Group #13018790 — subgroups

Subgroups — #13018790

Order 1280.

462 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 212 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×#64884	C₂	1	4	yes	maximal
2	#10604190	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#170	—	2	4	no	—
5	C₂×C₄⋊#170	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₄²×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×#168	C₄	1	1	yes	characteristic
11	#64884	—	2	8	no	—
12	#8806	—	5	1	no	maximal, Sylow
13	C₂²×#170	—	2	4	no	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅	D₄	1	2	yes	—
18	C₂×C₄×Dic₅	Q₈	1	2	yes	—
19	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
20	C₂²×#168	C₂×C₄	1	2	yes	—
21	C₄⋊#170	—	2	8	no	—
22	C₄⋊#170	—	4	8	no	—
23	C₄×C₅×C₈	—	2	2	no	—
24	C₈×Dic₅	—	2	4	no	—
25	C₈×Dic₅	D₄	1	4	yes	—
26	C₈×Dic₅	Q₈	1	4	yes	—
27	C₄×#168	C₈	1	4	yes	complemented
28	#1842	—	5	2	no	—
29	C₄²×C₈	—	5	1	no	—
30	C₂×#335	—	5	4	no	—
31	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
34	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
36	C₂×#170	—	4	4	no	—
37	C₂×#170	—	4	8	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	—	2	2	no	—
40	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
41	C₄²×C₅	C₂×C₈	1	2	yes	—
42	C₄²×C₅	M₁₆	1	2	yes	—
43	C₄×Dic₅	—	2	4	no	—
44	C₄×Dic₅	—	2	2	no	—
45	C₄×Dic₅	D₈	1	4	yes	—
46	C₄×Dic₅	Q₁₆	1	4	yes	—
47	C₄×Dic₅	C₂×D₄	1	2	yes	—
48	C₄×Dic₅	C₂×Q₈	1	2	yes	—
49	C₂×#168	C₄⋊C₄	1	4	yes	—
50	C₂×#168	C₂×C₈	1	4	yes	—
51	C₂×#168	M₁₆	1	4	yes	—
52	C₂×C₄⋊C₈	—	5	4	no	—
53	C₂×C₄⋊C₈	—	10	4	no	—
54	C₂×C₄×C₈	—	5	2	no	—
55	C₂×C₄×C₈	—	5	1	no	—
56	C₂×C₄×C₈	—	5	2	no	—
57	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₄³	—	5	1	no	—
59	#335	—	10	8	no	—
60	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
62	C₂×C₄×C₅	#107	1	4	yes	—
63	C₂×C₄×C₅	#108	1	4	yes	—
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₅	—	2	2	no	—
69	C₂×Dic₅	—	2	2	no	—
70	C₂×Dic₅	—	2	2	no	—
71	C₂×Dic₅	#107	1	4	yes	—
72	C₂×Dic₅	#127	1	1	yes	characteristic
73	C₂×Dic₅	C₄⋊Q₈	1	2	yes	—
74	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×Dic₅	C₂×D₈	1	2	yes	—
76	C₂×Dic₅	C₂×Q₁₆	1	2	yes	—
77	#170	—	8	4	no	—
78	C₅×C₈	—	2	4	no	—
79	#168	C₄⋊C₈	1	8	yes	complemented
80	C₂²×C₈	—	5	2	no	—
81	C₂²×C₈	—	10	4	no	—
82	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
83	C₂×C₄²	—	5	2	no	—
84	C₂×C₄²	—	5	1	no	—
85	C₂×C₄²	—	5	2	no	—
86	C₂×C₄²	—	5	1	no	—
87	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₄⋊C₈	—	10	8	no	—
89	C₄⋊C₈	—	20	8	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	5	4	no	—
92	C₄×C₈	—	5	4	no	—
93	C₄×C₈	—	5	4	no	—
94	C₄×C₈	—	10	4	no	—
95	C₂²×C₅	#384	1	1	yes	characteristic
96	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₅	#423	1	1	yes	characteristic
98	C₂²×C₅	C₂×#107	1	2	yes	—
99	C₂²×C₅	C₂×#108	1	2	yes	—
100	C₄×C₅	#335	1	8	yes	complemented
101	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
102	C₄×C₅	#423	1	2	yes	—
103	Dic₅	—	2	4	no	—
104	Dic₅	—	2	2	no	—
105	Dic₅	#493	1	1	yes	characteristic
106	Dic₅	#494	1	1	yes	characteristic
107	Dic₅	#500	1	2	yes	—
108	C₂²×C₄	—	5	1	no	—
109	C₂²×C₄	—	5	2	no	—
110	C₂²×C₄	—	5	1	no	—
111	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
112	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	5	4	no	—
116	C₂×C₈	—	5	4	no	—
117	C₂×C₈	—	5	4	no	—
118	C₂×C₈	—	20	4	no	—
119	C₂×C₈	—	20	8	no	—
120	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
121	C₄²	—	5	2	no	—
122	C₄²	—	5	2	no	—
123	C₄²	—	5	4	no	—
124	C₄²	—	5	4	no	—
125	C₄²	—	10	2	no	—
126	C₄²	—	10	4	no	—
127	C₄²	D₅⋊C₈	1	2	yes	—
128	C₄²	#704	1	2	yes	—
129	C₂×C₅	C₂×#335	1	2	yes	—
130	C₂×C₅	#1567	1	2	yes	—
131	C₂×C₅	#1842	1	1	yes	characteristic
132	C₂×C₅	#1845	1	1	yes	characteristic
133	C₂×C₅	#1848	1	1	yes	characteristic
134	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×C₄	—	5	1	no	—
136	C₂×C₄	—	5	1	no	—
137	C₂×C₄	—	5	2	no	—
138	C₂×C₄	—	5	2	no	—
139	C₂×C₄	—	5	2	no	—
140	C₂×C₄	—	5	4	no	—
141	C₂×C₄	—	10	2	no	—
142	C₂×C₄	—	10	2	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	#3994	1	4	yes	—
145	C₂×C₄	#3995	1	4	yes	—
146	C₂×C₄	#4004	1	4	yes	—
147	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
148	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
149	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
150	C₂×C₄	C₂×#704	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	5	8	no	—
153	C₈	—	40	4	no	—
154	C₅	#8806	1	1	yes	characteristic, complemented, Sylow
155	C₂²	#65682	1	1	yes	characteristic
156	C₂²	#65688	1	1	yes	characteristic
157	C₂²	C₂×#3994	1	2	yes	—
158	C₂²	C₂×#3995	1	2	yes	—
159	C₂²	C₂×#4004	1	1	yes	characteristic
160	C₄	—	5	1	no	—
161	C₄	—	5	1	no	—
162	C₄	—	5	2	no	—
163	C₄	—	10	4	no	—
164	C₄	—	10	2	no	—
165	C₄	#64884	1	8	yes	complemented
166	C₄	D₅⋊(C₄⋊C₈)	1	2	yes	—
167	C₄	#65758	1	2	yes	—
168	C₂	#10603819	1	2	yes	—
169	C₂	#10603848	1	1	yes	characteristic
170	C₂	#10603858	1	1	yes	characteristic
171	C₂	#10604191	1	1	yes	characteristic
172	C₂	C₂×#64884	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64884

