C₅	#2171	1	2	yes	—
153	C₂³	#4133	1	1	yes	characteristic
154	C₂×C₄	—	10	4	no	—
155	C₂×C₄	—	10	2	no	—
156	C₂×C₄	—	20	2	no	—
157	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₄	#4131	1	1	yes	characteristic
160	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₂×C₄	C₂×#704	1	1	yes	characteristic
163	C₈	—	4	2	no	—
164	C₈	—	5	1	no	—
165	C₈	—	5	1	no	—
166	C₈	—	5	1	no	—
167	C₈	—	5	1	no	—
168	C₈	#4004	1	2	yes	—
169	C₈	#4004	1	2	yes	—
170	C₅	#13162	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	2	no	—
172	C₂²	#65702	1	1	yes	characteristic
173	C₂²	#65708	1	1	yes	characteristic
174	C₂²	#65788	1	1	yes	characteristic
175	C₄	—	20	2	no	—
176	C₄	—	20	4	no	—
177	C₄	#65703	1	1	yes	characteristic
178	C₄	#65712	1	1	yes	characteristic
179	C₄	C₂×#4004	1	1	yes	characteristic
180	C₄	#65774	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#10604084	1	2	yes	—
184	C₂	#10604325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604746

C₂

yes

maximal

C₂×#64889

C₂

yes

maximal, characteristic

C₂×#64886

C₂

yes

maximal, characteristic

#10601495

C₂

yes

maximal, characteristic

#10598796

C₂

yes

maximal

C₂×#3998

C₂²

yes

—

C₂²×#674

C₂²

yes

characteristic

C₂²×#674

C₂²

yes

characteristic

C₂×C₅×#108

C₄

yes

characteristic

#65411

C₂²

yes

—

#65395

C₄

yes

characteristic

C₂×C₈×Dic₅

C₂²

yes

—

—

—

C₄

yes

—

#13162

—

maximal, Sylow

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

#4072

C₂×C₄

yes

—

C₂×C₄×Dic₅

C₂×C₄

yes

characteristic

C₂²×#168

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#674

Q₈

yes

—

C₅×#108

—

#3950

C₈

yes

—

#3949

C₈

yes

—

#3947

—

C₈×Dic₅

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

M₁₆

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

C₂×C₈

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₄⋊C₄

yes

—

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

characteristic

C₂×#168

C₂×Q₈

yes

yes

yes

—

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

yes

—

—

C₂×C₄×C₈

—

#364

—

C₄⋊C₁₆

—

#346

—

#339

—

C₂³×C₅

#131

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

#131

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

C₄⋊C₈

yes

—

C₅×C₈

C₄⋊C₈

yes

—

yes

C₂²⋊Q₈

yes

yes

C₄⋊Q₈

yes

characteristic

C₂×M₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

#117

—

C₂×C₄²

—

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

#108

—

110

#107

—

111

#106

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₂²×C₅

—

117

C₂²×C₅

#424

yes

characteristic

118

C₂²×C₅

C₈×D₄

yes

characteristic

119

C₂²×C₅

#446

yes

characteristic

120

C₄×C₅

#389

yes

characteristic

121

C₄×C₅

C₂×C₄⋊C₈

yes

characteristic

122

C₄×C₅

#436

yes

characteristic

123

C₄×C₅

C₈×Q₈

yes

characteristic

124

Dic₅

—

125

Dic₅

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

D₅⋊C₈

yes

characteristic

138

C₂×C₈

D₅⋊C₈

yes

characteristic

139

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

140

C₂×C₈

#704

yes

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₁₆

—

147

C₁₆

—

148

C₁₆

—

149

C₂×C₅

—

150

C₂×C₅

—

151

C₂×C₅

#1939

yes

characteristic

152

C₂×C₅

#2171

yes

—

153

C₂³

#4133

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

158

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

159

C₂×C₄

#4131

yes

characteristic

160

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

161

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

162

C₂×C₄

C₂×#704

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

#4004

yes

—

169

C₈

#4004

yes

—

170

C₅

#13162

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

#65702

yes

characteristic

173

C₂²

#65708

yes

characteristic

174

C₂²

#65788

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

#65703

yes

characteristic

178

C₄

#65712

yes

characteristic

179

C₄

C₂×#4004

yes

characteristic

180

C₄

#65774

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

#10604084

yes

—

184

C₂

#10604325

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×#108	C₄
socle	C₂²×C₅	C₈×D₄
prime radical	C₂³×C₅	#131
cosocle	C₂²×#168	C₂³
hypercenter	C₂×#108	AGL(1,𝔽₅)
hypocenter	C₅	#13162
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×#108

C₄

socle

C₂²×C₅

C₈×D₄

prime radical

C₂³×C₅

#131

cosocle

C₂²×#168

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
170. C₅	#13162	semidirect

H ≅ G/N

type

170. C₅

#13162

semidirect

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

subgroup	index
#10604746	2
C₂×#64889	2
C₂×#64886	2
#10601495	2
#10598796	2
#13162	5

subgroup

index

Subgroups — #13027190

Conjugacy classes of subgroups

Complements

Maximal subgroups