Group #13013747 — subgroups

Subgroups — #13013747

Order 1280.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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	#10603846	C₂	1	1	yes	maximal, characteristic
2	#10603831	C₂	1	1	yes	maximal, characteristic
3	#10599322	C₂	1	1	yes	maximal, characteristic
4	#65689	—	2	1	no	—
5	#65126	—	2	1	no	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64893	—	2	1	no	—
8	#64893	C₄	1	2	yes	—
9	#64887	—	2	1	no	—
10	#64886	C₄	1	2	yes	complemented
11	C₄⋊#674	—	2	1	no	—
12	C₅×#334	C₄	1	1	yes	characteristic, complemented
13	#64700	—	2	1	no	—
14	#64677	C₄	1	1	yes	characteristic, complemented
15	#7221	—	5	1	no	maximal, Sylow
16	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
17	C₂×#676	—	4	1	no	—
18	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₅	D₄	1	1	yes	characteristic
20	C₄²×D₅	D₄	1	1	yes	characteristic
21	(C₂×AGL(1,𝔽₅))⋊C₄	—	4	1	no	—
22	(C₂×AGL(1,𝔽₅))⋊C₄	—	4	1	no	—
23	C₂×#674	—	2	2	no	—
24	C₅×C₄⋊C₈	—	2	1	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	#3946	—	4	1	no	—
27	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅	Q₈	1	1	yes	characteristic
29	C₄⋊#168	—	2	1	no	—
30	C₄×#168	D₄	1	1	yes	characteristic
31	#2163	—	5	1	no	—
32	#1835	—	5	1	no	—
33	#1566	—	5	1	no	—
34	C₂²×AGL(1,𝔽₅)	—	4	2	no	—
35	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₅	D₈	1	1	yes	characteristic
37	C₂×C₄×D₅	SD₁₆	1	1	yes	characteristic
38	C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
39	C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
40	C₂×C₅×C₈	—	4	1	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅	SD₁₆	1	1	yes	characteristic
46	C₄×Dic₅	Q₁₆	1	1	yes	characteristic
47	C₂×#168	—	4	1	no	—
48	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
50	#676	—	8	2	no	—
51	C₈×D₅	—	4	1	no	—
52	C₈×D₅	—	4	1	no	—
53	#674	—	4	2	no	—
54	#423	—	10	1	no	—
55	C₂×C₄×C₈	—	5	1	no	—
56	#382	—	10	1	no	—
57	C₄⋊C₁₆	—	10	1	no	—
58	C₂²⋊C₁₆	—	10	1	no	—
59	#346	—	5	2	no	—
60	#340	—	5	2	no	—
61	#340	—	10	1	no	—
62	#334	—	5	1	no	—
63	#334	—	10	1	no	—
64	#334	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
65	C₂²×D₅	#102	1	1	yes	characteristic
66	C₂×AGL(1,𝔽₅)	—	8	2	no	—
67	C₂×AGL(1,𝔽₅)	—	8	2	no	—
68	C₂×AGL(1,𝔽₅)	—	8	2	no	—
69	C₂×C₄×C₅	#95	1	1	yes	characteristic
70	C₂×C₄×C₅	#102	1	1	yes	characteristic
71	C₂×C₄×C₅	#103	1	1	yes	characteristic
72	C₂×Dic₅	#103	1	1	yes	characteristic
73	C₂×Dic₅	#106	1	1	yes	characteristic
74	C₂×Dic₅	#107	1	1	yes	characteristic
75	C₄×D₅	—	2	2	no	—
76	C₄×D₅	—	2	1	no	—
77	C₄×D₅	—	2	1	no	—
78	C₄×D₅	—	4	1	no	—
79	C₅×C₈	—	2	1	no	—
80	C₅×C₈	—	2	1	no	—
81	C₅×C₈	—	8	1	no	—
82	#168	—	2	1	no	—
83	#168	—	2	1	no	—
84	#168	—	8	1	no	—
85	C₂×M₁₆	—	20	1	no	—
86	C₂²×C₈	—	5	1	no	—
87	C₂²×C₈	—	5	1	no	—
88	C₂×C₄⋊C₄	—	10	2	no	—
89	C₂×C₄²	—	5	1	no	—
90	C₂×C₁₆	—	10	2	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	10	1	no	—
93	C₄⋊C₈	—	20	1	no	—
94	C₄×C₈	—	5	1	no	—
95	C₄×C₈	—	5	1	no	—
96	C₄×C₈	—	5	1	no	—
97	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	#95	—	20	1	no	—
99	#95	—	20	1	no	—
100	C₂²×C₅	#340	1	1	yes	characteristic
101	C₂×D₅	—	2	1	no	—
102	C₂×D₅	—	2	1	no	—
103	C₂×D₅	#360	1	2	yes	—
104	AGL(1,𝔽₅)	—	16	2	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
108	C₄×C₅	#344	1	1	yes	characteristic
109	Dic₅	—	2	1	no	—
110	Dic₅	—	2	1	no	—
111	Dic₅	—	2	1	no	—
112	C₂²×C₄	—	5	1	no	—
113	C₂²×C₄	—	5	1	no	—
114	C₂²×C₄	—	5	1	no	—
115	C₂²×C₄	—	20	2	no	—
116	M₁₆	—	40	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	5	1	no	—
119	C₂×C₈	—	5	1	no	—
120	C₂×C₈	—	20	1	no	—
121	C₂×C₈	—	20	1	no	—
122	C₂×C₈	—	20	1	no	—
123	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₄⋊C₄	—	20	2	no	—
126	C₄⋊C₄	—	20	2	no	—
127	C₄²	—	5	1	no	—
128	C₄²	—	5	1	no	—
129	C₄²	—	5	1	no	—
130	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₁₆	—	20	2	no	—
132	C₂×C₅	#1286	1	1	yes	characteristic
133	C₂×C₅	#1386	1	1	yes	characteristic
134	C₂×C₅	#1389	1	1	yes	characteristic
135	D₅	—	2	2	no	—
136	C₂³	—	5	1	no	—
137	C₂×C₄	—	5	1	no	—
138	C₂×C₄	—	5	1	no	—
139	C₂×C₄	—	5	1	no	—
140	C₂×C₄	—	10	2	no	—
141	C₂×C₄	—	10	1	no	—
142	C₂×C₄	—	10	1	no	—
143	C₂×C₄	—	20	1	no	—
144	C₂×C₄	—	40	2	no	—
145	C₂×C₄	—	40	2	no	—
146	C₂×C₄	—	40	2	no	—
147	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#4007	1	1	yes	characteristic
149	C₂×C₄	#4009	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	8	1	no	—
153	C₈	—	10	1	no	—
154	C₈	—	10	1	no	—
155	C₈	—	40	1	no	—
156	C₅	#7221	1	1	yes	characteristic, complemented, Sylow
157	C₂²	—	5	2	no	—
158	C₂²	—	10	1	no	—
159	C₂²	—	10	1	no	—
160	C₂²	#64874	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	10	1	no	—
164	C₄	—	10	1	no	—
165	C₄	—	10	1	no	—
166	C₄	—	80	2	no	—
167	C₄	#64895	1	1	yes	characteristic
168	C₄	#64902	1	1	yes	characteristic
169	C₂	—	10	2	no	—
170	C₂	#10598595	1	1	yes	characteristic
171	C₂	#10598605	1	1	yes	characteristic
172	C₂	#10598632	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603846