C₂

yes

maximal

#10604190

C₂

yes

maximal

C₄×C₈×Dic₅

C₂

yes

maximal, characteristic

C₂×C₄⋊#170

—

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₈×Dic₅

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₄²×Dic₅

C₂²

yes

characteristic

C₂×C₄×#168

C₄

yes

—

—

C₂²×#170

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

D₄

yes

—

C₂×C₄×Dic₅

Q₈

yes

—

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₄⋊#170

—

C₄⋊#170

—

C₄×C₅×C₈

—

C₈×Dic₅

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₈

yes

complemented

#1842

—

C₄²×C₈

—

C₂×#335

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₄⋊C₄

yes

—

C₂²×Dic₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

C₂×Q₈

yes

characteristic

C₂×#170

—

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂×C₈

yes

—

C₄²×C₅

M₁₆

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

D₈

yes

—

C₄×Dic₅

Q₁₆

yes

—

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₂×C₈

yes

—

C₂×#168

M₁₆

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄³

—

#335

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

#107

yes

—

C₂×C₄×C₅

#108

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#107

yes

—

C₂×Dic₅

#127

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

C₂×Dic₅

C₄⋊Q₈

yes

characteristic

C₂×Dic₅

C₂×D₈

yes

—

C₂×Dic₅

C₂×Q₁₆

yes

—

#170

—

C₅×C₈

—

#168

C₄⋊C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

#384

yes

characteristic

C₂²×C₅

C₂×C₄⋊C₈

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

C₂×#108

yes

—

100

C₄×C₅

#335

yes

complemented

101

C₄×C₅

C₂×C₄⋊C₈

yes

—

102

C₄×C₅

#423

yes

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

#493

yes

characteristic

106

Dic₅

#494

yes

characteristic

107

Dic₅

#500

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

112

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

D₅⋊C₈

yes

—

128

C₄²

#704

yes

—

129

C₂×C₅

C₂×#335

yes

—

130

C₂×C₅

#1567

yes

—

131

C₂×C₅

#1842

yes

characteristic

132

C₂×C₅

#1845

yes

characteristic

133

C₂×C₅

#1848

yes

characteristic

134

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#3994

yes

—

145

C₂×C₄

#3995

yes

—

146

C₂×C₄

#4004

yes

—

147

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

148

C₂×C₄

D₅⋊M₁₆

yes

—

149

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

150

C₂×C₄

C₂×#704

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₅

#8806

yes

characteristic, complemented, Sylow

155

C₂²

#65682

yes

characteristic

156

C₂²

#65688

yes

characteristic

157

C₂²

C₂×#3994

yes

—

158

C₂²

C₂×#3995

yes

—

159

C₂²

C₂×#4004

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#64884

yes

complemented

166

C₄

D₅⋊(C₄⋊C₈)

yes

—

167

C₄

#65758

yes

—

168

C₂

#10603819

yes

—

169

C₂

#10603848

yes

characteristic

170

C₂

#10603858

yes

characteristic

171

C₂

#10604191

yes

characteristic

172

C₂

C₂×#64884

yes

—

	subgroup	G/H
center	C₂³	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#8806
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅×C₈

C₄

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅

C₂³

C₂×C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
27. C₄×#168	C₈	semidirect
79. #168	C₄⋊C₈	semidirect
100. C₄×C₅	#335	semidirect
154. C₅	#8806	semidirect
165. C₄	#64884	semidirect

H ≅ G/N

type

27. C₄×#168

C₈

semidirect

79. #168

C₄⋊C₈

semidirect

100. C₄×C₅

#335

semidirect

154. C₅

#8806

semidirect

165. C₄

#64884

semidirect

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

subgroup	index
C₂×#64884	2
#10604190	2
C₄×C₈×Dic₅	2
#8806	5

subgroup

index

C₂×#64884

#10604190

C₄×C₈×Dic₅

#8806

Subgroups — #13018790

Conjugacy classes of subgroups

Complements

Maximal subgroups