Group #20282783 — subgroups

Subgroups — #20282783

Order 6784.

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

normal subgroups only

This group has 380 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	C₂×#13133209	C₂	1	2	yes	maximal
2	#16810563	C₂	1	4	yes	maximal
3	#16810555	C₂	1	4	yes	maximal
4	#16810553	C₂	1	4	yes	maximal
5	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	8	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
10	C₂×#11719919	C₂²	1	4	yes	—
11	C₂×#11719919	C₂²	1	4	yes	—
12	#13133124	C₂²	1	8	yes	—
13	#13133124	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	C₂²	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂³	1	4	yes	—
21	C₂²×Dic₅₃	C₂³	1	4	yes	—
22	C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₄²×C₅₃	D₄	1	4	yes	—
24	C₄⋊Dic₅₃	—	2	4	no	—
25	C₄⋊Dic₅₃	—	2	4	no	—
26	C₄⋊Dic₅₃	—	4	4	no	—
27	#11719919	—	2	8	no	—
28	#11719919	—	2	8	no	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	—	2	8	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
37	C₂×Dic₅₃	—	2	4	no	—
38	C₂×Dic₅₃	—	2	4	no	—
39	C₂×Dic₅₃	—	2	4	no	—
40	C₂×Dic₅₃	—	2	8	no	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
43	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
45	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
46	C₄×C₅₃	—	2	4	no	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	4	4	no	—
49	Dic₅₃	—	4	4	no	—
50	Dic₅₃	—	4	4	no	—
51	#2822	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#536	1	2	yes	—
53	C₂×C₅₃	#537	1	1	yes	characteristic
54	C₂×C₅₃	#563	1	2	yes	—
55	C₂×C₅₃	#564	1	2	yes	—
56	C₂×#125	—	53	2	no	—
57	#400	—	53	4	no	—
58	#398	—	53	4	no	—
59	#398	—	53	4	no	—
60	#382	D₅₃	1	1	yes	characteristic
61	C₅₃	#2822	1	1	yes	characteristic, complemented, Sylow
62	#125	—	106	8	no	—
63	C₂×C₄⋊C₄	—	53	4	no	—
64	C₂×C₄⋊C₄	—	53	4	no	—
65	C₂×C₄⋊C₄	—	53	2	no	—
66	C₂×C₄⋊C₄	—	53	2	no	—
67	C₂×C₄⋊C₄	—	53	4	no	—
68	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
69	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
70	#95	—	53	8	no	—
71	#95	—	53	4	no	—
72	#95	C₂×D₅₃	1	4	yes	—
73	C₂²×C₄	—	53	4	no	—
74	C₂²×C₄	—	53	4	no	—
75	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
77	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	106	8	no	—
80	C₄⋊C₄	—	106	8	no	—
81	C₄⋊C₄	—	106	4	no	—
82	C₄⋊C₄	—	106	4	no	—
83	C₄⋊C₄	—	212	4	no	—
84	C₄²	D₂₁₂	1	4	yes	—
85	C₂³	C₂³×D₅₃	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	106	4	no	—
92	C₂×C₄	—	106	4	no	—
93	C₂×C₄	—	106	4	no	—
94	C₂×C₄	—	106	8	no	—
95	C₂×C₄	—	106	4	no	—
96	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
97	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
98	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
99	C₂²	#13133262	1	1	yes	characteristic
100	C₂²	#13133264	1	4	yes	—
101	C₂²	#13133265	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	—	212	4	no	—
106	C₄	—	212	4	no	—
107	C₂	#16811199	1	2	yes	—
108	C₂	#16811204	1	1	yes	characteristic
109	C₂	#16811373	1	2	yes	—
110	C₂	C₅₃⋊#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal

#16810563

C₂

yes

maximal

#16810555

C₂

yes

maximal

#16810553

C₂

yes

maximal

C₅₃×#382

C₂

yes

maximal, characteristic

#13133209

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#2822

—

maximal, Sylow

C₂×C₅₃

#536

yes

—

C₂×C₅₃

#537

yes

characteristic

C₂×C₅₃

#563

yes

—

C₂×C₅₃

#564

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2822

yes

characteristic, complemented, Sylow

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

—

C₂×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

D₂₁₂

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂²

C₂²×D₂₁₂

yes

characteristic

C₂²

#13133262

yes

characteristic

100

C₂²

#13133264

yes

—

101

C₂²

#13133265

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

212

—

106

C₄

—

212

—

107

C₂

#16811199

yes

—

108

C₂

#16811204

yes

characteristic

109

C₂

#16811373

yes

—

110

C₂

C₅₃⋊#564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#382	C₂
hypercenter	#382	D₅₃
hypocenter	C₅₃	#2822
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#382

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
61. C₅₃	#2822	semidirect

H ≅ G/N

type

61. C₅₃

#2822

semidirect

Subgroups — #20282783

Conjugacy classes of subgroups

Complements