Group #13104856 — subgroups

Subgroups — #13104856

Order 1600.

336 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 84 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	#11717451	C₂	1	1	yes	maximal, characteristic
2	#11717448	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
4	#11716885	C₂	1	2	yes	maximal, complemented
5	C₈×#1044	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂²⋊#1044	C₄	1	2	yes	—
8	#87867	C₄	1	2	yes	complemented
9	C₄⋊#1044	C₄	1	2	yes	—
10	C₄×#1044	C₂²	1	2	yes	—
11	#87863	C₄	1	2	yes	complemented
12	#87862	C₂²	1	2	yes	—
13	C₂×C₅²×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×#6029	C₄	1	1	yes	characteristic, complemented
15	C₈×C₅⋊D₅	C₄	1	1	yes	characteristic, complemented
16	C₈×C₅⋊D₅	C₄	1	1	yes	characteristic, complemented
17	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
18	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
19	#65718	—	5	2	no	maximal
20	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×#1044	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
23	C₂×#1039	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
28	#6034	C₂×C₄	1	4	yes	—
29	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
31	#6029	C₂×C₄	1	1	yes	characteristic
32	#6029	C₂×C₄	1	1	yes	characteristic
33	#4130	—	5	2	no	—
34	D₅⋊M₁₆	—	5	2	no	—
35	C₂×C₈×D₅	—	5	2	no	—
36	C₂×C₈×D₅	—	10	2	no	—
37	#3992	—	5	4	no	—
38	C₈×AGL(1,𝔽₅)	—	5	4	no	—
39	C₂²×C₅²	C₄²	1	1	yes	characteristic
40	C₂×C₅⋊D₅	—	2	2	no	—
41	C₂×C₅⋊D₅	C₄²	1	1	yes	characteristic
42	C₂×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
43	#1044	—	2	4	no	—
44	C₄×C₅²	C₄²	1	1	yes	characteristic
45	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
46	#1039	C₄²	1	1	yes	characteristic
47	#1039	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	—	5	2	no	—
49	C₂×C₄×D₅	—	10	2	no	—
50	C₂²⋊AGL(1,𝔽₅)	—	5	4	no	—
51	#704	—	5	4	no	—
52	C₄⋊AGL(1,𝔽₅)	—	5	4	no	—
53	C₄×AGL(1,𝔽₅)	—	5	4	no	—
54	#700	—	5	4	no	—
55	D₅⋊C₈	—	5	4	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	AGL(1,𝔽₅)	1	2	yes	complemented
58	C₂×#168	—	5	2	no	—
59	C₂×#168	—	10	2	no	—
60	C₈×D₅	—	5	2	no	—
61	C₈×D₅	—	5	2	no	—
62	C₈×D₅	—	5	2	no	—
63	C₈×D₅	—	5	2	no	—
64	C₈×D₅	—	10	2	no	—
65	C₈×D₅	—	10	2	no	—
66	C₈×D₅	—	10	2	no	—
67	C₈×D₅	—	10	2	no	—
68	#405	—	25	1	no	Sylow
69	C₂×C₅²	—	2	1	no	—
70	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
71	C₅⋊D₅	—	2	1	no	—
72	C₅⋊D₅	#131	1	2	yes	—
73	C₂²×D₅	—	5	2	no	—
74	C₂²×D₅	—	10	2	no	—
75	C₂×AGL(1,𝔽₅)	—	5	8	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
78	C₂×Dic₅	—	5	2	no	—
79	C₂×Dic₅	—	10	2	no	—
80	C₄×D₅	—	5	2	no	—
81	C₄×D₅	—	5	2	no	—
82	C₄×D₅	—	5	2	no	—
83	C₄×D₅	—	5	2	no	—
84	C₄×D₅	—	10	2	no	—
85	C₄×D₅	—	10	2	no	—
86	C₄×D₅	—	10	2	no	—
87	C₄×D₅	—	10	2	no	—
88	#170	—	5	8	no	—
89	C₅×C₈	—	2	2	no	—
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
92	C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
93	#168	—	5	2	no	—
94	#168	—	5	2	no	—
95	#168	—	10	2	no	—
96	#168	—	10	2	no	—
97	C₂×M₁₆	—	25	1	no	—
98	C₂²×C₈	—	25	1	no	—
99	#117	—	25	1	no	—
100	#97	—	25	2	no	—
101	C₄×C₈	—	25	2	no	—
102	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
105	C₂×D₅	—	5	2	no	—
106	C₂×D₅	—	5	2	no	—
107	C₂×D₅	—	10	2	no	—
108	C₂×D₅	—	10	2	no	—
109	C₂×D₅	—	10	4	no	—
110	C₂×D₅	—	10	8	no	—
111	AGL(1,𝔽₅)	—	10	8	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
115	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
116	Dic₅	—	5	2	no	—
117	Dic₅	—	5	2	no	—
118	Dic₅	—	10	2	no	—
119	Dic₅	—	10	2	no	—
120	C₂²×C₄	—	25	1	no	—
121	M₁₆	—	25	2	no	—
122	M₁₆	—	25	2	no	—
123	C₂×C₈	—	25	2	no	—
124	C₂×C₈	—	25	1	no	—
125	C₂×C₈	—	25	1	no	—
126	C₂×C₈	—	25	1	no	—
127	C₂×C₈	—	25	1	no	—
128	C₂×C₈	—	25	1	no	—
129	C₂×C₈	#1044	1	1	yes	characteristic, complemented
130	C₄⋊C₄	—	25	2	no	—
131	C₂²⋊C₄	—	25	2	no	—
132	C₄²	—	25	2	no	—
133	C₂×C₅	—	2	4	no	—
134	C₂×C₅	—	2	2	no	—
135	C₂×C₅	—	2	2	no	—
136	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
137	D₅	—	5	4	no	—
138	D₅	—	10	4	no	—
139	D₅	—	10	4	no	—
140	D₅	—	10	2	no	—
141	C₂³	—	25	1	no	—
142	C₂×C₄	—	25	1	no	—
143	C₂×C₄	—	25	1	no	—
144	C₂×C₄	—	25	1	no	—
145	C₂×C₄	—	25	1	no	—
146	C₂×C₄	—	25	1	no	—
147	C₂×C₄	—	25	4	no	—
148	C₂×C₄	C₂×#1044	1	1	yes	characteristic
149	C₈	—	25	4	no	—
150	C₈	—	25	1	no	—
151	C₈	—	25	1	no	—
152	C₈	C₂×#1044	1	1	yes	characteristic
153	C₈	C₂×#1044	1	1	yes	characteristic
154	C₅	—	2	2	no	—
155	C₅	#65718	1	2	yes	complemented
156	C₂²	—	25	1	no	—
157	C₂²	—	25	1	no	—
158	C₂²	—	50	2	no	—
159	C₂²	C₄×#1044	1	1	yes	characteristic
160	C₄	—	25	1	no	—
161	C₄	—	25	1	no	—
162	C₄	—	50	4	no	—
163	C₄	C₄×#1044	1	1	yes	characteristic
164	C₄	C₂²×#1044	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	25	2	no	—
167	C₂	—	50	1	no	—
168	C₂	C₂×C₄×#1044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717451

