Group #28330815 — subgroups

Subgroups — C₂⁴×D₅₃⋊C₈ #28330815

Order 13568.

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	C₂³×D₅₃⋊C₈	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×#88324	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₅₃⋊C₈	C₂²	1	560	yes	direct product factor
5	C₂³×#88324	C₂²	1	60	yes	complemented
6	C₂⁵×D₅₃	C₄	1	1	yes	characteristic
7	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₅₃	C₄	1	30	yes	—
9	C₂³×C₄×D₅₃	C₂²	1	30	yes	—
10	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
11	C₂×D₅₃⋊C₈	C₂³	1	960	yes	direct product factor
12	C₂²×#88324	C₂³	1	280	yes	complemented
13	C₂⁴×D₅₃	C₈	1	32	yes	complemented
14	C₂⁴×D₅₃	C₂×C₄	1	30	yes	—
15	C₂³×Dic₅₃	C₂×C₄	1	15	yes	—
16	C₂³×Dic₅₃	C₂³	1	15	yes	—
17	C₂²×C₄×D₅₃	C₂×C₄	1	420	yes	—
18	C₂²×C₄×D₅₃	C₂³	1	140	yes	—
19	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂×C₄	1	30	yes	—
21	C₂⁴×C₅₃	C₂×C₈	1	16	yes	complemented
22	C₂⁴×C₅₃	C₂²×C₄	1	15	yes	—
23	C₂³×D₅₃	C₂×C₈	1	480	yes	complemented
24	C₂³×D₅₃	C₂²×C₄	1	140	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	140	yes	—
26	C₂²×Dic₅₃	C₂²×C₄	1	105	yes	—
27	C₂²×Dic₅₃	C₂⁴	1	35	yes	—
28	C₂×C₄×D₅₃	C₂²×C₄	1	840	yes	—
29	C₂×C₄×D₅₃	C₂⁴	1	120	yes	—
30	C₂×#88324	C₂⁴	1	240	yes	complemented
31	D₅₃⋊C₈	C₂⁴	1	256	yes	direct product factor
32	C₂³×C₅₃	C₂²×C₈	1	120	yes	complemented
33	C₂³×C₅₃	C₂³×C₄	1	35	yes	—
34	C₂²×D₅₃	C₂²×C₈	1	1120	yes	complemented
35	C₂²×D₅₃	C₂³×C₄	1	120	yes	—
36	C₂×C₄×C₅₃	C₂³×C₄	1	120	yes	—
37	C₂×Dic₅₃	C₂³×C₄	1	105	yes	—
38	C₂×Dic₅₃	C₂⁵	1	15	yes	—
39	C₄×D₅₃	C₂³×C₄	1	240	yes	—
40	C₄×D₅₃	C₂⁵	1	16	yes	—
41	#88324	C₂⁵	1	32	yes	complemented
42	C₂⁵×C₈	—	53	1	no	maximal, Sylow
43	C₂²×C₅₃	C₂³×C₈	1	140	yes	complemented
44	C₂²×C₅₃	C₂⁴×C₄	1	15	yes	—
