Group #27215363 — subgroups

Subgroups — C₅₃×#7851 #27215363

Order 13568.

610 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 286 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	C₂×C₅₃×#431	C₂	1	3	yes	maximal
2	C₅₃×#1273	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
4	C₅₃×#431	—	2	12	no	—
5	C₂×C₄⋊C₈×C₅₃	C₄	1	6	yes	—
6	C₂×C₂²⋊C₈×C₅₃	C₄	1	6	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
8	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
10	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
11	C₂×C₄²×C₅₃	D₄	1	6	yes	—
12	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	12	no	—
15	C₂²⋊C₈×C₅₃	—	2	12	no	—
16	C₄×C₈×C₅₃	—	2	12	no	—
17	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	12	no	—
19	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
20	C₂²×C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
21	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
22	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
23	C₂×C₈×C₅₃	—	2	12	no	—
24	C₂×C₈×C₅₃	—	2	24	no	—
25	C₄²×C₅₃	C₂²⋊C₄	1	6	yes	—
26	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
27	C₄²×C₅₃	C₂×D₄	1	3	yes	—
28	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₂³×C₅₃	—	2	6	no	—
30	C₂³×C₅₃	#95	1	2	yes	—
31	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	—	2	12	no	—
34	C₂×C₄×C₅₃	—	2	12	no	—
35	C₂×C₄×C₅₃	#95	1	6	yes	—
36	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
37	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
38	C₂×C₄×C₅₃	#131	1	12	yes	—
39	C₈×C₅₃	—	4	12	no	—
40	#7851	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₅₃	—	2	6	no	—
42	C₂²×C₅₃	—	4	4	no	—
43	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
44	C₂²×C₅₃	#405	1	6	yes	—
45	C₄×C₅₃	—	2	6	no	—
46	C₄×C₅₃	#408	1	6	yes	—
47	C₄×C₅₃	#424	1	6	yes	—
48	C₂×#431	C₂×C₅₃	1	3	yes	—
49	#1273	C₂×C₅₃	1	4	yes	complemented
50	C₂×C₅₃	—	4	2	no	—
51	C₂×C₅₃	#1454	1	4	yes	—
52	C₂×C₅₃	#1729	1	3	yes	—
53	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
54	#431	—	2	12	no	—
55	C₂×C₄⋊C₈	C₄×C₅₃	1	6	yes	—
56	C₂×C₂²⋊C₈	C₄×C₅₃	1	6	yes	—
57	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
58	C₅₃	#7851	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
60	C₂²×C₈	C₂×C₄×C₅₃	1	12	yes	—
61	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
62	C₂×C₄²	D₄×C₅₃	1	6	yes	—
63	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
64	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
65	C₄⋊C₈	—	2	12	no	—
66	C₂²⋊C₈	—	2	12	no	—
67	C₄×C₈	—	2	12	no	—
68	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
69	C₂²×C₄	—	2	12	no	—
70	C₂²×C₄	C₄²×C₅₃	1	3	yes	—
71	C₂²×C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
72	C₂²×C₄	C₄⋊C₄×C₅₃	1	6	yes	—
73	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
74	C₂×C₈	—	2	12	no	—
75	C₂×C₈	—	2	24	no	—
76	C₄²	C₂²⋊C₄×C₅₃	1	6	yes	—
77	C₄²	C₄⋊C₄×C₅₃	1	6	yes	—
78	C₄²	C₂×D₄×C₅₃	1	3	yes	—
79	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
80	C₂³	—	2	6	no	—
81	C₂³	#95×C₅₃	1	2	yes	—
82	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	2	12	no	—
85	C₂×C₄	—	2	12	no	—
86	C₂×C₄	#95×C₅₃	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
88	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
89	C₂×C₄	#131×C₅₃	1	12	yes	—
90	C₈	—	4	12	no	—
91	C₂²	—	2	6	no	—
92	C₂²	—	4	4	no	—
93	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
94	C₂²	C₅₃×#405	1	6	yes	—
95	C₄	—	2	6	no	—
96	C₄	C₅₃×#408	1	6	yes	—
97	C₄	C₅₃×#424	1	6	yes	—
98	C₂	—	4	2	no	—
99	C₂	C₅₃×#1454	1	4	yes	—
100	C₂	C₅₃×#1729	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#431

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, complemented

C₂²×C₄²×C₅₃

C₂²

yes

characteristic

C₅₃×#431

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₂×C₂²⋊C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#95

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

#7851

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#95

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

#408

yes

—

C₄×C₅₃

#424

yes

—

C₂×#431

C₂×C₅₃

yes

—

#1273

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

#1454

yes

—

C₂×C₅₃

#1729

yes

—

C₂²×C₄²

C₂²×C₅₃

yes

characteristic

#431

—

C₂×C₄⋊C₈

C₄×C₅₃

yes

—

C₂×C₂²⋊C₈

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

C₅₃

#7851

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

D₄×C₅₃

yes

—

C₂×C₄²

Q₈×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

C₄²×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₅₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

C₄⋊C₄×C₅₃

yes

—

C₄²

C₂×D₄×C₅₃

yes

—

C₄²

C₂×Q₈×C₅₃

yes

characteristic

C₂³

—

C₂³

#95×C₅₃

yes

—

C₂³

C₂×C₄²×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

—

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

—

C₂×C₄

#131×C₅₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×#95×C₅₃

yes

characteristic

C₂²

C₅₃×#405

yes

—

C₄

—

C₄

C₅₃×#408

yes

—

C₄

C₅₃×#424

yes

—

C₂

—

C₂

C₅₃×#1454

yes

—

100

C₂

C₅₃×#1729

yes

—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄²×C₅₃

C₂³

commutator

C₂³

C₂×C₄²×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂³×C₅₃

C₂×C₄²

prime radical

C₂⁴×C₅₃

C₄²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1273	C₂	semidirect
40. #7851	C₅₃	direct (N × H)
49. #1273	C₂×C₅₃	semidirect
58. C₅₃	#7851	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1273

C₂

semidirect

40. #7851

C₅₃

direct (N × H)

49. #1273

C₂×C₅₃

semidirect

58. C₅₃

#7851

direct (N × H)

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

subgroup	index
C₂×C₅₃×#431	2
C₅₃×#1273	2
#7851	53

subgroup

index

C₂×C₅₃×#431

C₅₃×#1273

#7851

Subgroups — C₅₃×#7851 #27215363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7851 #27215363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7851 #27215363