Group #13100558 — subgroups

Subgroups — #13100558

Order 1600.

256 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₄×C₅×Dic₁₀	C₂	1	2	yes	maximal
2	C₂×#87775	C₂	1	1	yes	maximal, characteristic
3	#11716670	C₂	1	1	yes	maximal, characteristic
4	#11716615	C₂	1	1	yes	maximal, characteristic
5	#11716606	C₂	1	2	yes	maximal
6	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	C₂×#6029	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₅×#683	—	2	2	no	—
12	C₄×C₅×Dic₅	—	2	2	no	—
13	#87775	C₄	1	4	yes	complemented
14	#65319	—	5	2	no	maximal
15	C₂×C₄×C₅²	D₄	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×C₅×Dic₅	—	2	2	no	—
18	C₂×C₅×Dic₅	—	2	2	no	—
19	C₅×Dic₁₀	—	2	2	no	—
20	C₅×Dic₁₀	C₂×C₄	1	4	yes	—
21	#6029	—	2	1	no	—
22	#6029	C₂×C₄	1	2	yes	—
23	C₄×C₅×Q₈	—	5	2	no	—
24	C₂×#689	—	5	2	no	—
25	C₄×Dic₁₀	D₅	1	2	yes	—
26	#3967	—	5	2	no	—
27	#3942	—	5	2	no	—
28	#3940	—	5	2	no	—
29	#3935	—	5	2	no	—
30	#3935	—	10	2	no	—
31	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	—	2	2	no	—
33	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₅²	#41	1	1	yes	characteristic
35	C₅×Dic₅	—	2	4	no	—
36	C₅×Dic₅	—	4	2	no	—
37	C₂×C₅×Q₈	—	5	2	no	—
38	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
39	C₅×C₄⋊C₄	—	5	2	no	—
40	C₅×C₄⋊C₄	—	10	2	no	—
41	C₄²×C₅	—	2	2	no	—
42	C₄²×C₅	—	10	2	no	—
43	C₄²×C₅	C₂×D₅	1	2	yes	—
44	#689	—	5	8	no	—
45	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
46	#683	—	2	2	no	—
47	C₄×Dic₅	—	2	2	no	—
48	C₂×#168	—	5	2	no	—
49	C₂×#168	—	5	2	no	—
50	C₂×#168	—	10	2	no	—
51	C₂×#168	—	10	2	no	—
52	#441	—	25	1	no	Sylow
53	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
54	C₂×C₅²	#137	1	2	yes	—
55	C₅×Q₈	—	5	4	no	—
56	C₅×Q₈	—	10	2	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	10	2	no	—
60	C₂×C₄×C₅	—	10	2	no	—
61	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
62	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
63	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	—	2	2	no	—
66	Dic₁₀	—	2	2	no	—
67	Dic₁₀	C₄×D₅	1	4	yes	—
68	#168	—	5	4	no	—
69	#168	—	10	4	no	—
70	#168	—	10	4	no	—
71	#168	—	10	2	no	—
72	C₂×Q₁₆	—	25	1	no	—
73	C₄×Q₈	—	5	2	no	—
74	#106	—	25	1	no	—
75	#103	—	25	2	no	—
76	#97	—	25	1	no	—
77	C₅²	#441	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
80	C₄×C₅	—	2	2	no	—
81	C₄×C₅	—	2	2	no	—
82	C₄×C₅	—	2	2	no	—
83	C₄×C₅	—	2	2	no	—
84	C₄×C₅	—	4	4	no	—
85	C₄×C₅	—	10	4	no	—
86	C₄×C₅	—	20	2	no	—
87	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
88	C₄×C₅	#709	1	2	yes	—
89	Dic₅	—	2	4	no	—
90	Dic₅	—	4	2	no	—
91	C₂×Q₈	—	5	2	no	—
92	Q₁₆	—	25	4	no	—
93	C₂×C₈	—	25	1	no	—
94	C₂×C₈	—	25	1	no	—
95	C₄⋊C₄	—	5	2	no	—
96	C₄⋊C₄	—	10	2	no	—
97	C₄²	—	10	2	no	—
98	C₄²	D₅²	1	1	yes	characteristic
99	C₂×C₅	—	2	4	no	—
100	C₂×C₅	—	2	2	no	—
101	C₂×C₅	C₄×C₂²⋊D₅	1	2	yes	—
102	C₂×C₅	#4088	1	2	yes	—
103	C₂×C₅	#4097	1	2	yes	—
104	Q₈	—	5	4	no	—
105	Q₈	—	10	2	no	—
106	C₂×C₄	—	10	2	no	—
107	C₂×C₄	—	10	2	no	—
108	C₂×C₄	#6037	1	2	yes	—
109	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
110	C₈	—	25	2	no	—
111	C₈	—	50	1	no	—
112	C₅	—	2	2	no	—
113	C₅	#65319	1	2	yes	complemented
114	C₂²	C₂×#6037	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	10	4	no	—
117	C₄	—	20	2	no	—
118	C₄	#87873	1	1	yes	characteristic
119	C₄	C₄×D₅²	1	1	yes	characteristic
120	C₂	#11716945	1	2	yes	—
121	C₂	C₄×#6037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×Dic₁₀

C₂

yes

maximal

C₂×#87775

C₂

yes

maximal, characteristic

#11716670

C₂

yes

maximal, characteristic

#11716615

C₂

yes

maximal, characteristic

#11716606

C₂

yes

maximal

C₂×C₅×Dic₁₀

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

—

C₄×C₅×Dic₅

—

#87775

C₄

yes

complemented

#65319

—

maximal

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

C₂×C₄

yes

—

#6029

—

#6029

C₂×C₄

yes

—

C₄×C₅×Q₈

—

C₂×#689

—

C₄×Dic₁₀

D₅

yes

—

—

—

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

#41

yes

characteristic

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

#689

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

—

—

—

—

—

C₂×C₅²

C₄×D₄

yes

characteristic

C₂×C₅²

#137

yes

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

C₄×D₅

yes

—

—

—

—

—

C₂×Q₁₆

—

C₄×Q₈

—

#106

—

#103

—

#97

—

C₅²

#441

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

#709

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₅²

yes

characteristic

C₂×C₅

—

100

C₂×C₅

—

101

C₂×C₅

C₄×C₂²⋊D₅

yes

—

102

C₂×C₅

#4088

yes

—

103

C₂×C₅

#4097

yes

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#6037

yes

—

109

C₂×C₄

C₂×D₅²

yes

characteristic

110

C₈

—

111

C₈

—

112

C₅

—

113

C₅

#65319

yes

complemented

114

C₂²

C₂×#6037

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#87873

yes

characteristic

119

C₄

C₄×D₅²

yes

characteristic

120

C₂

#11716945

yes

—

121

C₂

C₄×#6037

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6037
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₄²×C₅²	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	#441
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6037

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₄²×C₅²

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₄²

D₅²

hypocenter

C₅²

#441

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
13. #87775	C₄	semidirect
77. C₅²	#441	semidirect
113. C₅	#65319	semidirect

H ≅ G/N

type

13. #87775

C₄

semidirect

77. C₅²

#441

semidirect

113. C₅

#65319

semidirect

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

subgroup	index
C₄×C₅×Dic₁₀	2
C₂×#87775	2
#11716670	2
#11716615	2
#11716606	2
#65319	5

subgroup

index

C₄×C₅×Dic₁₀

Subgroups — #13100558

Conjugacy classes of subgroups

Complements

Maximal subgroups