45	C₂×D₅₃	C₂³×C₈	1	480	yes	complemented
46	C₂×D₅₃	C₂⁴×C₄	1	16	yes	—
47	C₄×C₅₃	C₂⁴×C₄	1	16	yes	—
48	Dic₅₃	C₂⁴×C₄	1	15	yes	—
49	Dic₅₃	C₂⁶	1	1	yes	characteristic
50	C₂⁵×C₄	—	53	1	no	—
51	C₂⁴×C₈	—	53	2	no	—
52	C₂⁴×C₈	—	53	60	no	—
53	C₂×C₅₃	C₂⁴×C₈	1	30	yes	complemented
54	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
55	D₅₃	C₂⁴×C₈	1	32	yes	complemented
56	C₂⁶	—	53	1	no	—
57	C₂⁴×C₄	—	53	1	no	—
58	C₂⁴×C₄	—	53	30	no	—
59	C₂⁴×C₄	—	53	30	no	—
60	C₂⁴×C₄	#6157	1	1	yes	characteristic
61	C₂³×C₈	—	53	60	no	—
62	C₂³×C₈	—	53	560	no	—
63	C₅₃	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
64	C₂⁵	—	53	30	no	—
65	C₂⁵	—	53	32	no	—
66	C₂⁵	C₂×#6157	1	1	yes	characteristic
67	C₂³×C₄	—	53	15	no	—
68	C₂³×C₄	—	53	15	no	—
69	C₂³×C₄	—	53	140	no	—
70	C₂³×C₄	—	53	420	no	—
71	C₂³×C₄	C₂×#6157	1	30	yes	—
72	C₂²×C₈	—	53	280	no	—
73	C₂²×C₈	—	53	960	no	—
74	C₂⁴	—	53	480	no	—
75	C₂⁴	—	53	140	no	—
76	C₂⁴	D₅₃⋊C₈	1	16	yes	direct product factor
77	C₂⁴	C₂²×#6157	1	15	yes	—
78	C₂²×C₄	—	53	105	no	—
79	C₂²×C₄	—	53	35	no	—
80	C₂²×C₄	—	53	840	no	—
81	C₂²×C₄	—	53	120	no	—
82	C₂²×C₄	C₂²×#6157	1	140	yes	—
83	C₂×C₈	—	53	240	no	—
84	C₂×C₈	—	53	256	no	—
85	C₂³	—	53	1120	no	—
86	C₂³	—	53	120	no	—
87	C₂³	C₂×D₅₃⋊C₈	1	120	yes	direct product factor
88	C₂³	C₂³×#6157	1	35	yes	—
89	C₂×C₄	—	53	105	no	—
90	C₂×C₄	—	53	15	no	—
91	C₂×C₄	—	53	240	no	—
92	C₂×C₄	—	53	16	no	—
93	C₂×C₄	C₂³×#6157	1	120	yes	—
94	C₈	—	53	32	no	—
95	C₂²	—	53	480	no	—
96	C₂²	—	53	16	no	—
97	C₂²	C₂²×D₅₃⋊C₈	1	140	yes	direct product factor
98	C₂²	C₂⁴×#6157	1	15	yes	—
99	C₄	—	53	1	no	—
100	C₄	—	53	15	no	—
101	C₄	C₂⁴×#6157	1	16	yes	—
102	C₂	—	53	32	no	—
103	C₂	C₂³×D₅₃⋊C₈	1	30	yes	direct product factor
104	C₂	C₂⁵×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×D₅₃⋊C₈

