Group #13046023 — subgroups

Subgroups — C₂⁴×D₅⋊C₈ #13046023

Order 1280.

15454 conjugacy classes of subgroups; only the 8408 normal subgroups are shown.

normal subgroups only

This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 8408 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	C₂⁴×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#170	C₂	1	2	yes	maximal, complemented
3	C₂³×D₅⋊C₈	C₂	1	60	yes	maximal, direct product factor
4	C₂⁵×D₅	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₅	C₂²	1	1	yes	characteristic
7	C₂³×C₄×D₅	C₄	1	30	yes	—
8	C₂³×C₄×D₅	C₂²	1	30	yes	—
9	C₂³×#170	C₂²	1	60	yes	complemented
10	C₂²×D₅⋊C₈	C₂²	1	560	yes	direct product factor
11	C₂⁵×C₈	—	5	1	no	maximal, Sylow
12	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂⁴×D₅	C₈	1	32	yes	complemented
14	C₂⁴×D₅	C₂×C₄	1	30	yes	—
15	C₂³×C₄×C₅	C₂×C₄	1	30	yes	—
16	C₂³×Dic₅	C₂×C₄	1	15	yes	—
17	C₂³×Dic₅	C₂³	1	15	yes	—
18	C₂²×C₄×D₅	C₂×C₄	1	420	yes	—
19	C₂²×C₄×D₅	C₂³	1	140	yes	—
20	C₂²×#170	C₂³	1	280	yes	complemented
21	C₂×D₅⋊C₈	C₂³	1	960	yes	direct product factor
22	C₂⁵×C₄	—	5	1	no	—
23	C₂⁴×C₈	—	5	60	no	—
24	C₂⁴×C₈	—	5	2	no	—
25	C₂⁴×C₅	C₂×C₈	1	16	yes	complemented
26	C₂⁴×C₅	C₂²×C₄	1	15	yes	—
27	C₂³×D₅	C₂×C₈	1	480	yes	complemented
28	C₂³×D₅	C₂²×C₄	1	140	yes	—
29	C₂²×C₄×C₅	C₂²×C₄	1	140	yes	—
30	C₂²×Dic₅	C₂²×C₄	1	105	yes	—
31	C₂²×Dic₅	C₂⁴	1	35	yes	—
32	C₂×C₄×D₅	C₂²×C₄	1	840	yes	—
33	C₂×C₄×D₅	C₂⁴	1	120	yes	—
34	C₂×#170	C₂⁴	1	240	yes	complemented
35	D₅⋊C₈	C₂⁴	1	256	yes	direct product factor
36	C₂⁶	—	5	1	no	—
37	C₂⁴×C₄	—	5	1	no	—
38	C₂⁴×C₄	—	5	30	no	—
39	C₂⁴×C₄	—	5	30	no	—
40	C₂⁴×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₂³×C₈	—	5	560	no	—
42	C₂³×C₈	—	5	60	no	—
43	C₂³×C₅	C₂²×C₈	1	120	yes	complemented
44	C₂³×C₅	C₂³×C₄	1	35	yes	—
45	C₂²×D₅	C₂²×C₈	1	1120	yes	complemented
46	C₂²×D₅	C₂³×C₄	1	120	yes	—