C₂

yes

maximal, characteristic

#10603831

C₂

yes

maximal, characteristic

#10599322

C₂

yes

maximal, characteristic

#65689

—

#65126

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64893

—

#64893

C₄

yes

—

#64887

—

#64886

C₄

yes

complemented

C₄⋊#674

—

C₅×#334

C₄

yes

characteristic, complemented

#64700

—

#64677

C₄

yes

characteristic, complemented

#7221

—

maximal, Sylow

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×#676

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

D₄

yes

characteristic

C₄²×D₅

D₄

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

(C₂×AGL(1,𝔽₅))⋊C₄

—

C₂×#674

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3946

—

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

Q₈

yes

characteristic

C₄⋊#168

—

C₄×#168

D₄

yes

—

—

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅

D₈

yes

characteristic

C₂×C₄×D₅

SD₁₆

yes

characteristic

C₄⋊AGL(1,𝔽₅)

—

C₄⋊AGL(1,𝔽₅)

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂²⋊C₄

yes

characteristic

C₄×Dic₅

C₄⋊C₄

yes

characteristic

C₄×Dic₅

SD₁₆

yes

characteristic

C₄×Dic₅

Q₁₆

yes

characteristic

C₂×#168

—

C₂×#168

C₂²⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

—

#674

—

#423

—

C₂×C₄×C₈

—

#382

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

yes

characteristic, complemented

C₂²×D₅

yes

—

—

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

—

—

100

C₂²×C₅

#340

yes

characteristic

101

C₂×D₅

—

102

C₂×D₅

—

103

C₂×D₅

#360

yes

—

104

AGL(1,𝔽₅)

—

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

H(3,ℤ/4)

yes

characteristic

108

C₄×C₅

#344

yes

characteristic

109

Dic₅

—

110

Dic₅

—

111

Dic₅

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

124

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

131

C₁₆

—

132

C₂×C₅

#1286

yes

characteristic

133

C₂×C₅

#1386

yes

characteristic

134

C₂×C₅

#1389

yes

characteristic

135

D₅

—

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

148

C₂×C₄

#4007

yes

characteristic

149

C₂×C₄

#4009

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₅

#7221

yes

characteristic, complemented, Sylow

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#64874

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#64895

yes

characteristic

168

C₄

#64902

yes

characteristic

169

C₂

—

170

C₂

#10598595

yes

characteristic

171

C₂

#10598605

yes

characteristic

172

C₂

#10598632

yes

characteristic

	subgroup	G/H
center	C₂²	#64874
commutator	C₂×C₅×C₈	C₄²
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#334	C₄
socle	C₂²×C₅	#340
prime radical	C₂²×D₅	#102
cosocle	C₄×C₈×D₅	C₂²
hypercenter	#334	AGL(1,𝔽₅)
hypocenter	C₅	#7221
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#64874

commutator

C₂×C₅×C₈

C₄²

C₄×C₈

C₄

C₂²×C₅

C₂²×D₅

C₄×C₈×D₅

C₂²

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
10. #64886	C₄	semidirect
12. C₅×#334	C₄	semidirect
14. #64677	C₄	semidirect
64. #334	AGL(1,𝔽₅)	semidirect
156. C₅	#7221	semidirect

H ≅ G/N

type

10. #64886

C₄

semidirect

12. C₅×#334

C₄

semidirect

14. #64677

C₄

semidirect

64. #334

AGL(1,𝔽₅)

semidirect

156. C₅

#7221

semidirect

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

subgroup	index
#10603846	2
#10603831	2
#10599322	2
#7221	5

subgroup

index

Subgroups — #13013747

Conjugacy classes of subgroups

Complements

Maximal subgroups