Group #13104863 — subgroups

Subgroups — #13104863

Order 1600.

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

normal subgroups only

This group has 230 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	#11717227	C₂	1	1	yes	maximal, characteristic
2	#11716891	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87765	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716883	C₂	1	2	yes	maximal, complemented
5	#11716882	C₂	1	2	yes	maximal, complemented
6	#87900	C₄	1	1	yes	characteristic
7	C₅²×M₁₆	C₄	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	#87799	C₄	1	2	yes	—
10	C₈×C₅⋊D₅	C₂²	1	2	yes	—
11	#87765	C₂²	1	1	yes	characteristic
12	#87765	C₂²	1	2	yes	—
13	#87765	C₂²	1	1	yes	characteristic
14	#65725	—	5	2	no	maximal
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊(C₅⋊D₅)	C₈	1	2	yes	—
17	C₄⋊(C₅⋊D₅)	C₈	1	1	yes	characteristic
18	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
19	#6045	C₈	1	1	yes	characteristic
20	C₅²×C₈	C₂×C₄	1	2	yes	—
21	#6029	C₂×C₄	1	1	yes	characteristic
22	#6029	C₂³	1	1	yes	characteristic
23	#4053	—	5	2	no	—
24	#4053	—	10	2	no	—
25	#3998	—	5	2	no	—
26	C₂×#674	—	5	2	no	—
27	#3990	—	5	4	no	—
28	D₅⋊C₁₆	—	5	4	no	—
29	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
30	C₂×C₅⋊D₅	C₂×C₈	1	2	yes	—
31	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
32	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
33	#1039	C₂×C₈	1	2	yes	—
34	#709	—	5	2	no	—
35	#709	—	10	2	no	—
36	C₅×M₁₆	—	2	2	no	—
37	C₅×M₁₆	AGL(1,𝔽₅)	1	2	yes	—
38	C₂×#168	—	5	2	no	—
39	C₂×#168	—	10	2	no	—
40	#676	—	5	4	no	—
41	#676	—	10	4	no	—
42	C₈×D₅	—	5	4	no	—
43	C₈×D₅	—	10	4	no	—
44	#674	—	5	2	no	—
45	#674	—	5	4	no	—
46	#674	—	5	2	no	—
47	#504	—	25	1	no	Sylow
48	C₂×C₅²	—	2	1	no	—
49	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
50	C₅⋊D₅	—	2	2	no	—
51	C₂×C₄×C₅	—	2	2	no	—
52	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
53	C₂²⋊D₅	—	5	4	no	—
54	C₂²⋊D₅	—	10	4	no	—
55	D₂₀	—	5	2	no	—
56	D₂₀	—	10	2	no	—
57	C₄×D₅	—	5	4	no	—
58	C₄×D₅	—	10	4	no	—
59	Dic₁₀	—	5	2	no	—
60	Dic₁₀	—	10	2	no	—
61	C₅×C₈	—	2	4	no	—
62	C₅×C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
63	#168	—	5	2	no	—
64	#168	—	5	2	no	—
65	#168	—	10	2	no	—
66	#168	—	10	2	no	—
67	#131	—	25	1	no	—
68	M₃₂	—	25	2	no	—
69	M₃₂	—	25	1	no	—
70	C₂×C₁₆	—	25	2	no	—
71	C₂×C₁₆	—	25	1	no	—
72	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₅	—	2	2	no	—
74	C₂²×C₅	D₅⋊C₈	1	2	yes	—
75	C₂×D₅	—	5	4	no	—
76	C₂×D₅	—	10	4	no	—
77	C₄×C₅	—	2	2	no	—
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	D₅⋊C₈	1	2	yes	—
80	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
81	Dic₅	—	5	4	no	—
82	Dic₅	—	10	4	no	—
83	#41	—	25	1	no	—
84	M₁₆	—	25	2	no	—
85	M₁₆	#1044	1	1	yes	characteristic
86	C₂×C₈	—	25	1	no	—
87	C₂×C₈	—	25	2	no	—
88	C₁₆	—	25	1	no	—
89	C₁₆	—	25	2	no	—
90	C₁₆	—	25	1	no	—
91	C₂×C₅	—	2	2	no	—
92	C₂×C₅	—	2	2	no	—
93	C₂×C₅	—	4	2	no	—
94	C₂×C₅	C₂×D₅⋊C₈	1	2	yes	—
95	D₅	—	10	4	no	—
96	D₅	—	20	4	no	—
97	Q₈	—	25	1	no	—
98	D₄	—	25	2	no	—
99	D₄	—	25	1	no	—
100	C₂×C₄	—	25	2	no	—
101	C₂×C₄	C₂×#1044	1	1	yes	characteristic
102	C₈	—	25	1	no	—
103	C₈	—	25	1	no	—
104	C₈	C₂×#1044	1	2	yes	—
105	C₅	—	2	2	no	—
106	C₅	#65725	1	2	yes	complemented
107	C₂²	—	25	2	no	—
108	C₂²	#87862	1	1	yes	characteristic
109	C₄	—	25	2	no	—
110	C₄	#87862	1	1	yes	characteristic
111	C₄	C₂²×#1044	1	1	yes	characteristic
112	C₂	—	2	1	no	—
113	C₂	—	50	2	no	—
114	C₂	C₂×#87862	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717227