47	C₂×C₄×C₅	C₂³×C₄	1	120	yes	—
48	C₂×Dic₅	C₂³×C₄	1	105	yes	—
49	C₂×Dic₅	C₂⁵	1	15	yes	—
50	C₄×D₅	C₂³×C₄	1	240	yes	—
51	C₄×D₅	C₂⁵	1	16	yes	—
52	#170	C₂⁵	1	32	yes	complemented
53	C₂⁵	—	5	32	no	—
54	C₂⁵	—	5	30	no	—
55	C₂⁵	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂³×C₄	—	5	15	no	—
57	C₂³×C₄	—	5	15	no	—
58	C₂³×C₄	—	5	420	no	—
59	C₂³×C₄	—	5	140	no	—
60	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	30	yes	—
61	C₂²×C₈	—	5	960	no	—
62	C₂²×C₈	—	5	280	no	—
63	C₂²×C₅	C₂³×C₈	1	140	yes	complemented
64	C₂²×C₅	C₂⁴×C₄	1	15	yes	—
65	C₂×D₅	C₂³×C₈	1	480	yes	complemented
66	C₂×D₅	C₂⁴×C₄	1	16	yes	—
67	C₄×C₅	C₂⁴×C₄	1	16	yes	—
68	Dic₅	C₂⁴×C₄	1	15	yes	—
69	Dic₅	C₂⁶	1	1	yes	characteristic
70	C₂⁴	—	5	480	no	—
71	C₂⁴	—	5	140	no	—
72	C₂⁴	D₅⋊C₈	1	16	yes	direct product factor
73	C₂⁴	C₂²×AGL(1,𝔽₅)	1	15	yes	—
74	C₂²×C₄	—	5	35	no	—
75	C₂²×C₄	—	5	105	no	—
76	C₂²×C₄	—	5	120	no	—
77	C₂²×C₄	—	5	840	no	—
78	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	140	yes	—
79	C₂×C₈	—	5	256	no	—
80	C₂×C₈	—	5	240	no	—
81	C₂×C₅	C₂⁴×C₈	1	30	yes	complemented
82	C₂×C₅	C₂⁵×C₄	1	1	yes	characteristic
83	D₅	C₂⁴×C₈	1	32	yes	complemented
84	C₂³	—	5	1120	no	—
85	C₂³	—	5	120	no	—
86	C₂³	C₂×D₅⋊C₈	1	120	yes	direct product factor
87	C₂³	C₂³×AGL(1,𝔽₅)	1	35	yes	—
88	C₂×C₄	—	5	15	no	—
89	C₂×C₄	—	5	105	no	—
90	C₂×C₄	—	5	16	no	—
91	C₂×C₄	—	5	240	no	—
92	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	120	yes	—
93	C₈	—	5	32	no	—
94	C₅	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
95	C₂²	—	5	480	no	—
96	C₂²	—	5	16	no	—
97	C₂²	C₂²×D₅⋊C₈	1	140	yes	direct product factor
98	C₂²	C₂⁴×AGL(1,𝔽₅)	1	15	yes	—
99	C₄	—	5	1	no	—
100	C₄	—	5	15	no	—
101	C₄	C₂⁴×AGL(1,𝔽₅)	1	16	yes	—
102	C₂	—	5	32	no	—
103	C₂	C₂³×D₅⋊C₈	1	30	yes	direct product factor
104	C₂	C₂⁵×AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₄×D₅

