Group #13099910 — subgroups

Subgroups — #13099910

Order 1600.

256 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×#87855	C₂	1	1	yes	maximal, characteristic
3	#11716936	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	#87867	—	2	1	no	—
6	#87863	C₄	1	2	yes	—
7	#87862	—	2	1	no	—
8	C₂×#6033	—	2	1	no	—
9	#87855	—	2	1	no	—
10	#87855	C₄	1	2	yes	—
11	C₅×#681	C₄	1	2	yes	—
12	#87769	—	2	1	no	—
13	#87767	—	2	1	no	—
14	#64901	—	5	2	no	maximal
15	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×#1039	D₄	1	1	yes	characteristic
18	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
22	#6034	—	2	2	no	—
23	#6033	—	2	2	no	—
24	C₅×#168	—	2	2	no	—
25	D₅⋊M₁₆	—	5	2	no	—
26	C₂×D₅⋊C₈	—	5	2	no	—
27	C₂×D₅⋊C₈	—	10	2	no	—
28	D₅×M₁₆	—	5	2	no	—
29	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₅⋊D₅	—	2	2	no	—
31	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₅²	C₄²	1	1	yes	characteristic
34	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
35	#1039	C₂²⋊C₄	1	1	yes	characteristic
36	#1039	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₅	—	5	2	no	—
38	C₂×C₄×D₅	—	10	2	no	—
39	#704	—	10	2	no	—
40	C₂×#170	—	10	2	no	—
41	C₂×#170	—	10	4	no	—
42	#700	—	5	4	no	—
43	D₅⋊C₈	—	5	4	no	—
44	D₅⋊C₈	—	10	4	no	—
45	D₅⋊C₈	—	10	2	no	—
46	D₅⋊C₈	—	10	2	no	—
47	D₅⋊C₈	—	10	4	no	—
48	C₅×M₁₆	—	5	2	no	—
49	#681	AGL(1,𝔽₅)	1	2	yes	—
50	#676	—	10	2	no	—
51	C₈×D₅	—	10	2	no	—
52	#341	—	25	1	no	Sylow
53	C₂×C₅²	—	2	1	no	—
54	C₂×C₅²	#95	1	1	yes	characteristic
55	C₅⋊D₅	—	2	1	no	—
56	C₅⋊D₅	#108	1	2	yes	—
57	C₂²×D₅	—	5	2	no	—
58	C₂²×D₅	—	10	2	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
61	C₂×Dic₅	—	5	2	no	—
62	C₂×Dic₅	—	10	2	no	—
63	C₄×D₅	—	5	2	no	—
64	C₄×D₅	—	5	2	no	—
65	C₄×D₅	—	5	2	no	—
66	C₄×D₅	—	5	2	no	—
67	C₄×D₅	—	10	2	no	—
68	C₄×D₅	—	10	2	no	—
69	C₄×D₅	—	10	2	no	—
70	C₄×D₅	—	10	2	no	—
71	#170	—	10	4	no	—
72	#170	—	10	4	no	—
73	#170	—	10	8	no	—
74	C₅×C₈	—	10	2	no	—
75	#168	—	2	2	no	—
76	C₂×M₁₆	—	25	1	no	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂²×C₈	—	25	1	no	—
79	C₅²	#341	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₅	—	2	2	no	—
81	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
82	C₂×D₅	—	5	2	no	—
83	C₂×D₅	—	5	2	no	—
84	C₂×D₅	—	10	2	no	—
85	C₂×D₅	—	10	2	no	—
86	C₂×D₅	—	10	4	no	—
87	C₂×D₅	—	10	8	no	—
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	—	2	2	no	—
90	C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
91	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
92	Dic₅	—	5	2	no	—
93	Dic₅	—	5	2	no	—
94	Dic₅	—	10	2	no	—
95	Dic₅	—	10	2	no	—
96	C₂²×C₄	—	25	1	no	—
97	M₁₆	—	5	2	no	—
98	M₁₆	—	25	2	no	—
99	M₁₆	—	50	1	no	—
100	M₁₆	—	50	1	no	—
101	C₂×C₈	—	25	2	no	—
102	C₂×C₈	—	50	1	no	—
103	C₂×C₈	—	50	1	no	—
104	C₂×C₈	—	50	1	no	—
105	C₂×C₈	—	50	1	no	—
106	C₂×C₅	—	2	4	no	—
107	C₂×C₅	—	2	2	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
110	D₅	—	5	4	no	—
111	D₅	—	10	4	no	—
112	D₅	—	10	4	no	—
113	D₅	—	10	2	no	—
114	C₂³	—	25	1	no	—
115	C₂×C₄	—	25	1	no	—
116	C₂×C₄	—	25	1	no	—
117	C₂×C₄	—	25	1	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₈	—	10	2	no	—
122	C₈	—	50	2	no	—
123	C₈	—	50	2	no	—
124	C₅	—	2	2	no	—
125	C₅	#64901	1	2	yes	complemented
126	C₂²	—	25	1	no	—
127	C₂²	—	25	1	no	—
128	C₂²	—	50	2	no	—
129	C₂²	#87832	1	1	yes	characteristic
130	C₄	—	25	1	no	—
131	C₄	—	25	1	no	—
132	C₄	#87830	1	1	yes	characteristic
133	C₄	#87831	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	25	2	no	—
136	C₂	—	50	1	no	—
137	C₂	#11716776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717448

C₂

yes

maximal, characteristic

C₂×#87855

C₂

yes

maximal, characteristic

#11716936

C₂

yes

maximal, characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

#87867

—

#87863

C₄

yes

—

—

—

—

C₄

yes

—

C₅×#681

C₄

yes

—

—

—

—

C₂²×C₅⋊D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

—

—

—

D₅⋊M₁₆

—

C₂×D₅⋊C₈

—

C₂×D₅⋊C₈

—

D₅×M₁₆

—

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₄×C₅²

C₄²

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

characteristic

#1039

C₂²⋊C₄

yes

characteristic

#1039

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

—

—

—

—

D₅⋊C₈

—

D₅⋊C₈

—

D₅⋊C₈

—

D₅⋊C₈

—

D₅⋊C₈

—

C₅×M₁₆

—

#681

AGL(1,𝔽₅)

yes

—

#676

—

C₈×D₅

—

#341

—

Sylow

C₂×C₅²

—

C₂×C₅²

#95

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

#108

yes

—

C₂²×D₅

—

C₂²×D₅

—

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

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₅²

#341

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

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₅

—

107

C₂×C₅

—

108

C₂×C₅

—

109

C₂×C₅

(C₂×AGL(1,𝔽₅))⋊C₄

yes

—

110

D₅

—

111

D₅

—

112

D₅

—

113

D₅

—

114

C₂³

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₅

—

125

C₅

#64901

yes

complemented

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#87832

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#87830

yes

characteristic

133

C₄

#87831

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

#11716776

yes

characteristic

	subgroup	G/H
center	C₄	#87831
commutator	C₄×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	Q₈
cosocle	C₂×C₄×C₅⋊D₅	C₂²
hypocenter	C₅²	#341
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center

C₄

#87831

commutator

C₄×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

Q₈

cosocle

C₂×C₄×C₅⋊D₅

C₂²

hypocenter

C₅²

#341

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
79. C₅²	#341	semidirect
125. C₅	#64901	semidirect

H ≅ G/N

type

79. C₅²

#341

semidirect

125. C₅

#64901

semidirect

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

subgroup	index
#11717448	2
C₂×#87855	2
#11716936	2
#64901	5

subgroup

index

Subgroups — #13099910

Conjugacy classes of subgroups

Complements

Maximal subgroups