Group #13100721 — subgroups

Subgroups — #13100721

Order 1600.

236 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 66 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	#11717210	C₂	1	1	yes	maximal, characteristic
2	#11717037	C₂	1	1	yes	maximal, characteristic
3	C₅×#3940	C₂	1	2	yes	maximal
4	#11716616	C₂	1	2	yes	maximal
5	#11716596	C₂	1	1	yes	maximal, characteristic
6	C₂×#6039	—	2	1	no	—
7	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₄⋊#1039	—	2	1	no	—
9	C₄⋊#1039	C₂²	1	1	yes	characteristic
10	#87806	—	2	1	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₅×#168	C₂²	1	2	yes	—
14	#87781	—	2	1	no	—
15	#65145	—	5	2	no	maximal
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	1	no	—
19	C₂×#1039	D₄	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₂×C₅×Dic₅	—	2	2	no	—
22	#6039	—	4	2	no	—
23	C₅×#168	Q₈	1	4	yes	complemented
24	#4036	—	5	2	no	—
25	#4036	—	10	2	no	—
26	#4034	—	5	2	no	—
27	C₅×#106	—	5	2	no	—
28	#3950	—	5	2	no	—
29	#3947	—	5	2	no	—
30	#3940	D₅	1	2	yes	—
31	#3939	—	5	2	no	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅²	—	4	1	no	—
34	C₄×C₅²	C₂×Q₈	1	2	yes	—
35	#1039	—	2	1	no	—
36	#1039	—	2	1	no	—
37	#1039	—	4	1	no	—
38	C₅×Dic₅	—	4	2	no	—
39	C₂×Dic₁₀	—	10	2	no	—
40	C₂×C₅×C₈	—	5	2	no	—
41	C₅×C₄⋊C₄	—	2	2	no	—
42	C₅×C₄⋊C₄	—	5	2	no	—
43	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
44	C₄⋊Dic₅	—	5	2	no	—
45	C₄⋊Dic₅	—	10	2	no	—
46	C₄⋊Dic₅	—	10	2	no	—
47	C₄⋊Dic₅	—	10	4	no	—
48	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
49	#683	—	10	2	no	—
50	#683	—	10	2	no	—
51	#683	—	10	4	no	—
52	C₄×Dic₅	—	5	2	no	—
53	C₄×Dic₅	—	10	2	no	—
54	C₂×#168	C₂×D₅	1	2	yes	—
55	#501	—	25	1	no	Sylow
56	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×C₅²	#137	1	1	yes	characteristic
59	C₂×C₄×C₅	—	2	4	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	10	2	no	—
63	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	—	5	2	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	10	2	no	—
68	C₂×Dic₅	—	10	2	no	—
69	C₂×Dic₅	—	10	4	no	—
70	C₂×Dic₅	—	10	2	no	—
71	Dic₁₀	—	20	4	no	—
72	C₅×C₈	—	5	4	no	—
73	#168	Dic₁₀	1	4	yes	complemented
74	C₄⋊Q₈	—	25	1	no	—
75	#125	—	25	1	no	—
76	#107	—	25	2	no	—
77	#106	—	5	2	no	—
78	#97	—	25	1	no	—
79	C₅²	#501	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₅	—	2	2	no	—
81	C₂²×C₅	D₄×D₅	1	2	yes	—
82	C₄×C₅	—	2	4	no	—
83	C₄×C₅	—	4	4	no	—
84	C₄×C₅	—	4	2	no	—
85	C₄×C₅	—	20	2	no	—
86	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
87	C₄×C₅	Q₈×D₅	1	2	yes	—
88	Dic₅	—	4	2	no	—
89	Dic₅	—	10	2	no	—
90	Dic₅	—	10	2	no	—
91	Dic₅	—	20	2	no	—
92	Dic₅	—	20	2	no	—
93	Dic₅	—	20	4	no	—
94	Dic₅	—	20	2	no	—
95	C₂×Q₈	—	50	1	no	—
96	C₂×C₈	—	5	2	no	—
97	C₄⋊C₄	—	5	2	no	—
98	C₄⋊C₄	—	25	1	no	—
99	C₄⋊C₄	—	50	1	no	—
100	C₄⋊C₄	—	50	1	no	—
101	C₄⋊C₄	—	50	1	no	—
102	C₄⋊C₄	D₅²	1	1	yes	characteristic
103	C₄²	—	25	1	no	—
104	C₂×C₅	—	2	2	no	—
105	C₂×C₅	—	2	2	no	—
106	C₂×C₅	—	2	2	no	—
107	C₂×C₅	#4034	1	2	yes	—
108	C₂×C₅	#4060	1	2	yes	—
109	C₂×C₅	#4061	1	2	yes	—
110	Q₈	—	100	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	10	2	no	—
113	C₂×C₄	—	25	1	no	—
114	C₂×C₄	—	25	1	no	—
115	C₂×C₄	—	50	1	no	—
116	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
117	C₈	—	5	4	no	—
118	C₅	—	2	2	no	—
119	C₅	#65145	1	2	yes	complemented
120	C₂²	#87886	1	1	yes	characteristic
121	C₄	—	4	1	no	—
122	C₄	—	20	2	no	—
123	C₄	—	50	1	no	—
124	C₄	—	50	1	no	—
125	C₄	—	100	1	no	—
126	C₄	D₅×Dic₁₀	1	2	yes	—
127	C₂	#11716958	1	1	yes	characteristic
128	C₂	#11717049	1	1	yes	characteristic
129	C₂	#11717059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717210

C₂

yes

maximal, characteristic

#11717037

C₂

yes

maximal, characteristic

C₅×#3940

C₂

yes

maximal

#11716616

C₂

yes

maximal

#11716596

C₂

yes

maximal, characteristic

C₂×#6039

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₄⋊#1039

—

C₄⋊#1039

C₂²

yes

characteristic

#87806

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

—

#65145

—

maximal

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×C₅×Dic₅

—

#6039

—

C₅×#168

Q₈

yes

—

—

—

—

—

—

D₅

yes

—

#3939

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

—

#1039

—

#1039

—

C₅×Dic₅

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

#501

—

Sylow

C₂×C₅²

C₄⋊Q₈

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

Dic₁₀

yes

complemented

C₄⋊Q₈

—

—

—

—

—

C₅²

#501

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

D₅²

yes

characteristic

103

C₄²

—

104

C₂×C₅

—

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

#4034

yes

—

108

C₂×C₅

#4060

yes

—

109

C₂×C₅

#4061

yes

—

110

Q₈

—

100

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₅²

yes

characteristic

117

C₈

—

118

C₅

—

119

C₅

#65145

yes

complemented

120

C₂²

#87886

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

100

—

126

C₄

D₅×Dic₁₀

yes

—

127

C₂

#11716958

yes

characteristic

128

C₂

#11717049

yes

characteristic

129

C₂

#11717059

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×C₄⋊C₄	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#501
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×C₄⋊C₄

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊C₄

D₅²

hypocenter

C₅²

#501

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
23. C₅×#168	Q₈	semidirect
73. #168	Dic₁₀	semidirect
79. C₅²	#501	semidirect
119. C₅	#65145	semidirect

H ≅ G/N

type

23. C₅×#168

Q₈

semidirect

73. #168

Dic₁₀

semidirect

79. C₅²

#501

semidirect

119. C₅

#65145

semidirect

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

subgroup	index
#11717210	2
#11717037	2
C₅×#3940	2
#11716616	2
#11716596	2
#65145	5

subgroup

index

Subgroups — #13100721

Conjugacy classes of subgroups

Complements

Maximal subgroups