Group #20271551 — subgroups

Subgroups — C₅₃×#2614 #20271551

Order 6784.

782 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 232 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×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#123×C₅₃	C₂	1	3	yes	maximal
3	C₅₃×#395	C₂	1	4	yes	maximal, complemented
4	C₅₃×#393	C₂	1	4	yes	maximal, complemented
5	C₅₃×#381	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×C₅₃	—	2	12	no	—
7	C₂²×Q₈×C₅₃	C₂²	1	4	yes	—
8	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
10	#123×C₅₃	—	2	12	no	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
12	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
14	#95×C₅₃	C₂²	1	12	yes	—
15	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
16	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	32	no	—
18	C₂×Q₈×C₅₃	—	2	24	no	—
19	C₂×D₄×C₅₃	—	2	6	no	—
20	C₂²×C₄×C₅₃	—	2	12	no	—
21	C₂²×C₄×C₅₃	D₄	1	12	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
25	C₄⋊C₄×C₅₃	—	2	12	no	—
26	C₂²⋊C₄×C₅₃	—	2	6	no	—
27	C₂²⋊C₄×C₅₃	—	2	12	no	—
28	C₂³×C₅₃	—	2	4	no	—
29	C₂³×C₅₃	—	2	3	no	—
30	C₂³×C₅₃	—	2	4	no	—
31	C₂³×C₅₃	C₂×D₄	1	6	yes	—
32	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
33	Q₈×C₅₃	—	2	32	no	—
34	D₄×C₅₃	—	4	4	no	—
35	C₂×C₄×C₅₃	—	2	24	no	—
36	C₂×C₄×C₅₃	—	2	12	no	—
37	C₂×C₄×C₅₃	—	2	3	no	—
38	C₂×C₄×C₅₃	—	2	6	no	—
39	C₂×C₄×C₅₃	—	2	12	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
41	C₂²×C₅₃	—	2	12	no	—
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	—	4	6	no	—
44	C₂²×C₅₃	#120	1	4	yes	—
45	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
46	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
47	C₄×C₅₃	—	2	12	no	—
48	C₄×C₅₃	—	4	1	no	—
49	C₄×C₅₃	—	4	6	no	—
50	#2614	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₅₃	—	2	4	no	—
52	C₂×C₅₃	—	8	1	no	—
53	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
54	C₂×C₅₃	#536	1	6	yes	—
55	C₂³×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
56	C₂×#123	C₂×C₅₃	1	3	yes	—
57	#395	C₂×C₅₃	1	4	yes	complemented
58	#393	C₂×C₅₃	1	4	yes	complemented
59	#381	C₂×C₅₃	1	3	yes	complemented
60	C₅₃	#2614	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂²×C₅₃	1	4	yes	—
63	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
64	C₂³×C₄	C₂²×C₅₃	1	3	yes	—
65	#123	—	2	12	no	—
66	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
67	C₂×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
68	C₂×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
69	#95	C₂²×C₅₃	1	12	yes	—
70	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
71	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	2	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	D₄×C₅₃	1	12	yes	—
77	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
78	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
80	C₄⋊C₄	—	2	12	no	—
81	C₂²⋊C₄	—	2	6	no	—
82	C₂²⋊C₄	—	2	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	2	3	no	—
85	C₂³	—	2	4	no	—
86	C₂³	C₂×D₄×C₅₃	1	6	yes	—
87	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	4	4	no	—
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	12	no	—
95	C₂×C₄	C₂×D₄×C₅₃	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	4	6	no	—
99	C₂²	#120×C₅₃	1	4	yes	—
100	C₂²	C₂²×D₄×C₅₃	1	3	yes	—
101	C₂²	2₋¹⁺⁴×C₅₃	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	4	1	no	—
104	C₄	—	4	6	no	—
105	C₂	—	2	4	no	—
106	C₂	—	8	1	no	—
107	C₂	C₂×#120×C₅₃	1	1	yes	characteristic
108	C₂	C₅₃×#536	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#123×C₅₃

C₂

yes

maximal

C₅₃×#395

C₂

yes

maximal, complemented

C₅₃×#393

C₂

yes

maximal, complemented

C₅₃×#381

C₂

yes

maximal, complemented

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

#123×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#2614

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#120

yes

characteristic

C₂×C₅₃

#536

yes

—

C₂³×Q₈

C₂×C₅₃

yes

characteristic, complemented

C₂×#123

C₂×C₅₃

yes

—

#395

C₂×C₅₃

yes

complemented

#393

C₂×C₅₃

yes

complemented

#381

C₂×C₅₃

yes

complemented

C₅₃

#2614

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂²×C₅₃

yes

—

C₂²×D₄

C₂²×C₅₃

yes

characteristic

C₂³×C₄

C₂²×C₅₃

yes

—

#123

—

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₅₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₅₃

yes

—

#95

C₂²×C₅₃

yes

—

C₂⁴

C₂³×C₅₃

yes

characteristic

C₂⁴

C₂³×C₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

C₂²×C₄

C₂³×C₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₅₃

yes

—

C₂³

C₂⁴×C₅₃

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₅₃

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#120×C₅₃

yes

—

100

C₂²

C₂²×D₄×C₅₃

yes

—

101

C₂²

2₋¹⁺⁴×C₅₃

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

—

107

C₂

C₂×#120×C₅₃

yes

characteristic

108

C₂

C₅₃×#536

yes

—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₅₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₅₃

prime radical

C₂²×D₄×C₅₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₂³×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#395	C₂	semidirect
4. C₅₃×#393	C₂	semidirect
5. C₅₃×#381	C₂	semidirect
50. #2614	C₅₃	direct (N × H)
55. C₂³×Q₈	C₂×C₅₃	semidirect
57. #395	C₂×C₅₃	semidirect
58. #393	C₂×C₅₃	semidirect
59. #381	C₂×C₅₃	semidirect
60. C₅₃	#2614	direct (N × H)

H ≅ G/N

type

1. C₂³×Q₈×C₅₃

C₂

semidirect

3. C₅₃×#395

C₂

semidirect

4. C₅₃×#393

C₂

semidirect

5. C₅₃×#381

C₂

semidirect

50. #2614

C₅₃

direct (N × H)

55. C₂³×Q₈

C₂×C₅₃

semidirect

57. #395

C₂×C₅₃

semidirect

58. #393

C₂×C₅₃

semidirect

59. #381

C₂×C₅₃

semidirect

60. C₅₃

#2614

direct (N × H)

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

subgroup	index
C₂³×Q₈×C₅₃	2
C₂×#123×C₅₃	2
C₅₃×#395	2
C₅₃×#393	2
C₅₃×#381	2
#2614	53

subgroup

index

C₂³×Q₈×C₅₃

C₂×#123×C₅₃

C₅₃×#395

C₅₃×#393

C₅₃×#381

#2614

Subgroups — C₅₃×#2614 #20271551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2614 #20271551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2614 #20271551