Group #13103909 — subgroups

Subgroups — #13103909

Order 1600.

328 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 81 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	#11717435	C₂	1	1	yes	maximal, characteristic
3	C₂×#87767	C₂	1	1	yes	maximal, characteristic
4	#11716764	C₂	1	2	yes	maximal, complemented
5	#11716762	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂²⋊#1044	C₄	1	2	yes	—
8	C₄⋊#1044	C₄	1	2	yes	—
9	C₄×#1044	C₂²	1	2	yes	—
10	#87860	C₄	1	2	yes	complemented
11	#87856	C₄	1	2	yes	complemented
12	#87855	C₂²	1	2	yes	—
13	C₂×C₅×#168	C₄	1	2	yes	complemented
14	#87767	C₄	1	2	yes	complemented
15	#87767	C₂²	1	1	yes	characteristic
16	#87767	C₂²	1	1	yes	characteristic
17	#65718	—	5	2	no	maximal
18	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
19	C₂×#1044	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
21	C₂×#1039	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
26	#6033	C₂×C₄	1	4	yes	—
27	C₅×#168	C₂×C₄	1	2	yes	—
28	C₅×#168	C₂×C₄	1	2	yes	—
29	#4130	—	5	2	no	—
30	D₅⋊M₁₆	—	5	2	no	—
31	D₅⋊M₁₆	—	10	2	no	—
32	C₂×C₈×D₅	—	5	2	no	—
33	#3992	—	5	4	no	—
34	C₈×AGL(1,𝔽₅)	—	5	4	no	—
35	C₂²×C₅²	C₄²	1	1	yes	characteristic
36	C₂×C₅⋊D₅	—	2	2	no	—
37	C₂×C₅⋊D₅	C₄²	1	1	yes	characteristic
38	C₂×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
39	#1044	—	2	4	no	—
40	C₄×C₅²	C₄²	1	1	yes	characteristic
41	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
42	#1039	C₄²	1	1	yes	characteristic
43	#1039	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅	—	5	2	no	—
45	C₂×C₄×D₅	—	10	2	no	—
46	C₂²⋊AGL(1,𝔽₅)	—	5	4	no	—
47	#704	—	5	4	no	—
48	#704	—	10	4	no	—
49	C₄⋊AGL(1,𝔽₅)	—	5	4	no	—
50	C₄×AGL(1,𝔽₅)	—	5	4	no	—
51	#700	—	5	4	no	—
52	#700	—	10	4	no	—
53	D₅⋊C₈	—	5	4	no	—
54	D₅⋊C₈	—	10	4	no	—
55	C₂×C₅×C₈	—	5	2	no	—
56	C₂×#168	AGL(1,𝔽₅)	1	2	yes	complemented
57	C₈×D₅	—	5	2	no	—
58	C₈×D₅	—	5	2	no	—
59	C₈×D₅	—	5	2	no	—
60	C₈×D₅	—	5	2	no	—
61	#405	—	25	1	no	Sylow
62	C₂×C₅²	—	2	1	no	—
63	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
64	C₅⋊D₅	—	2	1	no	—
65	C₅⋊D₅	#131	1	2	yes	—
66	C₂²×D₅	—	5	2	no	—
67	C₂²×D₅	—	10	2	no	—
68	C₂×AGL(1,𝔽₅)	—	5	8	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₂×Dic₅	—	5	2	no	—
72	C₂×Dic₅	—	10	2	no	—
73	C₄×D₅	—	5	2	no	—
74	C₄×D₅	—	5	2	no	—
75	C₄×D₅	—	5	2	no	—
76	C₄×D₅	—	5	2	no	—
77	C₄×D₅	—	10	2	no	—
78	C₄×D₅	—	10	2	no	—
79	C₄×D₅	—	10	2	no	—
80	C₄×D₅	—	10	2	no	—
81	#170	—	5	8	no	—
82	#170	—	10	8	no	—
83	C₅×C₈	—	5	2	no	—
84	C₅×C₈	—	5	2	no	—
85	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
86	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₂×M₁₆	—	25	1	no	—
88	C₂²×C₈	—	25	1	no	—
89	#117	—	25	1	no	—
90	#97	—	25	2	no	—
91	C₄×C₈	—	25	2	no	—
92	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
95	C₂×D₅	—	5	2	no	—
96	C₂×D₅	—	5	2	no	—
97	C₂×D₅	—	10	2	no	—
98	C₂×D₅	—	10	2	no	—
99	C₂×D₅	—	10	4	no	—
100	C₂×D₅	—	20	4	no	—
101	AGL(1,𝔽₅)	—	10	8	no	—
102	C₄×C₅	—	2	2	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
105	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
106	Dic₅	—	5	2	no	—
107	Dic₅	—	5	2	no	—
108	Dic₅	—	10	2	no	—
109	Dic₅	—	10	2	no	—
110	C₂²×C₄	—	25	1	no	—
111	M₁₆	—	25	2	no	—
112	M₁₆	—	25	2	no	—
113	C₂×C₈	—	5	2	no	—
114	C₂×C₈	—	25	1	no	—
115	C₂×C₈	—	25	1	no	—
116	C₂×C₈	—	25	2	no	—
117	C₂×C₈	—	25	2	no	—
118	C₄⋊C₄	—	25	2	no	—
119	C₂²⋊C₄	—	25	2	no	—
120	C₄²	—	25	2	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	4	2	no	—
124	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
125	D₅	—	5	4	no	—
126	D₅	—	10	4	no	—
127	D₅	—	10	2	no	—
128	D₅	—	20	2	no	—
129	C₂³	—	25	1	no	—
130	C₂×C₄	—	25	1	no	—
131	C₂×C₄	—	25	1	no	—
132	C₂×C₄	—	25	1	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	—	25	1	no	—
135	C₂×C₄	—	25	4	no	—
136	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₈	—	5	2	no	—
138	C₈	—	5	2	no	—
139	C₈	—	25	4	no	—
140	C₅	—	2	2	no	—
141	C₅	#65718	1	2	yes	complemented
142	C₂²	—	25	1	no	—
143	C₂²	—	25	1	no	—
144	C₂²	—	50	2	no	—
145	C₂²	#87830	1	1	yes	characteristic
146	C₄	—	25	1	no	—
147	C₄	—	25	1	no	—
148	C₄	—	50	4	no	—
149	C₄	#87830	1	1	yes	characteristic
150	C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	25	2	no	—
153	C₂	—	50	1	no	—
154	C₂	C₂×#87830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717451

