Group #13027417 — subgroups

Subgroups — #13027417

Order 1280.

386 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂×#65119	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic
3	C₂×#64859	C₂	1	1	yes	maximal, characteristic
4	#10598777	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₁₀	C₄	1	1	yes	characteristic
6	C₂²×#674	C₂²	1	2	yes	—
7	C₂²×#674	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₂×#3946	C₄	1	2	yes	—
10	C₂×C₈×Dic₅	C₂²	1	2	yes	—
11	C₂×C₄×#168	C₂²	1	1	yes	characteristic
12	#65119	—	2	2	no	—
13	#65119	—	2	2	no	—
14	#64859	—	2	2	no	—
15	C₄⋊#674	—	2	2	no	—
16	#13264	—	5	1	no	maximal, Sylow
17	C₂²×Dic₁₀	C₈	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
21	C₂×#683	C₈	1	2	yes	—
22	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
23	C₂²×#168	C₂×C₄	1	1	yes	characteristic
24	C₂²×#168	C₂³	1	1	yes	characteristic
25	C₄×Dic₁₀	—	2	2	no	—
26	C₄×Dic₁₀	—	2	2	no	—
27	C₂×#674	—	2	2	no	—
28	C₂×#674	—	2	2	no	—
29	C₂×#674	—	2	2	no	—
30	C₂×#674	—	2	4	no	—
31	C₂×#674	Q₈	1	4	yes	complemented
32	C₅×C₄⋊C₈	—	2	2	no	—
33	#3946	—	2	4	no	—
34	C₈×Dic₅	—	2	2	no	—
35	C₈×Dic₅	—	2	2	no	—
36	C₄×#168	D₄	1	2	yes	—
37	C₄×#168	D₄	1	2	yes	—
38	C₂×C₈×Q₈	—	5	1	no	—
39	C₂×C₄⋊C₁₆	—	5	1	no	—
40	C₂×#346	—	5	1	no	—
41	#1374	—	5	4	no	—
42	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₂×C₈	1	2	yes	—
46	C₂²×Dic₅	C₂×C₈	1	2	yes	—
47	C₂×Dic₁₀	—	2	2	no	—
48	C₂×Dic₁₀	—	2	4	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₅	M₁₆	1	2	yes	—
53	C₄⋊Dic₅	—	2	2	no	—
54	#683	—	2	4	no	—
55	C₄×Dic₅	—	2	2	no	—
56	C₄×Dic₅	—	2	2	no	—
57	C₂×#168	—	2	2	no	—
58	C₂×#168	C₂²⋊C₄	1	2	yes	—
59	C₂×#168	C₂×D₄	1	1	yes	characteristic
60	C₂×#168	C₂×D₄	1	1	yes	characteristic
61	C₂×#168	C₂×Q₈	1	2	yes	—
62	C₂×#168	#41	1	2	yes	—
63	#674	—	2	4	no	—
64	#674	—	4	2	no	—
65	C₂×C₄×Q₈	—	5	1	no	—
66	C₂²×C₁₆	—	5	2	no	—
67	C₂²×C₁₆	—	5	2	no	—
68	C₈×Q₈	—	10	2	no	—
69	C₈×Q₈	—	10	2	no	—
70	C₂×C₄⋊C₈	—	5	2	no	—
71	C₂×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×C₄×C₈	—	5	1	no	—
73	C₂×C₄×C₈	—	5	2	no	—
74	C₄⋊C₁₆	—	10	2	no	—
75	#346	—	10	2	no	—
76	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	#131	1	2	yes	—
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	—	2	2	no	—
87	C₂×Dic₅	—	2	2	no	—
88	C₂×Dic₅	—	2	4	no	—
89	C₂×Dic₅	M₃₂	1	4	yes	—
90	Dic₁₀	—	2	8	no	—
91	C₅×C₈	—	4	2	no	—
92	#168	—	2	2	no	—
93	#168	C₂²⋊Q₈	1	2	yes	—
94	#168	#124	1	1	yes	characteristic
95	#168	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×Q₈	—	5	1	no	—
97	C₂²×C₈	—	5	1	no	—
98	C₂²×C₈	—	5	1	no	—
99	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₄×Q₈	—	10	2	no	—
