Group #13104751 — subgroups

Subgroups — #13104751

Order 1600.

432 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 84 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	Q₈×#1043	C₂	1	2	yes	maximal
2	Q₁₆×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
3	#11716877	C₂	1	2	yes	maximal
4	#11716861	C₂	1	1	yes	maximal, characteristic
5	#11716860	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₅⋊D₅	C₂²	1	2	yes	—
7	C₄⋊#1043	—	2	2	no	—
8	C₄⋊#1043	C₂²	1	2	yes	—
9	C₄×#1043	—	2	2	no	—
10	C₄×#1043	C₂²	1	1	yes	characteristic
11	#87855	C₂²	1	1	yes	characteristic
12	C₅²×Q₁₆	C₄	1	1	yes	characteristic, complemented
13	#87812	C₄	1	2	yes	complemented
14	#87802	C₄	1	1	yes	characteristic, complemented
15	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
16	#65739	—	5	6	no	maximal
17	C₂×#1043	—	2	2	no	—
18	C₂×#1043	D₄	1	2	yes	—
19	C₅²×Q₈	C₂×C₄	1	2	yes	—
20	C₄×C₅⋊D₅	—	2	2	no	—
21	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
22	#6045	—	2	2	no	—
23	#6045	C₂×C₄	1	2	yes	—
24	#6033	—	2	1	no	—
25	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
26	#6029	C₂×C₄	1	1	yes	characteristic
27	Q₈×AGL(1,𝔽₅)	—	5	12	no	—
28	D₅×Q₁₆	—	5	6	no	—
29	#4009	—	5	12	no	—
30	#3993	—	5	6	no	—
31	#3992	—	5	6	no	—
32	C₂×C₅⋊D₅	C₂×D₄	1	1	yes	characteristic
33	#1043	—	2	2	no	—
34	#1043	—	4	2	no	—
35	C₄×C₅²	—	2	2	no	—
36	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
37	#1039	—	2	2	no	—
38	#1039	#41	1	1	yes	characteristic
39	Q₈×D₅	—	5	12	no	—
40	C₄⋊AGL(1,𝔽₅)	—	5	12	no	—
41	C₄⋊AGL(1,𝔽₅)	—	10	12	no	—
42	C₄×AGL(1,𝔽₅)	—	5	6	no	—
43	C₄×AGL(1,𝔽₅)	—	10	12	no	—
44	D₅⋊C₈	—	5	6	no	—
45	C₅×Q₁₆	AGL(1,𝔽₅)	1	6	yes	complemented
46	#689	—	5	12	no	—
47	Dic₂₀	—	5	6	no	—
48	C₈×D₅	—	5	6	no	—
49	#441	—	25	1	no	Sylow
50	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
51	C₅⋊D₅	#137	1	2	yes	—
52	C₂×AGL(1,𝔽₅)	—	5	12	no	—
53	C₂×AGL(1,𝔽₅)	—	10	12	no	—
54	C₅×Q₈	C₂×AGL(1,𝔽₅)	1	12	yes	—
55	C₄×D₅	—	5	6	no	—
56	C₄×D₅	—	10	12	no	—
57	Dic₁₀	—	5	12	no	—
58	Dic₁₀	—	10	12	no	—
59	#170	—	10	6	no	—
60	C₅×C₈	C₂×AGL(1,𝔽₅)	1	6	yes	—
61	#168	—	5	6	no	—
62	C₂×Q₁₆	—	25	1	no	—
63	C₄×Q₈	—	25	2	no	—
64	#106	—	25	1	no	—
65	#103	—	25	2	no	—
66	#97	—	25	1	no	—
67	C₅²	#441	1	1	yes	characteristic, complemented, Sylow
68	C₂×D₅	—	5	6	no	—
69	AGL(1,𝔽₅)	—	10	12	no	—
70	AGL(1,𝔽₅)	—	20	12	no	—
71	C₄×C₅	—	2	12	no	—
72	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	6	yes	—
73	Dic₅	—	5	6	no	—
74	Dic₅	—	10	12	no	—
75	C₂×Q₈	—	25	2	no	—
76	Q₁₆	—	25	2	no	—
77	Q₁₆	—	25	1	no	—
78	Q₁₆	#1043	1	1	yes	characteristic, complemented
79	C₂×C₈	—	25	1	no	—
80	C₂×C₈	—	25	1	no	—
81	C₄⋊C₄	—	25	2	no	—
82	C₄⋊C₄	—	50	2	no	—
83	C₄²	—	25	1	no	—
84	C₄²	—	50	2	no	—
85	C₂×C₅	D₄×AGL(1,𝔽₅)	1	6	yes	—
86	D₅	—	5	12	no	—
87	Q₈	—	25	2	no	—
88	Q₈	—	50	2	no	—
89	Q₈	C₂×#1043	1	2	yes	—
90	C₂×C₄	—	25	1	no	—
91	C₂×C₄	—	25	2	no	—
92	C₂×C₄	—	50	2	no	—
93	C₂×C₄	—	50	2	no	—
94	C₈	—	25	1	no	—
95	C₈	—	50	1	no	—
96	C₈	C₂×#1043	1	1	yes	characteristic
97	C₅	#65739	1	6	yes	complemented
98	C₂²	—	25	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	25	1	no	—
101	C₄	—	50	2	no	—
102	C₄	—	50	2	no	—
103	C₄	—	100	2	no	—
104	C₄	C₂²×#1043	1	1	yes	characteristic
105	C₂	—	25	2	no	—
106	C₂	D₄×#1043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#1043