C₂

yes

maximal, characteristic

#11717435

C₂

yes

maximal, characteristic

C₂×#87767

C₂

yes

maximal, characteristic

#11716764

C₂

yes

maximal, complemented

#11716762

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂²⋊#1044

C₄

yes

—

C₄⋊#1044

C₄

yes

—

C₄×#1044

C₂²

yes

—

#87860

C₄

yes

complemented

#87856

C₄

yes

complemented

#87855

C₂²

yes

—

C₂×C₅×#168

C₄

yes

complemented

#87767

C₄

yes

complemented

#87767

C₂²

yes

characteristic

#87767

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

#6033

C₂×C₄

yes

—

C₅×#168

C₂×C₄

yes

—

C₅×#168

C₂×C₄

yes

—

#4130

—

D₅⋊M₁₆

—

D₅⋊M₁₆

—

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

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×#168

AGL(1,𝔽₅)

yes

complemented

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

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

C₂×AGL(1,𝔽₅)

yes

—

#168

C₂×AGL(1,𝔽₅)

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#97

—

C₄×C₈

—

C₅²

#405

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

100

C₂×D₅

—

101

AGL(1,𝔽₅)

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

C₄×AGL(1,𝔽₅)

yes

—

105

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

C₂²×C₄

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

C₂×C₄×AGL(1,𝔽₅)

yes

—

125

D₅

—

126

D₅

—

127

D₅

—

128

D₅

—

129

C₂³

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₅

—

141

C₅

#65718

yes

complemented

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#87830

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#87830

yes

characteristic

150

C₄

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#87830

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂×C₄²
Fitting	C₂×C₄×C₅²	C₂×C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₄

C₂×D₅⋊AGL(1,𝔽₅)

commutator, socle

C₂×C₅²

C₂×C₄²

Fitting

C₂×C₄×C₅²

C₂×C₄

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₄×C₅⋊D₅

C₂³

C₂×C₄

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
4. #11716764	C₂	semidirect
5. #11716762	C₂	semidirect
10. #87860	C₄	semidirect
11. #87856	C₄	semidirect
13. C₂×C₅×#168	C₄	semidirect
14. #87767	C₄	semidirect
56. C₂×#168	AGL(1,𝔽₅)	semidirect
92. C₅²	#405	semidirect
141. C₅	#65718	semidirect

H ≅ G/N

type

4. #11716764

C₂

semidirect

5. #11716762

C₂

semidirect

10. #87860

C₄

semidirect

11. #87856

C₄

semidirect

13. C₂×C₅×#168

C₄

semidirect

14. #87767

C₄

semidirect

56. C₂×#168

AGL(1,𝔽₅)

semidirect

92. C₅²

#405

semidirect

141. C₅

#65718

semidirect

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

subgroup	index
#11717451	2
#11717435	2
C₂×#87767	2
#11716764	2
#11716762	2
#65718	5

subgroup

index

Subgroups — #13103909

Conjugacy classes of subgroups

Complements

Maximal subgroups