C₂

yes

maximal, characteristic

C₂⁴×#170

C₂

yes

maximal, complemented

C₂³×D₅⋊C₈

C₂

yes

maximal, direct product factor

C₂⁵×D₅

C₄

yes

characteristic

C₂⁴×C₄×C₅

C₄

yes

characteristic

C₂⁴×Dic₅

C₂²

yes

characteristic

C₂³×C₄×D₅

C₄

yes

—

C₂³×C₄×D₅

C₂²

yes

—

C₂³×#170

C₂²

yes

complemented

C₂²×D₅⋊C₈

C₂²

560

yes

direct product factor

C₂⁵×C₈

—

maximal, Sylow

C₂⁵×C₅

C₂×C₄

yes

characteristic

C₂⁴×D₅

C₈

yes

complemented

C₂⁴×D₅

C₂×C₄

yes

—

C₂³×C₄×C₅

C₂×C₄

yes

—

C₂³×Dic₅

C₂×C₄

yes

—

C₂³×Dic₅

C₂³

yes

—

C₂²×C₄×D₅

C₂×C₄

420

yes

—

C₂²×C₄×D₅

C₂³

140

yes

—

C₂²×#170

C₂³

280

yes

complemented

C₂×D₅⋊C₈

C₂³

960

yes

direct product factor

C₂⁵×C₄

—

C₂⁴×C₈

—

C₂⁴×C₈

—

C₂⁴×C₅

C₂×C₈

yes

complemented

C₂⁴×C₅

C₂²×C₄

yes

—

C₂³×D₅

C₂×C₈

480

yes

complemented

C₂³×D₅

C₂²×C₄

140

yes

—

C₂²×C₄×C₅

C₂²×C₄

140

yes

—

C₂²×Dic₅

C₂²×C₄

105

yes

—

C₂²×Dic₅

C₂⁴

yes

—

C₂×C₄×D₅

C₂²×C₄

840

yes

—

C₂×C₄×D₅

C₂⁴

120

yes

—

C₂×#170

C₂⁴

240

yes

complemented

D₅⋊C₈

C₂⁴

256

yes

direct product factor

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

AGL(1,𝔽₅)

yes

characteristic

C₂³×C₈

—

560

—

C₂³×C₈

—

C₂³×C₅

C₂²×C₈

120

yes

complemented

C₂³×C₅

C₂³×C₄

yes

—

C₂²×D₅

C₂²×C₈

1120

yes

complemented

C₂²×D₅

C₂³×C₄

120

yes

—

C₂×C₄×C₅

C₂³×C₄

120

yes

—

C₂×Dic₅

C₂³×C₄

105

yes

—

C₂×Dic₅

C₂⁵

yes

—

C₄×D₅

C₂³×C₄

240

yes

—

C₄×D₅

C₂⁵

yes

—

#170

C₂⁵

yes

complemented

C₂⁵

—

C₂⁵

—

C₂⁵

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

420

—

C₂³×C₄

—

140

—

C₂³×C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₈

—

960

—

C₂²×C₈

—

280

—

C₂²×C₅

C₂³×C₈

140

yes

complemented

C₂²×C₅

C₂⁴×C₄

yes

—

C₂×D₅

C₂³×C₈

480

yes

complemented

C₂×D₅

C₂⁴×C₄

yes

—

C₄×C₅

C₂⁴×C₄

yes

—

Dic₅

C₂⁴×C₄

yes

—

Dic₅

C₂⁶

yes

characteristic

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

D₅⋊C₈

yes

direct product factor

C₂⁴

C₂²×AGL(1,𝔽₅)

yes

—

C₂²×C₄

—

C₂²×C₄

—

105

—

C₂²×C₄

—

120

—

C₂²×C₄

—

840

—

C₂²×C₄

C₂²×AGL(1,𝔽₅)

140

yes

—

C₂×C₈

—

256

—

C₂×C₈

—

240

—

C₂×C₅

C₂⁴×C₈

yes

complemented

C₂×C₅

C₂⁵×C₄

yes

characteristic

D₅

C₂⁴×C₈

yes

complemented

C₂³

—

1120

—

C₂³

—

120

—

C₂³

C₂×D₅⋊C₈

120

yes

direct product factor

C₂³

C₂³×AGL(1,𝔽₅)

yes

—

C₂×C₄

—

C₂×C₄

—

105

—

C₂×C₄

—

C₂×C₄

—

240

—

C₂×C₄

C₂³×AGL(1,𝔽₅)

120

yes

—

C₈

—

C₅

C₂⁵×C₈

yes

characteristic, complemented, Sylow

C₂²

—

480

—

C₂²

—

C₂²

C₂²×D₅⋊C₈

140

yes

direct product factor

C₂²

C₂⁴×AGL(1,𝔽₅)

yes

—

C₄

—

100

C₄

—

101

C₄

C₂⁴×AGL(1,𝔽₅)

yes

—

102

C₂

—

103

C₂

C₂³×D₅⋊C₈

yes

direct product factor

104

C₂

C₂⁵×AGL(1,𝔽₅)