C₂

yes

maximal

Q₁₆×C₅⋊D₅

C₂

yes

maximal, characteristic

#11716877

C₂

yes

maximal

#11716861

C₂

yes

maximal, characteristic

#11716860

C₂

yes

maximal, characteristic

Q₈×C₅⋊D₅

C₂²

yes

—

C₄⋊#1043

—

C₄⋊#1043

C₂²

yes

—

C₄×#1043

—

C₄×#1043

C₂²

yes

characteristic

#87855

C₂²

yes

characteristic

C₅²×Q₁₆

C₄

yes

characteristic, complemented

#87812

C₄

yes

complemented

#87802

C₄

yes

characteristic, complemented

C₈×C₅⋊D₅

C₂²

yes

—

—

D₄

yes

—

C₅²×Q₈

C₂×C₄

yes

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

C₂³

yes

characteristic

#6045

—

#6045

C₂×C₄

yes

—

#6033

—

C₅²×C₈

C₂×C₄

yes

characteristic

#6029

C₂×C₄

yes

characteristic

Q₈×AGL(1,𝔽₅)

—

D₅×Q₁₆

—

#4009

—

#3993

—

#3992

—

C₂×C₅⋊D₅

C₂×D₄

yes

characteristic

#1043

—

#1043

—

C₄×C₅²

—

C₄×C₅²

C₂²×C₄

yes

—

yes

Q₈×D₅

—

—

—

—

—

D₅⋊C₈

—

C₅×Q₁₆

AGL(1,𝔽₅)

yes

complemented

#689

—

Dic₂₀

—

C₈×D₅

—

#441

—

Sylow

C₂×C₅²

C₄×D₄

yes

characteristic

C₅⋊D₅

#137

yes

—

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₅×Q₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

#170

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

#168

—

C₂×Q₁₆

—

C₄×Q₈

—

#106

—

#103

—

#97

—

C₅²

#441

yes

characteristic, complemented, Sylow

C₂×D₅

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

#1043

yes

characteristic, complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₅

D₄×AGL(1,𝔽₅)

yes

—

D₅

—

Q₈

—

Q₈

—

Q₈

C₂×#1043

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

C₈

yes

C₅

yes

C₂²

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

100

—

104

C₄

C₂²×#1043

yes

characteristic

105

C₂

—

106

C₂

D₄×#1043

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#1043
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₄	C₂²×#1043
Fitting	C₅²×Q₁₆	C₄
socle	C₂×C₅²	C₄×D₄
prime radical	C₂×C₅⋊D₅	C₂×D₄
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	Q₁₆	#1043
hypocenter	C₅²	#441
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

subgroup

G/H

center

C₂

D₄×#1043

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₄

C₂²×#1043

Fitting

C₅²×Q₁₆

C₄

socle

C₂×C₅²

C₄×D₄

prime radical

C₂×C₅⋊D₅

C₂×D₄

cosocle

C₄×C₅⋊D₅

C₂³

Q₁₆

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

N	H ≅ G/N	type
12. C₅²×Q₁₆	C₄	semidirect
13. #87812	C₄	semidirect
14. #87802	C₄	semidirect
45. C₅×Q₁₆	AGL(1,𝔽₅)	semidirect
67. C₅²	#441	semidirect
78. Q₁₆	#1043	semidirect
97. C₅	#65739	semidirect

H ≅ G/N

type

12. C₅²×Q₁₆

C₄

semidirect

13. #87812

C₄

semidirect

14. #87802

C₄

semidirect

45. C₅×Q₁₆

AGL(1,𝔽₅)

semidirect

67. C₅²

#441

semidirect

78. Q₁₆

#1043

semidirect

97. C₅

#65739

semidirect

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

subgroup	index
Q₈×#1043	2
Q₁₆×C₅⋊D₅	2
#11716877	2
#11716861	2
#11716860	2
#65739	5

subgroup

index

Q₁₆×C₅⋊D₅

Subgroups — #13104751

Conjugacy classes of subgroups

Complements

Maximal subgroups