C₂

yes

maximal, direct product factor

C₂⁴×#88324

C₂

yes

maximal, complemented

C₂⁴×C₄×D₅₃

C₂

yes

maximal, characteristic

C₂²×D₅₃⋊C₈

C₂²

560

yes

direct product factor

C₂³×#88324

C₂²

yes

complemented

C₂⁵×D₅₃

C₄

yes

characteristic

C₂⁴×Dic₅₃

C₂²

yes

characteristic

C₂³×C₄×D₅₃

C₄

yes

—

C₂³×C₄×D₅₃

C₂²

yes

—

C₂⁴×C₄×C₅₃

C₄

yes

characteristic

C₂×D₅₃⋊C₈

C₂³

960

yes

direct product factor

C₂²×#88324

C₂³

280

yes

complemented

C₂⁴×D₅₃

C₈

yes

complemented

C₂⁴×D₅₃

C₂×C₄

yes

—

C₂³×Dic₅₃

C₂×C₄

yes

—

C₂³×Dic₅₃

C₂³

yes

—

C₂²×C₄×D₅₃

C₂×C₄

420

yes

—

C₂²×C₄×D₅₃

C₂³

140

yes

—

C₂⁵×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂×C₈

yes

complemented

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂³×D₅₃

C₂×C₈

480

yes

complemented

C₂³×D₅₃

C₂²×C₄

140

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

140

yes

—

C₂²×Dic₅₃

C₂²×C₄

105

yes

—

C₂²×Dic₅₃

C₂⁴

yes

—

C₂×C₄×D₅₃

C₂²×C₄

840

yes

—

C₂×C₄×D₅₃

C₂⁴

120

yes

—

C₂×#88324

C₂⁴

240

yes

complemented

D₅₃⋊C₈

C₂⁴

256

yes

direct product factor

C₂³×C₅₃

C₂²×C₈

120

yes

complemented

C₂³×C₅₃

C₂³×C₄

yes

—

C₂²×D₅₃

C₂²×C₈

1120

yes

complemented

C₂²×D₅₃

C₂³×C₄

120

yes

—

C₂×C₄×C₅₃

C₂³×C₄

120

yes

—

C₂×Dic₅₃

C₂³×C₄

105

yes

—

C₂×Dic₅₃

C₂⁵

yes

—

C₄×D₅₃

C₂³×C₄

240

yes

—

C₄×D₅₃

C₂⁵

yes

—

#88324

C₂⁵

yes

complemented

C₂⁵×C₈

—

maximal, Sylow

C₂²×C₅₃

C₂³×C₈

140

yes

complemented

C₂²×C₅₃

C₂⁴×C₄

yes

—

C₂×D₅₃

C₂³×C₈

480

yes

complemented

C₂×D₅₃

C₂⁴×C₄

yes

—

C₄×C₅₃

C₂⁴×C₄

yes

—

Dic₅₃

C₂⁴×C₄

yes

—

Dic₅₃

C₂⁶

yes

characteristic

C₂⁵×C₄

—

C₂⁴×C₈

—

C₂⁴×C₈

—

C₂×C₅₃

C₂⁴×C₈

yes

complemented

C₂×C₅₃

C₂⁵×C₄

yes

characteristic

D₅₃

C₂⁴×C₈

yes

complemented

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

#6157

yes

characteristic

C₂³×C₈

—

C₂³×C₈

—

560

—

C₅₃

C₂⁵×C₈

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂⁵

—

C₂⁵

C₂×#6157

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

140

—

C₂³×C₄

—

420

—

C₂³×C₄

C₂×#6157

yes

—

C₂²×C₈

—

280

—

C₂²×C₈

—

960

—

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

D₅₃⋊C₈

yes

direct product factor

C₂⁴

C₂²×#6157

yes

—

C₂²×C₄

—

105

—

C₂²×C₄

—

C₂²×C₄

—

840

—

C₂²×C₄

—

120

—

C₂²×C₄

C₂²×#6157

140

yes

—

C₂×C₈

—

240

—

C₂×C₈

—

256

—

C₂³

—

1120

—

C₂³

—

120

—

C₂³

C₂×D₅₃⋊C₈

120

yes

direct product factor

C₂³

C₂³×#6157

yes

—

C₂×C₄

—

105

—

C₂×C₄

—

C₂×C₄

—

240

—

C₂×C₄

—

C₂×C₄

C₂³×#6157

120

yes

—

C₈

—

C₂²

—

480

—

C₂²

—

C₂²

C₂²×D₅₃⋊C₈

140

yes

direct product factor

C₂²

C₂⁴×#6157

yes

—

C₄

—

100

C₄

—

101

C₄

C₂⁴×#6157

yes

—

102

C₂

—

103

C₂

C₂³×D₅₃⋊C₈

yes

direct product factor

104

C₂

C₂⁵×#6157

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#6157
commutator, hypocenter	C₅₃	C₂⁵×C₈
Frattini	C₂	C₂⁵×#6157
Fitting	C₂⁴×C₄×C₅₃	C₄
socle	C₂⁵×C₅₃	C₂×C₄
prime radical	C₂⁵×D₅₃	C₄
cosocle	Dic₅₃	C₂⁶
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄

subgroup

G/H

center, hypercenter

C₂⁴×C₄

#6157

commutator, hypocenter

C₅₃

C₂⁵×C₈

Frattini

C₂

C₂⁵×#6157

Fitting

C₂⁴×C₄×C₅₃

C₄

socle

C₂⁵×C₅₃

C₂×C₄

prime radical

C₂⁵×D₅₃

C₄

cosocle

Dic₅₃

C₂⁶

LCS

G ▷ C₅₃

UCS

C₁ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂³×D₅₃⋊C₈	C₂	direct (N × H)
2. C₂⁴×#88324	C₂	semidirect
4. C₂²×D₅₃⋊C₈	C₂²	direct (N × H)
5. C₂³×#88324	C₂²	semidirect
11. C₂×D₅₃⋊C₈	C₂³	direct (N × H)
12. C₂²×#88324	C₂³	semidirect
13. C₂⁴×D₅₃	C₈	semidirect
21. C₂⁴×C₅₃	C₂×C₈	semidirect
23. C₂³×D₅₃	C₂×C₈	semidirect
30. C₂×#88324	C₂⁴	semidirect
31. D₅₃⋊C₈	C₂⁴	direct (N × H)
32. C₂³×C₅₃	C₂²×C₈	semidirect
34. C₂²×D₅₃	C₂²×C₈	semidirect
41. #88324	C₂⁵	semidirect
43. C₂²×C₅₃	C₂³×C₈	semidirect
45. C₂×D₅₃	C₂³×C₈	semidirect
53. C₂×C₅₃	C₂⁴×C₈	semidirect
55. D₅₃	C₂⁴×C₈	semidirect
63. C₅₃	C₂⁵×C₈	semidirect
76. C₂⁴	D₅₃⋊C₈	direct (N × H)
87. C₂³	C₂×D₅₃⋊C₈	direct (N × H)
97. C₂²	C₂²×D₅₃⋊C₈	direct (N × H)
103. C₂	C₂³×D₅₃⋊C₈	direct (N × H)

H ≅ G/N

type

1. C₂³×D₅₃⋊C₈

C₂

direct (N × H)

2. C₂⁴×#88324

C₂

semidirect

4. C₂²×D₅₃⋊C₈

C₂²

direct (N × H)

5. C₂³×#88324

C₂²

semidirect

11. C₂×D₅₃⋊C₈

C₂³

direct (N × H)

12. C₂²×#88324

C₂³

semidirect

13. C₂⁴×D₅₃

C₈

semidirect

21. C₂⁴×C₅₃

C₂×C₈

semidirect

23. C₂³×D₅₃

C₂×C₈

semidirect

30. C₂×#88324

C₂⁴

semidirect

31. D₅₃⋊C₈

C₂⁴

direct (N × H)

32. C₂³×C₅₃

C₂²×C₈

semidirect

34. C₂²×D₅₃

C₂²×C₈

semidirect

41. #88324

C₂⁵

semidirect

43. C₂²×C₅₃

C₂³×C₈

semidirect

45. C₂×D₅₃

C₂³×C₈

semidirect

53. C₂×C₅₃

C₂⁴×C₈

semidirect

55. D₅₃

C₂⁴×C₈

semidirect

63. C₅₃

C₂⁵×C₈

semidirect

76. C₂⁴

D₅₃⋊C₈

direct (N × H)

87. C₂³

C₂×D₅₃⋊C₈

direct (N × H)

97. C₂²

C₂²×D₅₃⋊C₈

direct (N × H)

103. C₂

C₂³×D₅₃⋊C₈

direct (N × H)

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

subgroup	index
C₂³×D₅₃⋊C₈	2
C₂⁴×#88324	2
C₂⁴×C₄×D₅₃	2
C₂⁵×C₈	53

subgroup

index

C₂³×D₅₃⋊C₈

C₂⁴×#88324

C₂⁴×C₄×D₅₃

C₂⁵×C₈

Subgroups — C₂⁴×D₅₃⋊C₈ #28330815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×D53⋊C8 #28330815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₅₃⋊C₈ #28330815