Group #13103837 — subgroups

Subgroups — #13103837

Order 1600.

196 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11716923	C₂	1	1	yes	maximal, characteristic
2	#11716883	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716857	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716775	C₂	1	2	yes	maximal, complemented
5	D₅⋊#674	C₂	1	2	yes	maximal, complemented
6	#87873	C₄	1	1	yes	characteristic
7	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
8	C₅×#676	C₄	1	2	yes	—
9	#87769	C₄	1	1	yes	characteristic
10	D₅×#168	C₂²	1	2	yes	—
11	#87765	C₂²	1	2	yes	—
12	#87764	C₂²	1	1	yes	characteristic
13	#87764	C₂²	1	1	yes	characteristic
14	#65725	—	5	2	no	maximal
15	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
16	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
17	#6039	C₈	1	1	yes	characteristic
18	#6038	C₈	1	2	yes	—
19	#6037	C₈	1	1	yes	characteristic
20	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
21	#6029	C₂³	1	1	yes	characteristic
22	C₅×#168	C₂×C₄	1	2	yes	—
23	#4053	—	5	2	no	—
24	#3998	—	5	2	no	—
25	C₂×#674	—	5	2	no	—
26	#3990	—	5	2	no	—
27	#3990	—	5	2	no	—
28	D₅⋊C₁₆	—	5	2	no	—
29	D₅⋊C₁₆	—	5	2	no	—
30	D₅⋊C₁₆	—	10	2	no	—
31	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
32	C₂×C₅×D₅	C₂×C₈	1	2	yes	—
33	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
34	#1039	C₂×C₈	1	1	yes	characteristic
35	C₅×Dic₅	C₂×C₈	1	2	yes	—
36	#709	—	5	2	no	—
37	C₅×M₁₆	—	5	2	no	—
38	C₂×#168	—	5	2	no	—
39	#676	—	5	2	no	—
40	#676	AGL(1,𝔽₅)	1	2	yes	—
41	C₈×D₅	—	5	2	no	—
42	C₈×D₅	—	5	2	no	—
43	C₈×D₅	—	10	2	no	—
44	#674	—	5	2	no	—
45	#674	—	5	2	no	—
46	#674	—	5	2	no	—
47	#674	—	5	2	no	—
48	#674	—	10	2	no	—
49	#674	—	10	2	no	—
50	#504	—	25	1	no	Sylow
51	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
52	C₅⋊D₅	—	2	1	no	—
53	C₅×D₅	—	2	2	no	—
54	C₂×C₄×C₅	—	5	2	no	—
55	C₂²⋊D₅	—	5	2	no	—
56	C₂²⋊D₅	—	5	2	no	—
57	D₂₀	—	5	2	no	—
58	C₄×D₅	—	5	2	no	—
59	C₄×D₅	—	10	2	no	—
60	C₄×D₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
61	Dic₁₀	—	5	2	no	—
62	C₅×C₈	—	2	2	no	—
63	C₅×C₈	—	5	2	no	—
64	C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
65	#168	—	5	2	no	—
66	#168	—	10	2	no	—
67	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
68	#131	—	25	1	no	—
69	M₃₂	—	25	1	no	—
70	M₃₂	—	25	2	no	—
71	C₂×C₁₆	—	25	1	no	—
72	C₂×C₁₆	—	25	2	no	—
73	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₅	—	5	2	no	—
75	C₂×D₅	—	5	2	no	—
76	C₂×D₅	—	10	2	no	—
77	C₂×D₅	D₅⋊C₈	1	2	yes	—
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	—	5	2	no	—
80	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
81	Dic₅	—	5	2	no	—
82	Dic₅	—	10	2	no	—
83	Dic₅	D₅⋊C₈	1	2	yes	—
84	#41	—	25	1	no	—
85	M₁₆	—	5	2	no	—
86	M₁₆	—	25	1	no	—
87	C₂×C₈	—	25	2	no	—
88	C₂×C₈	—	25	1	no	—
89	C₁₆	—	25	1	no	—
90	C₁₆	—	25	2	no	—
91	C₁₆	—	25	1	no	—
92	C₂×C₅	—	2	2	no	—
93	C₂×C₅	—	10	2	no	—
94	C₂×C₅	C₂×D₅⋊C₈	1	2	yes	—
95	D₅	—	2	2	no	—
96	D₅	—	10	4	no	—
97	D₅	—	10	2	no	—
98	Q₈	—	25	1	no	—
99	D₄	—	25	2	no	—
100	D₄	—	25	1	no	—
101	C₂×C₄	—	5	2	no	—
102	C₂×C₄	—	25	1	no	—
103	C₈	—	5	2	no	—
104	C₈	—	25	1	no	—
105	C₈	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₅	—	2	2	no	—
107	C₅	#65725	1	2	yes	complemented
108	C₂²	—	5	2	no	—
109	C₂²	—	25	1	no	—
110	C₄	—	5	2	no	—
111	C₄	—	25	1	no	—
112	C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂	—	10	2	no	—
114	C₂	—	50	1	no	—
115	C₂	D₅⋊(D₅⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11716923