C₂

yes

maximal, characteristic

#11717448

C₂

yes

maximal, characteristic

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic

#11716885

C₂

yes

maximal, complemented

C₈×#1044

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂²⋊#1044

C₄

yes

—

#87867

C₄

yes

complemented

C₄⋊#1044

C₄

yes

—

C₄×#1044

C₂²

yes

—

#87863

C₄

yes

complemented

#87862

C₂²

yes

—

C₂×C₅²×C₈

C₄

yes

characteristic, complemented

C₂×#6029

C₄

yes

characteristic, complemented

C₈×C₅⋊D₅

C₄

yes

characteristic, complemented

C₈×C₅⋊D₅

C₄

yes

characteristic, complemented

C₈×C₅⋊D₅

C₂²

yes

characteristic

C₈×C₅⋊D₅

C₂²

yes

characteristic

#65718

—

maximal

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×#1044

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂³

yes

characteristic

#6034

C₂×C₄

yes

—

C₅²×C₈

C₂×C₄

yes

characteristic

C₅²×C₈

C₂×C₄

yes

characteristic

#6029

C₂×C₄

yes

characteristic

#6029

C₂×C₄

yes

characteristic

#4130

—

D₅⋊M₁₆

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

#3992

—

C₈×AGL(1,𝔽₅)

