Group #20280675 — subgroups

Subgroups — #20280675

Order 6784.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16810552	C₂	1	6	yes	maximal, complemented
2	#16810547	C₂	1	2	yes	maximal, complemented
3	#16810525	C₂	1	3	yes	maximal
4	#16810521	C₂	1	3	yes	maximal, complemented
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
8	C₂×#11719919	C₂²	1	6	yes	—
9	C₂×#11719921	C₂²	1	6	yes	—
10	C₂×#11719921	C₂²	1	6	yes	—
11	#13133124	C₂²	1	6	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
13	#95×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	C₂²	1	3	yes	—
15	C₂³×D₅₃	C₂³	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂³	1	6	yes	—
20	C₂×D₂₁₂	—	2	6	no	—
21	C₄²×C₅₃	—	2	6	no	—
22	#11719921	—	4	6	no	—
23	#11719921	—	4	6	no	—
24	C₄⋊Dic₅₃	—	2	6	no	—
25	#11719919	—	4	6	no	—
26	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₂²×D₅₃	—	2	6	no	—
28	C₂²×D₅₃	—	2	6	no	—
29	C₂²×D₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	—	2	3	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	2	3	no	—
35	C₂×C₄×C₅₃	—	2	3	no	—
36	C₂×C₄×C₅₃	#41	1	6	yes	—
37	C₂×Dic₅₃	—	2	6	no	—
38	C₂×Dic₅₃	—	2	6	no	—
39	C₂×Dic₅₃	—	2	6	no	—
40	D₂₁₂	—	4	4	no	—
41	C₂²×C₅₃	C₂×#41	1	3	yes	—
42	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
43	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
44	C₂×D₅₃	—	4	2	no	—
45	C₂×D₅₃	—	4	6	no	—
46	C₂×D₅₃	—	4	6	no	—
47	C₄×C₅₃	—	4	3	no	—
48	C₄×C₅₃	—	4	3	no	—
49	C₄×C₅₃	#126	1	4	yes	—
50	Dic₅₃	—	4	6	no	—
51	#2644	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
53	C₂×C₅₃	#540	1	3	yes	—
54	C₂×C₅₃	#542	1	3	yes	—
55	D₅₃	—	8	2	no	—
56	#397	—	53	6	no	—
57	#396	—	53	2	no	—
58	#390	—	53	3	no	—
59	#389	—	53	3	no	—
60	#376	D₅₃	1	1	yes	characteristic, complemented
61	C₅₃	#2644	1	1	yes	characteristic, complemented, Sylow
62	C₂²×D₄	—	53	1	no	—
63	C₂×C₄⋊C₄	—	53	3	no	—
64	C₂×C₄⋊C₄	—	53	6	no	—
65	C₂×C₂²⋊C₄	—	53	6	no	—
66	C₂×C₂²⋊C₄	—	53	6	no	—
67	C₂×C₄²	C₂×D₅₃	1	3	yes	—
68	#95	—	53	6	no	—
69	#95	C₂×D₅₃	1	1	yes	characteristic
70	#95	C₂×D₅₃	1	3	yes	—
71	C₂⁴	—	53	2	no	—
72	C₂×D₄	—	106	6	no	—
73	C₂²×C₄	—	53	6	no	—
74	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
75	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
77	C₄⋊C₄	—	106	6	no	—
78	C₄⋊C₄	—	212	6	no	—
79	C₂²⋊C₄	—	212	6	no	—
80	C₂²⋊C₄	—	212	6	no	—
81	C₄²	—	2	6	no	—
82	C₂³	—	106	6	no	—
83	C₂³	—	106	6	no	—
84	C₂³	—	106	2	no	—
85	C₂³	C₂³×D₅₃	1	1	yes	characteristic
86	D₄	—	212	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	106	6	no	—
94	C₂×C₄	—	106	6	no	—
95	C₂×C₄	—	106	6	no	—
96	C₂×C₄	#11719945	1	6	yes	—
97	C₂²	—	212	2	no	—
98	C₂²	—	212	6	no	—
99	C₂²	—	212	6	no	—
100	C₂²	C₂×#11719945	1	3	yes	—
101	C₂²	#13133263	1	3	yes	—
102	C₂²	#13133264	1	1	yes	characteristic
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	212	6	no	—
106	C₄	#13133211	1	4	yes	—
107	C₂	—	424	2	no	—
108	C₂	C₂×#13133211	1	1	yes	characteristic
109	C₂	#16811228	1	3	yes	—
110	C₂	#16811244	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810552

C₂

yes

maximal, complemented

#16810547

C₂

yes

maximal, complemented

#16810525

C₂

yes

maximal

#16810521

C₂

yes

maximal, complemented

C₅₃×#376

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₄²×C₅₃

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#126

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#542

yes

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2644

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

#95

C₂×D₅₃

yes

characteristic

#95

C₂×D₅₃

yes

—

C₂⁴

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#11719945

yes

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

—

212

—

100

C₂²

C₂×#11719945

yes

—

101

C₂²

#13133263

yes

—

102

C₂²

#13133264

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

212

—

106

C₄

#13133211

yes

—

107

C₂

—

424

—

108

C₂

C₂×#13133211

yes

characteristic

109

C₂

#16811228

yes

—

110

C₂

#16811244

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#376	C₂
hypercenter	#376	D₅₃
hypocenter	C₅₃	#2644
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#376

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
1. #16810552	C₂	semidirect
2. #16810547	C₂	semidirect
4. #16810521	C₂	semidirect
5. C₅₃×#376	C₂	semidirect
60. #376	D₅₃	semidirect
61. C₅₃	#2644	semidirect

H ≅ G/N

type

1. #16810552

C₂

semidirect

2. #16810547

C₂

semidirect

4. #16810521

C₂

semidirect

5. C₅₃×#376

C₂

semidirect

60. #376

D₅₃

semidirect

61. C₅₃

#2644

semidirect

Subgroups — #20280675

Conjugacy classes of subgroups

Complements