101	C₄×Q₈	—	10	2	no	—
102	C₂×C₄⋊C₄	—	5	2	no	—
103	C₂×C₄⋊C₄	—	5	1	no	—
104	C₂×C₄²	—	5	2	no	—
105	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₂×C₁₆	—	5	4	no	—
107	C₂×C₁₆	—	10	2	no	—
108	C₂×C₁₆	—	10	2	no	—
109	C₂×C₁₆	—	10	2	no	—
110	C₂×C₁₆	—	10	4	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	10	4	no	—
113	C₄×C₈	—	5	2	no	—
114	C₄×C₈	—	5	2	no	—
115	C₄×C₈	—	10	2	no	—
116	C₄×C₈	—	10	2	no	—
117	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₅	C₈×Q₈	1	2	yes	—
119	C₂²×C₅	C₂×M₃₂	1	2	yes	—
120	C₂²×C₅	#504	1	2	yes	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	2	2	no	—
123	C₄×C₅	#391	1	1	yes	characteristic
124	C₄×C₅	#408	1	1	yes	characteristic
125	C₄×C₅	#446	1	2	yes	—
126	Dic₅	—	2	4	no	—
127	Dic₅	—	4	2	no	—
128	C₂×Q₈	—	10	4	no	—
129	C₂×Q₈	—	10	2	no	—
130	C₂²×C₄	—	5	2	no	—
131	C₂²×C₄	—	5	2	no	—
132	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
133	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
134	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	5	2	no	—
139	C₂×C₈	—	5	1	no	—
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	5	2	no	—
142	C₂×C₈	—	10	2	no	—
143	C₄⋊C₄	—	10	4	no	—
144	C₄⋊C₄	—	10	2	no	—
145	C₄²	—	10	2	no	—
146	C₄²	—	10	2	no	—
147	C₄²	#700	1	2	yes	—
148	C₄²	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
149	C₁₆	—	10	4	no	—
150	C₁₆	—	20	2	no	—
151	C₂×C₅	#1984	1	1	yes	characteristic
152	C₂×C₅	#2112	1	1	yes	characteristic
153	C₂×C₅	#2113	1	1	yes	characteristic
154	C₂×C₅	#2183	1	4	yes	—
155	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
156	Q₈	—	10	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	5	4	no	—
160	C₂×C₄	—	10	2	no	—
161	C₂×C₄	—	10	2	no	—
162	C₂×C₄	—	10	4	no	—
163	C₂×C₄	—	10	2	no	—
164	C₂×C₄	#4003	1	2	yes	—
165	C₂×C₄	#4003	1	2	yes	—
166	C₂×C₄	C₂×#700	1	1	yes	characteristic
167	C₂×C₄	#4131	1	2	yes	—
168	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₈	—	4	2	no	—
171	C₈	—	5	2	no	—
172	C₈	—	5	1	no	—
173	C₈	—	5	1	no	—
174	C₈	—	10	2	no	—
175	C₅	#13264	1	1	yes	characteristic, complemented, Sylow
176	C₂²	#65703	1	2	yes	—
177	C₂²	D₅⋊M₃₂	1	2	yes	—
178	C₂²	#65725	1	2	yes	—
179	C₂²	C₂×#4003	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	10	4	no	—
183	C₄	—	20	2	no	—
184	C₄	#65709	1	2	yes	—
185	C₄	#65776	1	1	yes	characteristic
186	C₄	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
187	C₂	#10604026	1	4	yes	—
188	C₂	#10604330	1	1	yes	characteristic
189	C₂	#10604705	1	1	yes	characteristic
190	C₂	#10604742	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#65119

