Group #20270778 — subgroups

Subgroups — C₅₃×#1841 #20270778

Order 6784.

162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#344	C₂	1	2	yes	maximal
3	C₅₃×#339	C₂	1	2	yes	maximal
4	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	#117×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#97×C₅₃	C₄	1	2	yes	complemented
12	C₄×C₈×C₅₃	C₄	1	2	yes	complemented
13	C₂²×C₄×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	M₁₆×C₅₃	—	2	2	no	—
16	M₁₆×C₅₃	—	2	2	no	—
17	M₁₆×C₅₃	—	2	2	no	—
18	M₁₆×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂²⋊C₄×C₅₃	—	2	2	no	—
24	C₄²×C₅₃	—	2	1	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₅₃	#41	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	#41	1	2	yes	—
35	C₂×C₄×C₅₃	#41	1	2	yes	—
36	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₈×C₅₃	—	2	2	no	—
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	—	4	2	no	—
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	#117	1	1	yes	characteristic
42	C₂²×C₅₃	#126	1	2	yes	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	4	1	no	—
45	C₄×C₅₃	—	8	2	no	—
46	C₄×C₅₃	#117	1	1	yes	characteristic
47	C₄×C₅₃	#117	1	1	yes	characteristic
48	C₄×C₅₃	#126	1	2	yes	—
49	#1841	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	2	2	no	—
52	C₂×C₅₃	#383	1	1	yes	characteristic
53	C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
54	#344	C₂×C₅₃	1	2	yes	—
55	#339	C₂×C₅₃	1	2	yes	—
56	#338	C₂×C₅₃	1	1	yes	characteristic
57	H(3,ℤ/4)	C₂×C₅₃	1	1	yes	characteristic
58	C₅₃	#1841	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
60	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
61	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
62	#117	C₂²×C₅₃	1	2	yes	—
63	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
64	#97	C₄×C₅₃	1	2	yes	complemented
65	C₄×C₈	C₄×C₅₃	1	2	yes	complemented
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
74	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	2	2	no	—
79	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
80	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
81	C₂³	#41×C₅₃	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
87	C₂×C₄	#41×C₅₃	1	2	yes	—
88	C₂×C₄	#41×C₅₃	1	2	yes	—
89	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	2	no	—
92	C₈	—	4	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#117×C₅₃	1	1	yes	characteristic
95	C₂²	#126×C₅₃	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	4	1	no	—
98	C₄	—	8	2	no	—
99	C₄	#117×C₅₃	1	1	yes	characteristic
100	C₄	#117×C₅₃	1	1	yes	characteristic
101	C₄	#126×C₅₃	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	C₅₃×#383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₁₆×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#344

C₂

yes

maximal

C₅₃×#339

C₂

yes

maximal

C₅₃×#338

C₂

yes

maximal, characteristic

C₅₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

complemented

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

#126

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1841

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#383

yes

characteristic

C₄×M₁₆

C₂×C₅₃

yes

characteristic

#344

C₂×C₅₃

yes

—

#339

C₂×C₅₃

yes

—

#338

C₂×C₅₃

yes

characteristic

H(3,ℤ/4)

C₂×C₅₃

yes

characteristic

C₅₃

#1841

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₂²×C₅₃

yes

—

C₂×M₁₆

C₂²×C₅₃

yes

characteristic

C₂×M₁₆

C₂²×C₅₃

yes

characteristic

#117

C₂²×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

#97

C₄×C₅₃

yes

complemented

C₄×C₈

C₄×C₅₃

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₈

C₂×C₄×C₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₂³

#41×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂×C₄

#41×C₅₃

yes

—

C₂×C₄

#41×C₅₃

yes

—

C₂×C₄

#41×C₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#117×C₅₃

yes

characteristic

C₂²

#126×C₅₃

yes

—

C₄

—

C₄

—

C₄

—

C₄

#117×C₅₃

yes

characteristic

100

C₄

#117×C₅₃

yes

characteristic

101

C₄

#126×C₅₃

yes

—

102

C₂

—

103

C₂

—

104

C₂

C₅₃×#383

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	#117
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂×C₅₃	#383
prime radical	C₂³×C₅₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

#117

commutator

C₂×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

C₂×C₅₃

C₂³×C₅₃

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

N	H ≅ G/N	type
11. #97×C₅₃	C₄	semidirect
12. C₄×C₈×C₅₃	C₄	semidirect
49. #1841	C₅₃	direct (N × H)
58. C₅₃	#1841	direct (N × H)
64. #97	C₄×C₅₃	semidirect
65. C₄×C₈	C₄×C₅₃	semidirect

H ≅ G/N

type

11. #97×C₅₃

C₄

semidirect

12. C₄×C₈×C₅₃

C₄

semidirect

49. #1841

C₅₃

direct (N × H)

58. C₅₃

#1841

direct (N × H)

64. #97

C₄×C₅₃

semidirect

65. C₄×C₈

C₄×C₅₃

semidirect

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

subgroup	index
C₄×M₁₆×C₅₃	2
C₅₃×#344	2
C₅₃×#339	2
C₅₃×#338	2
C₅₃×H(3,ℤ/4)	2
#1841	53

subgroup

index

C₄×M₁₆×C₅₃

C₅₃×#344

C₅₃×#339

C₅₃×#338

C₅₃×H(3,ℤ/4)

#1841

Subgroups — C₅₃×#1841 #20270778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1841 #20270778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1841 #20270778