Group #13044198 — subgroups

Subgroups — AGL(1,𝔽₅)×#564 #13044198

Order 1280.

1800 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 1800 conjugacy classes of subgroups — too many to draw. Only its 776 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	D₅×#564	C₂	1	1	yes	maximal, characteristic
2	#10617348	C₂	1	15	yes	maximal
3	AGL(1,𝔽₅)×#125	C₂	1	15	yes	maximal
4	C₅×#564	C₄	1	1	yes	characteristic, complemented
5	D₅×#125	C₂²	1	15	yes	—
6	C₅⋊#564	C₄	1	15	yes	complemented
7	#65712	C₂²	1	30	yes	—
8	#65711	C₂²	1	60	yes	—
9	C₄⋊C₄×AGL(1,𝔽₅)	C₂²	1	30	yes	—
10	#65689	C₂²	1	15	yes	—
11	C₄²×AGL(1,𝔽₅)	C₂²	1	5	yes	—
12	C₄×#564	—	5	1	no	maximal, Sylow
13	C₂×C₄⋊AGL(1,𝔽₅)	C₂³	1	30	yes	—
14	C₂×C₄×AGL(1,𝔽₅)	C₂³	1	15	yes	—
15	C₂×C₄×AGL(1,𝔽₅)	C₂³	1	15	yes	—
16	C₅×#125	C₂×C₄	1	15	yes	—
17	D₅×C₄⋊C₄	C₂³	1	30	yes	—
18	#4036	C₂×C₄	1	30	yes	—
19	#4035	C₂×C₄	1	60	yes	—
20	C₄²×D₅	C₂³	1	5	yes	—
21	#4016	C₂×C₄	1	15	yes	—
22	(C₂×AGL(1,𝔽₅))⋊C₄	C₂³	1	60	yes	—
23	C₂×#564	—	5	1	no	—
24	#2382	—	5	15	no	—
25	C₄×#125	—	5	15	no	—
26	C₂²×AGL(1,𝔽₅)	C₂⁴	1	15	yes	—
27	C₂²×AGL(1,𝔽₅)	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×D₅	C₂⁴	1	15	yes	—
29	C₄⋊AGL(1,𝔽₅)	—	4	30	no	—
30	C₄×AGL(1,𝔽₅)	—	2	30	no	—
31	C₄×AGL(1,𝔽₅)	—	4	15	no	—
32	C₅×C₄⋊C₄	C₂²×C₄	1	30	yes	—
33	C₄²×C₅	C₂²×C₄	1	5	yes	—
34	C₄⋊Dic₅	C₂²×C₄	1	30	yes	—
35	#683	C₂²×C₄	1	60	yes	—
36	C₄×Dic₅	C₂²×C₄	1	15	yes	—
37	#564	—	5	15	no	—
38	#564	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
39	C₂×#125	—	5	15	no	—
40	#389	—	5	30	no	—
41	#387	—	5	60	no	—
42	#382	—	5	15	no	—
43	C₄×C₄⋊C₄	—	5	30	no	—
44	C₄³	—	5	5	no	—
45	C₂²×D₅	C₂⁵	1	1	yes	characteristic
46	C₂×AGL(1,𝔽₅)	—	2	15	no	—
47	C₂×AGL(1,𝔽₅)	—	2	30	no	—
48	C₂×AGL(1,𝔽₅)	2₋¹⁺⁴	1	6	yes	—
49	C₂×C₄×C₅	C₂³×C₄	1	15	yes	—
50	C₂×Dic₅	C₂³×C₄	1	15	yes	—
51	C₄×D₅	—	2	15	no	—
52	C₄×D₅	—	2	15	no	—
53	#125	—	5	60	no	—
54	#125	—	5	30	no	—
55	#125	—	5	15	no	—
56	#125	C₂×AGL(1,𝔽₅)	1	15	yes	—
57	C₂×C₄⋊C₄	—	5	30	no	—
58	C₂×C₄⋊C₄	—	5	30	no	—
59	C₂×C₄²	—	5	5	no	—
60	C₂×C₄²	—	5	15	no	—
61	C₂×C₄²	—	5	15	no	—
62	#95	—	5	60	no	—
63	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
64	C₂×D₅	C₂×2₋¹⁺⁴	1	3	yes	—
65	C₂×D₅	#585	1	3	yes	—
66	AGL(1,𝔽₅)	—	4	15	no	—
67	AGL(1,𝔽₅)	#564	1	4	yes	direct product factor
68	C₄×C₅	—	2	15	no	—
69	Dic₅	—	2	15	no	—
70	C₂²×C₄	—	5	1	no	—
71	C₂²×C₄	—	5	15	no	—
72	C₂²×C₄	—	5	15	no	—
73	C₄⋊C₄	—	5	60	no	—
74	C₄⋊C₄	—	5	30	no	—
75	C₄⋊C₄	—	20	30	no	—
76	C₄⋊C₄	C₂²×AGL(1,𝔽₅)	1	30	yes	—
77	C₄²	—	5	15	no	—
78	C₄²	—	10	30	no	—
79	C₄²	—	20	15	no	—
80	C₄²	C₂²×AGL(1,𝔽₅)	1	5	yes	—
81	C₂×C₅	C₄×2₋¹⁺⁴	1	3	yes	—
82	D₅	C₂×#564	1	1	yes	characteristic
83	D₅	#3531	1	3	yes	—
84	C₂³	—	5	1	no	—
85	C₂×C₄	—	5	6	no	—
86	C₂×C₄	—	5	15	no	—
87	C₂×C₄	—	10	15	no	—
88	C₂×C₄	—	10	15	no	—
89	C₂×C₄	—	10	15	no	—
90	C₂×C₄	—	10	30	no	—
91	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	15	yes	—
92	C₅	C₄×#564	1	1	yes	characteristic, complemented, Sylow
93	C₂²	—	5	3	no	—
94	C₂²	—	5	3	no	—
95	C₂²	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₄	—	2	15	no	—
97	C₄	—	5	4	no	—
98	C₄	—	10	15	no	—
99	C₄	—	20	15	no	—
100	C₂	—	5	1	no	—
101	C₂	—	5	3	no	—
102	C₂	AGL(1,𝔽₅)×2₋¹⁺⁴	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×#564

