Group #13099917 — subgroups

Subgroups — #13099917

Order 1600.

256 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 47 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	#11717448	C₂	1	1	yes	maximal, characteristic
2	#11717438	C₂	1	1	yes	maximal, characteristic
3	C₂×#87769	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	#87867	C₄	1	2	yes	complemented
6	#87863	—	2	1	no	—
7	#87862	—	2	1	no	—
8	C₂²⋊#1043	C₄	1	2	yes	—
9	C₄⋊#1043	—	2	1	no	—
10	C₄×#1043	—	2	1	no	—
11	C₂×C₅×#168	C₄	1	2	yes	complemented
12	#87769	—	2	1	no	—
13	#87769	—	2	1	no	—
14	#64894	—	5	2	no	maximal
15	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×#1043	—	2	2	no	—
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
20	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	Q₈	1	1	yes	characteristic
23	#6034	—	2	2	no	—
24	C₅×#168	—	2	2	no	—
25	#4130	—	5	2	no	—
26	#4130	—	10	2	no	—
27	D₅⋊M₁₆	—	5	2	no	—
28	C₂×#676	—	5	2	no	—
29	C₂²×C₅²	C₄²	1	1	yes	characteristic
30	C₂×C₅⋊D₅	—	4	1	no	—
31	C₂×C₅⋊D₅	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
33	#1043	—	4	2	no	—
34	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
36	#1039	C₂²⋊C₄	1	1	yes	characteristic
37	#1039	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₅	—	5	2	no	—
39	C₂×C₄×D₅	—	10	2	no	—
40	C₂²⋊AGL(1,𝔽₅)	—	5	4	no	—
41	C₂²⋊AGL(1,𝔽₅)	—	10	4	no	—
42	#704	—	5	4	no	—
43	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
44	C₄⋊AGL(1,𝔽₅)	—	10	4	no	—
45	C₄×AGL(1,𝔽₅)	—	10	2	no	—
46	C₄×AGL(1,𝔽₅)	—	10	4	no	—
47	#700	—	10	2	no	—
48	D₅⋊C₈	—	10	2	no	—
49	C₂×C₅×C₈	—	5	2	no	—
50	C₂×#168	AGL(1,𝔽₅)	1	2	yes	complemented
51	#676	—	10	2	no	—
52	#676	—	10	2	no	—
53	#344	—	25	1	no	Sylow
54	C₂×C₅²	—	2	1	no	—
55	C₂×C₅²	#95	1	1	yes	characteristic
56	C₅⋊D₅	—	2	1	no	—
57	C₅⋊D₅	—	2	1	no	—
58	C₂²×D₅	—	5	2	no	—
59	C₂²×D₅	—	10	2	no	—
60	C₂×AGL(1,𝔽₅)	—	10	4	no	—
61	C₂×AGL(1,𝔽₅)	—	10	8	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
64	C₂×Dic₅	—	5	2	no	—
65	C₂×Dic₅	—	10	2	no	—
66	C₄×D₅	—	5	2	no	—
67	C₄×D₅	—	5	2	no	—
68	C₄×D₅	—	5	2	no	—
69	C₄×D₅	—	5	2	no	—
70	C₄×D₅	—	10	2	no	—
71	C₄×D₅	—	10	2	no	—
72	C₄×D₅	—	10	2	no	—
73	C₄×D₅	—	10	2	no	—
74	#170	—	10	4	no	—
75	C₅×C₈	—	10	2	no	—
76	#168	—	2	2	no	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂×M₁₆	—	25	1	no	—
79	#117	—	25	1	no	—
80	C₅²	#344	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	2	no	—
82	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
83	C₂×D₅	—	5	2	no	—
84	C₂×D₅	—	5	2	no	—
85	C₂×D₅	—	10	2	no	—
86	C₂×D₅	—	10	2	no	—
87	C₂×D₅	—	20	2	no	—
88	C₂×D₅	—	20	4	no	—
89	AGL(1,𝔽₅)	—	20	8	no	—
90	AGL(1,𝔽₅)	—	20	4	no	—
91	C₄×C₅	—	2	2	no	—
92	C₄×C₅	—	2	2	no	—
93	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
94	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
95	Dic₅	—	5	2	no	—
96	Dic₅	—	5	2	no	—
97	Dic₅	—	10	2	no	—
98	Dic₅	—	10	2	no	—
99	C₂²×C₄	—	25	1	no	—
100	M₁₆	—	25	2	no	—
101	M₁₆	—	50	1	no	—
102	M₁₆	—	50	1	no	—
103	M₁₆	—	50	1	no	—
104	C₂×C₈	—	5	2	no	—
105	C₂×C₈	—	50	1	no	—
106	C₄⋊C₄	—	50	1	no	—
107	C₂²⋊C₄	—	25	2	no	—
108	C₄²	—	50	1	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	—	4	2	no	—
112	C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
113	D₅	—	10	4	no	—
114	D₅	—	10	2	no	—
115	D₅	—	10	2	no	—
116	D₅	—	20	2	no	—
117	C₂³	—	25	1	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	—	25	1	no	—
121	C₂×C₄	—	25	1	no	—
122	C₂×C₄	—	25	1	no	—
123	C₂×C₄	—	50	2	no	—
124	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₈	—	10	2	no	—
126	C₈	—	50	2	no	—
127	C₅	—	2	2	no	—
128	C₅	#64894	1	2	yes	complemented
129	C₂²	—	25	1	no	—
130	C₂²	—	25	1	no	—
131	C₂²	—	100	1	no	—
132	C₂²	#87830	1	1	yes	characteristic
133	C₄	—	25	1	no	—
134	C₄	—	25	1	no	—
135	C₄	—	100	2	no	—
136	C₄	#87831	1	1	yes	characteristic
137	C₄	#87832	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	50	1	no	—
140	C₂	—	50	1	no	—
141	C₂	#11716776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717448

