Group #20282906 — subgroups

Subgroups — #20282906

Order 6784.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16810565	C₂	1	6	yes	maximal, complemented
2	#16810548	C₂	1	3	yes	maximal
3	#16810542	C₂	1	2	yes	maximal, complemented
4	#16810495	C₂	1	3	yes	maximal
5	C₅₃×#393	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
8	C₂×#11719919	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
10	#13133124	C₂²	1	6	yes	—
11	#13133124	C₂²	1	3	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
15	#95×C₅₃	C₂²	1	3	yes	—
16	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	4	3	no	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
20	C₂²×Dic₅₃	—	4	3	no	—
21	C₂²×Dic₅₃	C₂³	1	6	yes	—
22	C₂²×Dic₅₃	C₂³	1	2	yes	—
23	C₂²⋊C₄×C₅₃	—	4	3	no	—
24	C₂²⋊Dic₅₃	—	2	6	no	—
25	C₄⋊Dic₅₃	—	4	3	no	—
26	#11719919	—	4	6	no	—
27	C₂³×C₅₃	—	2	3	no	—
28	C₂³×C₅₃	—	2	3	no	—
29	C₂³×C₅₃	#41	1	2	yes	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	Q₈×C₅₃	—	4	4	no	—
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	2	3	no	—
35	C₂×C₄×C₅₃	—	2	6	no	—
36	C₂×Dic₅₃	—	2	6	no	—
37	C₂×Dic₅₃	—	2	6	no	—
38	C₂×Dic₅₃	—	2	6	no	—
39	C₂×Dic₅₃	—	2	6	no	—
40	C₂×Dic₅₃	—	4	4	no	—
41	C₂²×C₅₃	—	2	6	no	—
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	—	4	3	no	—
44	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
45	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
47	C₄×C₅₃	—	4	3	no	—
48	C₄×C₅₃	—	4	3	no	—
49	Dic₅₃	—	4	6	no	—
50	Dic₅₃	—	4	2	no	—
51	#2835	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	4	2	no	—
53	C₂×C₅₃	#539	1	3	yes	—
54	C₂×C₅₃	H(3,𝔽₄)	1	1	yes	characteristic
55	C₂×C₅₃	#563	1	3	yes	—
56	#400	—	53	6	no	—
57	#397	—	53	3	no	—
58	#395	—	53	2	no	—
59	#393	D₅₃	1	1	yes	characteristic
60	#385	—	53	3	no	—
61	C₅₃	#2835	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂³×C₄	—	53	1	no	—
64	C₂×C₄⋊C₄	—	53	6	no	—
65	C₂×C₄⋊C₄	—	53	3	no	—
66	C₂×C₂²⋊C₄	—	53	3	no	—
67	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
68	#95	—	53	6	no	—
69	#95	—	53	6	no	—
70	#95	—	53	3	no	—
71	#95	C₂×D₅₃	1	3	yes	—
72	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
73	C₂×Q₈	—	4	3	no	—
74	C₂²×C₄	—	53	6	no	—
75	C₂²×C₄	—	53	2	no	—
76	C₂²×C₄	—	212	3	no	—
77	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
78	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
79	C₄⋊C₄	—	212	6	no	—
80	C₄⋊C₄	—	212	3	no	—
81	C₂²⋊C₄	—	4	3	no	—
82	C₂²⋊C₄	—	106	6	no	—
83	C₂³	—	2	3	no	—
84	C₂³	—	2	3	no	—
85	C₂³	#11719949	1	2	yes	—
86	C₂³	C₂³×D₅₃	1	1	yes	characteristic
87	Q₈	—	4	4	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	106	6	no	—
93	C₂×C₄	—	106	6	no	—
94	C₂×C₄	—	106	6	no	—
95	C₂×C₄	—	106	6	no	—
96	C₂×C₄	—	212	4	no	—
97	C₂²	—	2	6	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	4	3	no	—
100	C₂²	C₂×#11719949	1	1	yes	characteristic
101	C₂²	#13133262	1	3	yes	—
102	C₂²	#13133264	1	3	yes	—
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	212	2	no	—
106	C₄	—	212	6	no	—
107	C₂	—	4	2	no	—
108	C₂	#16811222	1	3	yes	—
109	C₂	#16811366	1	1	yes	characteristic
110	C₂	#16811373	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810565

C₂

yes

maximal, complemented

#16810548

C₂

yes

maximal

#16810542

C₂

yes

maximal, complemented

#16810495

C₂

yes

maximal

C₅₃×#393

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#2835

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#539

yes

—

C₂×C₅₃

H(3,𝔽₄)

yes

characteristic

C₂×C₅₃

#563

yes

—

—

—

—

D₅₃

yes

characteristic

#385

—

C₅₃

#2835

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

—

—

—

C₂×D₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

C₂³

#11719949

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#11719949

yes

characteristic

101

C₂²

#13133262

yes

—

102

C₂²

#13133264

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

212

—

106

C₄

—

212

—

107

C₂

—

108

C₂

#16811222

yes

—

109

C₂

#16811366

yes

characteristic

110

C₂

#16811373

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#393	C₂
hypercenter	#393	D₅₃
hypocenter	C₅₃	#2835
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#393

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #16810565	C₂	semidirect
3. #16810542	C₂	semidirect
61. C₅₃	#2835	semidirect

H ≅ G/N

type

1. #16810565

C₂

semidirect

3. #16810542

C₂

semidirect

61. C₅₃

#2835

semidirect

Subgroups — #20282906

Conjugacy classes of subgroups

Complements