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂⁵×C₈
Frattini	C₂	C₂⁵×AGL(1,𝔽₅)
Fitting	C₂⁴×C₄×C₅	C₄
socle	C₂⁵×C₅	C₂×C₄
prime radical	C₂⁵×D₅	C₄
cosocle	Dic₅	C₂⁶
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴×C₄

subgroup

G/H

center, hypercenter

C₂⁴×C₄

AGL(1,𝔽₅)

commutator, hypocenter

C₅

C₂⁵×C₈

Frattini

C₂

C₂⁵×AGL(1,𝔽₅)

Fitting

C₂⁴×C₄×C₅

C₄

socle

C₂⁵×C₅

C₂×C₄

prime radical

C₂⁵×D₅

C₄

cosocle

Dic₅

C₂⁶

LCS

G ▷ C₅

UCS

C₁ ◁ C₂⁴×C₄

N	H ≅ G/N	type
2. C₂⁴×#170	C₂	semidirect
3. C₂³×D₅⋊C₈	C₂	direct (N × H)
9. C₂³×#170	C₂²	semidirect
10. C₂²×D₅⋊C₈	C₂²	direct (N × H)
13. C₂⁴×D₅	C₈	semidirect
20. C₂²×#170	C₂³	semidirect
21. C₂×D₅⋊C₈	C₂³	direct (N × H)
25. C₂⁴×C₅	C₂×C₈	semidirect
27. C₂³×D₅	C₂×C₈	semidirect
34. C₂×#170	C₂⁴	semidirect
35. D₅⋊C₈	C₂⁴	direct (N × H)
43. C₂³×C₅	C₂²×C₈	semidirect
45. C₂²×D₅	C₂²×C₈	semidirect
52. #170	C₂⁵	semidirect
63. C₂²×C₅	C₂³×C₈	semidirect
65. C₂×D₅	C₂³×C₈	semidirect
72. C₂⁴	D₅⋊C₈	direct (N × H)
81. C₂×C₅	C₂⁴×C₈	semidirect
83. D₅	C₂⁴×C₈	semidirect
86. C₂³	C₂×D₅⋊C₈	direct (N × H)
94. C₅	C₂⁵×C₈	semidirect
97. C₂²	C₂²×D₅⋊C₈	direct (N × H)
103. C₂	C₂³×D₅⋊C₈	direct (N × H)

H ≅ G/N

type

2. C₂⁴×#170

C₂

semidirect

3. C₂³×D₅⋊C₈

C₂

direct (N × H)

9. C₂³×#170

C₂²

semidirect

10. C₂²×D₅⋊C₈

C₂²

direct (N × H)

13. C₂⁴×D₅

C₈

semidirect

20. C₂²×#170

C₂³

semidirect

21. C₂×D₅⋊C₈

C₂³

direct (N × H)

25. C₂⁴×C₅

C₂×C₈

semidirect

27. C₂³×D₅

C₂×C₈

semidirect

34. C₂×#170

C₂⁴

semidirect

35. D₅⋊C₈

C₂⁴

direct (N × H)

43. C₂³×C₅

C₂²×C₈

semidirect

45. C₂²×D₅

C₂²×C₈

semidirect

52. #170

C₂⁵

semidirect

63. C₂²×C₅

C₂³×C₈

semidirect

65. C₂×D₅

C₂³×C₈

semidirect

72. C₂⁴

D₅⋊C₈

direct (N × H)

81. C₂×C₅

C₂⁴×C₈

semidirect

83. D₅

C₂⁴×C₈

semidirect

86. C₂³

C₂×D₅⋊C₈

direct (N × H)

94. C₅

C₂⁵×C₈

semidirect

97. C₂²

C₂²×D₅⋊C₈

direct (N × H)

103. C₂

C₂³×D₅⋊C₈

direct (N × H)

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

subgroup	index
C₂⁴×C₄×D₅	2
C₂⁴×#170	2
C₂³×D₅⋊C₈	2
C₂⁵×C₈	5

subgroup

index

C₂⁴×C₄×D₅

C₂⁴×#170

C₂³×D₅⋊C₈

C₂⁵×C₈

Subgroups — C₂⁴×D₅⋊C₈ #13046023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×D5⋊C8 #13046023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₅⋊C₈ #13046023