C₂

yes

maximal, characteristic

#11717438

C₂

yes

maximal, characteristic

C₂×#87769

C₂

yes

maximal, characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

#87867

C₄

yes

complemented

#87863

—

#87862

—

C₂²⋊#1043

C₄

yes

—

C₄⋊#1043

—

C₄×#1043

—

C₂×C₅×#168

C₄

yes

—

—

—

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×#1043

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

Q₈

yes

—

—

—

—

D₅⋊M₁₆

—

C₂×#676

—

C₂²×C₅²

C₄²

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₂²⋊C₄

yes

characteristic

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

#1043

—

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

characteristic

#1039

C₂²⋊C₄

yes

characteristic

#1039

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂²⋊AGL(1,𝔽₅)

—

C₂²⋊AGL(1,𝔽₅)

—

—

—

—

—

—

—

D₅⋊C₈

—

C₂×C₅×C₈

—

yes

—

—

—

C₂×C₅²

—

C₂×C₅²

#95

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×AGL(1,𝔽₅)

—

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

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

#117

—

C₅²

#344

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

—

112

C₂×C₅

(C₂×AGL(1,𝔽₅))⋊C₄

yes

—

113

D₅

—

114

D₅

—

115

D₅

—

116

D₅

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₅

—

128

C₅

#64894

yes

complemented

129

C₂²

—

130

C₂²

—

131

C₂²

—

100

—

132

C₂²

#87830

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

100

—

136

C₄

#87831

yes

characteristic

137

C₄

#87832

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

#11716776

yes

characteristic

	subgroup	G/H
center	C₄	#87831
commutator	C₂²×C₅²	C₄²
Frattini, hypercenter	C₂×C₄	D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅²	C₂×C₄
socle	C₂×C₅²	#95
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅⋊D₅	C₂²
hypocenter	C₅²	#344
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center

C₄

#87831

commutator

C₂²×C₅²

C₄²

Frattini, hypercenter

C₂×C₄

D₅⋊AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅²

C₂×C₄

socle

C₂×C₅²

#95

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₂×C₄×C₅⋊D₅

C₂²

hypocenter

C₅²

#344

LCS

G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
5. #87867	C₄	semidirect
11. C₂×C₅×#168	C₄	semidirect
50. C₂×#168	AGL(1,𝔽₅)	semidirect
80. C₅²	#344	semidirect
128. C₅	#64894	semidirect

H ≅ G/N

type

5. #87867

C₄

semidirect

11. C₂×C₅×#168

C₄

semidirect

50. C₂×#168

AGL(1,𝔽₅)

semidirect

80. C₅²

#344

semidirect

128. C₅

#64894

semidirect

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

subgroup	index
#11717448	2
#11717438	2
C₂×#87769	2
#64894	5

subgroup

index

Subgroups — #13099917

Conjugacy classes of subgroups

Complements

Maximal subgroups