—

C₂²×C₅²

C₄²

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄²

yes

characteristic

C₂×C₅⋊D₅

C₂²×C₄

yes

characteristic

#1044

—

C₄×C₅²

C₄²

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄²

yes

characteristic

#1039

C₂²×C₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂²⋊AGL(1,𝔽₅)

—

—

—

—

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

yes

—

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

#405

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄²

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

#131

yes

—

C₂²×D₅

—

C₂²×D₅

—

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

100

#97

—

101

C₄×C₈

—

102

C₅²

#405

yes

characteristic, complemented, Sylow

103

C₂²×C₅

—

104

C₂²×C₅

C₄×AGL(1,𝔽₅)

yes

—

105

C₂×D₅

—

106

C₂×D₅

—

107

C₂×D₅

—

108

C₂×D₅

—

109

C₂×D₅

—

110

C₂×D₅

—

111

AGL(1,𝔽₅)

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

C₄×AGL(1,𝔽₅)

yes

—

115

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

116

Dic₅

—

117

Dic₅

—

118

Dic₅

—

119

Dic₅

—

120

C₂²×C₄

—

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

#1044

yes

characteristic, complemented

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₂×C₅

—

134

C₂×C₅

—

135

C₂×C₅

—

136

C₂×C₅

C₂×C₄×AGL(1,𝔽₅)

yes

—

137

D₅

—

138

D₅

—

139

D₅

—

140

D₅

—

141

C₂³

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×#1044

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

C₂×#1044

yes

characteristic

153

C₈

C₂×#1044

yes

characteristic

154

C₅

—

155

C₅

#65718

yes

complemented

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₄×#1044

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₄×#1044

yes

characteristic

164

C₄

C₂²×#1044

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

C₂×C₄×#1044

yes

characteristic

	subgroup	G/H
center	C₈	C₂×#1044
commutator, socle	C₂×C₅²	C₂×C₄²
Frattini	C₄	C₂²×#1044
Fitting	C₂×C₅²×C₈	C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₈	#1044
hypocenter	C₅²	#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₂×C₈

subgroup

G/H

center

C₈

C₂×#1044

commutator, socle

C₂×C₅²

C₂×C₄²

Frattini

C₄

C₂²×#1044

Fitting

C₂×C₅²×C₈

C₄

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₄×C₅⋊D₅

C₂³

C₂×C₈

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₈ ◁ C₂×C₈

N	H ≅ G/N	type
4. #11716885	C₂	semidirect
5. C₈×#1044	C₂	semidirect
8. #87867	C₄	semidirect
11. #87863	C₄	semidirect
13. C₂×C₅²×C₈	C₄	semidirect
14. C₂×#6029	C₄	semidirect
15. C₈×C₅⋊D₅	C₄	semidirect
16. C₈×C₅⋊D₅	C₄	semidirect
57. C₂×C₅×C₈	AGL(1,𝔽₅)	semidirect
102. C₅²	#405	semidirect
129. C₂×C₈	#1044	semidirect
155. C₅	#65718	semidirect

H ≅ G/N

type

4. #11716885

C₂

semidirect

5. C₈×#1044

C₂

semidirect

8. #87867

C₄

semidirect

11. #87863

C₄

semidirect

13. C₂×C₅²×C₈

C₄

semidirect

14. C₂×#6029

C₄

semidirect

15. C₈×C₅⋊D₅

C₄

semidirect

16. C₈×C₅⋊D₅

C₄

semidirect

57. C₂×C₅×C₈

AGL(1,𝔽₅)

semidirect

102. C₅²

#405

semidirect

129. C₂×C₈

#1044

semidirect

155. C₅

#65718

semidirect

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

subgroup	index
#11717451	2
#11717448	2
C₂×C₈×C₅⋊D₅	2
#11716885	2
C₈×#1044	2
#65718	5

subgroup

index

C₂×C₈×C₅⋊D₅

Subgroups — #13104856

Conjugacy classes of subgroups

Complements

Maximal subgroups