C₂

yes

maximal, characteristic

#11716883

C₂

yes

maximal, characteristic, complemented

#11716857

C₂

yes

maximal, characteristic, complemented

#11716775

C₂

yes

maximal, complemented

D₅⋊#674

C₂

yes

maximal, complemented

#87873

C₄

yes

characteristic

C₈×C₅⋊D₅

C₂²

yes

characteristic

C₅×#676

C₄

yes

—

#87769

C₄

yes

characteristic

D₅×#168

C₂²

yes

—

#87765

C₂²

yes

—

#87764

C₂²

yes

characteristic

#87764

C₂²

yes

characteristic

#65725

—

maximal

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅×D₅

C₂×C₄

yes

—

#6039

C₈

yes

characteristic

#6038

C₈

yes

—

#6037

C₈

yes

characteristic

C₅²×C₈

C₂×C₄

yes

characteristic

#6029

C₂³

yes

characteristic

C₅×#168

C₂×C₄

yes

—

—

—

—

—

—

D₅⋊C₁₆

—

D₅⋊C₁₆

—

D₅⋊C₁₆

—

C₂×C₅⋊D₅

C₂×C₈

yes

characteristic

C₂×C₅×D₅

C₂×C₈

yes

—

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₂×C₈

yes

characteristic

C₅×Dic₅

C₂×C₈

yes

—

#709

—

C₅×M₁₆

—

—

—

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

—

—

—

—

—

—

—

C₂×C₅²

C₂²×C₈

yes

characteristic

C₅⋊D₅

—

C₅×D₅

—

C₂×C₄×C₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×AGL(1,𝔽₅)

yes

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

—

—

yes

—

#131

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₅²

#504

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

D₅⋊C₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

Dic₅

D₅⋊C₈

yes

—

#41

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×D₅⋊C₈

yes

—

D₅

—

D₅

—

D₅

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₈

—

104

C₈

—

105

C₈

D₅⋊AGL(1,𝔽₅)

yes

characteristic

106

C₅

—

107

C₅

#65725

yes

complemented

108

C₂²

—

109

C₂²

—

110

C₄

—

111

C₄

—

112

C₄

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

D₅⋊(D₅⋊C₈)

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting	C₅²×C₈	C₂×C₄
prime radical	#87873	C₄
cosocle	#6029	C₂³
hypercenter	C₈	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₈

subgroup

G/H

center, Frattini

C₄

C₂×D₅⋊AGL(1,𝔽₅)

commutator, socle

C₂×C₅²

C₂²×C₈

Fitting

C₅²×C₈

C₂×C₄

C₄

C₂³

C₈

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
2. #11716883	C₂	semidirect
3. #11716857	C₂	semidirect
4. #11716775	C₂	semidirect
5. D₅⋊#674	C₂	semidirect
73. C₅²	#504	semidirect
107. C₅	#65725	semidirect

H ≅ G/N

type

2. #11716883

C₂

semidirect

3. #11716857

C₂

semidirect

4. #11716775

C₂

semidirect

5. D₅⋊#674

C₂

semidirect

73. C₅²

#504

semidirect

107. C₅

#65725

semidirect

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

subgroup	index
#11716923	2
#11716883	2
#11716857	2
#11716775	2
D₅⋊#674	2
#65725	5

subgroup

index

Subgroups — #13103837

Conjugacy classes of subgroups

Complements

Maximal subgroups