C₂

yes

maximal, characteristic

#11716891

C₂

yes

maximal, characteristic, complemented

C₂×#87765

C₂

yes

maximal, characteristic, complemented

#11716883

C₂

yes

maximal, complemented

#11716882

C₂

yes

maximal, complemented

#87900

C₄

yes

characteristic

C₅²×M₁₆

C₄

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#87799

C₄

yes

—

C₈×C₅⋊D₅

C₂²

yes

—

#87765

C₂²

yes

characteristic

#87765

C₂²

yes

—

#87765

C₂²

yes

characteristic

#65725

—

maximal

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂²⋊(C₅⋊D₅)

C₈

yes

—

C₄⋊(C₅⋊D₅)

C₈

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

—

#6045

C₈

yes

characteristic

C₅²×C₈

C₂×C₄

yes

—

#6029

C₂×C₄

yes

characteristic

#6029

C₂³

yes

—

—

—

—

—

D₅⋊C₁₆

—

C₂²×C₅²

C₂×C₈

yes

characteristic

C₂×C₅⋊D₅

C₂×C₈

yes

—

C₄×C₅²

C₂×C₈

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₂×C₈

yes

—

#709

—

#709

—

C₅×M₁₆

—

C₅×M₁₆

AGL(1,𝔽₅)

yes

—

—

—

—

—

C₈×D₅

—

C₈×D₅

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂²×C₈

yes

characteristic

C₅⋊D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊D₅

—

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

—

—

—

—

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₅²

#504

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₅⋊C₈

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

D₅⋊C₈

yes

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

#41

—

M₁₆

—

M₁₆

#1044

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×D₅⋊C₈

yes

—

D₅

—

D₅

—

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×#1044

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

C₂×#1044

yes

—

105

C₅

—

106

C₅

#65725

yes

complemented

107

C₂²

—

108

C₂²

#87862

yes

characteristic

109

C₄

—

110

C₄

#87862

yes

characteristic

111

C₄

C₂²×#1044

yes

characteristic

112

C₂

—

113

C₂

—

114

C₂

C₂×#87862

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#1044
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting	C₅²×M₁₆	C₄
prime radical	#87900	C₄
cosocle	#6029	C₂³
hypercenter	M₁₆	#1044
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ M₁₆

subgroup

G/H

center, Frattini

C₄

C₂²×#1044

commutator, socle

C₂×C₅²

C₂²×C₈

Fitting

C₅²×M₁₆

C₄

C₄

C₂³

M₁₆

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₄ ◁ M₁₆

N	H ≅ G/N	type
2. #11716891	C₂	semidirect
3. C₂×#87765	C₂	semidirect
4. #11716883	C₂	semidirect
5. #11716882	C₂	semidirect
72. C₅²	#504	semidirect
106. C₅	#65725	semidirect

H ≅ G/N

type

2. #11716891

C₂

semidirect

3. C₂×#87765

C₂

semidirect

4. #11716883

C₂

semidirect

5. #11716882

C₂

semidirect

72. C₅²

#504

semidirect

106. C₅

#65725

semidirect

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

subgroup	index
#11717227	2
#11716891	2
C₂×#87765	2
#11716883	2
#11716882	2
#65725	5

subgroup

index

Subgroups — #13104863

Conjugacy classes of subgroups

Complements

Maximal subgroups