C₂

yes

maximal, characteristic

#10617348

C₂

yes

maximal

AGL(1,𝔽₅)×#125

C₂

yes

maximal

C₅×#564

C₄

yes

characteristic, complemented

D₅×#125

C₂²

yes

—

C₅⋊#564

C₄

yes

complemented

#65712

C₂²

yes

—

#65711

C₂²

yes

—

C₄⋊C₄×AGL(1,𝔽₅)

C₂²

yes

—

#65689

C₂²

yes

—

C₄²×AGL(1,𝔽₅)

C₂²

yes

—

C₄×#564

—

maximal, Sylow

C₂×C₄⋊AGL(1,𝔽₅)

C₂³

yes

—

C₂×C₄×AGL(1,𝔽₅)

C₂³

yes

—

C₂×C₄×AGL(1,𝔽₅)

C₂³

yes

—

C₅×#125

C₂×C₄

yes

—

D₅×C₄⋊C₄

C₂³

yes

—

#4036

C₂×C₄

yes

—

#4035

C₂×C₄

yes

—

C₄²×D₅

C₂³

yes

—

#4016

C₂×C₄

yes

—

(C₂×AGL(1,𝔽₅))⋊C₄

C₂³

yes

—

C₂×#564

—

#2382

—

C₄×#125

—

C₂²×AGL(1,𝔽₅)

C₂⁴

yes

—

C₂²×AGL(1,𝔽₅)

C₂⁴

yes

characteristic

C₂×C₄×D₅

C₂⁴

yes

—

C₄⋊AGL(1,𝔽₅)

—

C₄×AGL(1,𝔽₅)

—

C₄×AGL(1,𝔽₅)

—

C₅×C₄⋊C₄

C₂²×C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

—

C₄⋊Dic₅

C₂²×C₄

yes

—

#683

C₂²×C₄

yes

—

C₄×Dic₅

C₂²×C₄

yes

—

#564

—

#564

AGL(1,𝔽₅)

yes

characteristic, direct product factor

—

—

—

—

C₄×C₄⋊C₄

—

C₄³

—

C₂²×D₅

C₂⁵

yes

—

—

2₋¹⁺⁴

yes

—

C₂×C₄×C₅

C₂³×C₄

yes

—

C₂×Dic₅

C₂³×C₄

yes

—

C₄×D₅

—

C₄×D₅

—

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

C₂²×C₅

C₂⁴×C₄

yes

characteristic

C₂×D₅

C₂×2₋¹⁺⁴

yes

—

C₂×D₅

#585

yes

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

#564

yes

direct product factor

C₄×C₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

—

C₂×C₅

C₄×2₋¹⁺⁴

yes

—

D₅

C₂×#564

yes

characteristic

D₅

#3531

yes

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×AGL(1,𝔽₅)

yes

—

C₅

C₄×#564

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

C₂⁴×AGL(1,𝔽₅)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

—

101

C₂

—

102

C₂

AGL(1,𝔽₅)×2₋¹⁺⁴

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂⁴×C₄
Fitting	C₅×#564	C₄
prime radical, cosocle	C₂²×D₅	C₂⁵
hypercenter	#564	AGL(1,𝔽₅)
hypocenter	C₅	C₄×#564
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂⁴×C₄

Fitting

C₅×#564

C₄

prime radical, cosocle

C₂²×D₅

C₂⁵

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
4. C₅×#564	C₄	semidirect
6. C₅⋊#564	C₄	semidirect
38. #564	AGL(1,𝔽₅)	direct (N × H)
67. AGL(1,𝔽₅)	#564	direct (N × H)
92. C₅	C₄×#564	semidirect

H ≅ G/N

type

4. C₅×#564

C₄

semidirect

6. C₅⋊#564

C₄

semidirect

38. #564

AGL(1,𝔽₅)

direct (N × H)

67. AGL(1,𝔽₅)

#564

direct (N × H)

92. C₅

C₄×#564

semidirect

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

subgroup	index
D₅×#564	2
#10617348	2
AGL(1,𝔽₅)×#125	2
C₄×#564	5

subgroup

index

Subgroups — AGL(1,𝔽₅)×#564 #13044198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽5)×#564 #13044198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₅)×#564 #13044198