C₂

yes

maximal, characteristic

C₂×C₄⋊#674

C₂

yes

maximal, characteristic

C₂×#64859

C₂

yes

maximal, characteristic

#10598777

C₂

yes

maximal

C₂×C₄×Dic₁₀

C₄

yes

characteristic

C₂²×#674

C₂²

yes

—

C₂²×#674

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₄

yes

characteristic

C₂×#3946

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

—

—

—

—

C₂²×Dic₁₀

C₈

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₅

C₈

yes

characteristic

C₂×#683

C₈

yes

—

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂³

yes

characteristic

C₄×Dic₁₀

—

C₄×Dic₁₀

—

—

—

—

—

Q₈

yes

complemented

C₅×C₄⋊C₈

—

#3946

—

C₈×Dic₅

—

C₈×Dic₅

—

C₄×#168

D₄

yes

—

C₄×#168

D₄

yes

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

—

C₂×#346

—

#1374

—

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₈

yes

—

C₂²×Dic₅

C₂×C₈

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

M₁₆

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

C₂²⋊C₄

yes

—

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

—

C₂×#168

#41

yes

—

#674

—

#674

—

C₂×C₄×Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

#346

—

C₂³×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

#131

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

M₃₂

yes

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

C₂²⋊Q₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

—

100

C₄×Q₈

—

101

C₄×Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₂²×C₅

C₂×C₂²⋊C₈

yes

characteristic

118

C₂²×C₅

C₈×Q₈

yes

—

119

C₂²×C₅

C₂×M₃₂

yes

—

120

C₂²×C₅

#504

yes

—

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

#391

yes

characteristic

124

C₄×C₅

#408

yes

characteristic

125

C₄×C₅

#446

yes

—

126

Dic₅

—

127

Dic₅

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

D₅⋊C₈

yes

characteristic

133

C₂²×C₄

D₅⋊C₈

yes

characteristic

134

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

#700

yes

—

148

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

—

149

C₁₆

—

150

C₁₆

—

151

C₂×C₅

#1984

yes

characteristic

152

C₂×C₅

#2112

yes

characteristic

153

C₂×C₅

#2113

yes

characteristic

154

C₂×C₅

#2183

yes

—

155

C₂³

C₂×D₅⋊C₈

yes

characteristic

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#4003

yes

—

165

C₂×C₄

#4003

yes

—

166

C₂×C₄

C₂×#700

yes

characteristic

167

C₂×C₄

#4131

yes

—

168

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

—

169

C₂×C₄

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₅

#13264

yes

characteristic, complemented, Sylow

176

C₂²

#65703

yes

—

177

C₂²

D₅⋊M₃₂

yes

—

178

C₂²

#65725

yes

—

179

C₂²

C₂×#4003

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#65709

yes

—

185

C₄

#65776

yes

characteristic

186

C₄

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

187

C₂

#10604026

yes

—

188

C₂

#10604330

yes

characteristic

189

C₂

#10604705

yes

characteristic

190

C₂

#10604742

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×C₄⋊C₈	C₄
cosocle	C₂²×#168	C₂³
hypercenter	C₂×C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#13264
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

commutator, socle, prime radical

C₂³×C₅

C₂²×C₈

Fitting

C₂×C₅×C₄⋊C₈

C₄

cosocle

C₂²×#168

C₂³

C₂×C₄⋊C₈

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
31. C₂×#674	Q₈	semidirect
175. C₅	#13264	semidirect

H ≅ G/N

type

31. C₂×#674

Q₈

semidirect

175. C₅

#13264

semidirect

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

subgroup	index
C₂×#65119	2
C₂×C₄⋊#674	2
C₂×#64859	2
#10598777	2
#13264	5

subgroup

index

C₂×C₄⋊#674

Subgroups — #13027417

Conjugacy classes of subgroups

Complements

Maximal subgroups