Group #20272860 — subgroups

Subgroups — #20272860

Order 6784.

122 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 25 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	#16810416	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#453	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	1	no	—
5	#13133132	—	2	1	no	—
6	H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic, complemented
7	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
8	#127×C₅₃	C₂²	1	1	yes	characteristic
9	#104×C₅₃	—	2	1	no	—
10	#100×C₅₃	C₄	1	1	yes	characteristic, complemented
11	#41×C₅₃	—	2	1	no	—
12	#41×C₅₃	—	4	1	no	—
13	C₂×D₄×C₅₃	—	4	1	no	—
14	C₂×D₄×C₅₃	—	4	1	no	—
15	C₂×D₄×C₅₃	—	4	1	no	—
16	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
18	SD₁₆×C₅₃	—	4	1	no	—
19	D₈×C₅₃	—	4	1	no	—
20	M₁₆×C₅₃	—	2	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂²⋊Dic₅₃	—	4	1	no	—
23	C₂×#88322	—	4	1	no	—
24	C₂³×C₅₃	—	2	1	no	—
25	C₂³×C₅₃	—	2	1	no	—
26	C₂³×C₅₃	—	4	1	no	—
27	Q₈×C₅₃	—	2	1	no	—
28	D₄×C₅₃	—	2	1	no	—
29	D₄×C₅₃	—	4	1	no	—
30	D₄×C₅₃	—	4	1	no	—
31	D₄×C₅₃	—	4	2	no	—
32	D₄×C₅₃	—	4	1	no	—
33	D₄×C₅₃	—	8	1	no	—
34	D₄×C₅₃	—	8	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₅₃	—	8	1	no	—
40	C₈×C₅₃	—	4	1	no	—
41	#88322	—	8	1	no	—
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	—	4	1	no	—
44	C₂²×C₅₃	—	4	1	no	—
45	C₂²×C₅₃	—	8	1	no	—
46	C₂²×C₅₃	—	8	1	no	—
47	C₂²×C₅₃	—	16	1	no	—
48	C₂²×C₅₃	#99	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	4	1	no	—
53	Dic₅₃	—	16	1	no	—
54	#1402	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	8	1	no	—
57	C₂×C₅₃	—	8	1	no	—
58	C₂×C₅₃	—	8	1	no	—
59	C₂×C₅₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
60	#453	D₅₃	1	1	yes	characteristic
61	#353	—	53	1	no	—
62	#331	—	53	1	no	—
63	C₅₃	#1402	1	1	yes	characteristic, complemented, Sylow
64	H(5,𝔽₂)	Dic₅₃	1	1	yes	characteristic, complemented
65	AGL(1,ℤ/8)	—	2	1	no	—
66	#127	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	106	1	no	—
68	#104	—	2	1	no	—
69	#100	Dic₅₃	1	1	yes	characteristic, complemented
70	#99	—	106	1	no	—
71	#41	—	2	1	no	—
72	#41	—	4	1	no	—
73	C₂×D₄	—	4	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
78	SD₁₆	—	4	1	no	—
79	D₈	—	4	1	no	—
80	M₁₆	—	2	1	no	—
81	C₂×C₈	—	212	1	no	—
82	C₂²⋊C₄	—	212	1	no	—
83	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
84	C₂³	—	2	1	no	—
85	C₂³	—	2	1	no	—
86	C₂³	—	4	1	no	—
87	Q₈	—	2	1	no	—
88	D₄	—	2	1	no	—
89	D₄	—	4	1	no	—
90	D₄	—	4	1	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	1	no	—
93	D₄	—	8	1	no	—
94	D₄	—	8	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	424	1	no	—
99	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	424	1	no	—
102	C₂²	—	4	1	no	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	1	no	—
105	C₂²	—	8	1	no	—
106	C₂²	—	8	1	no	—
107	C₂²	—	16	1	no	—
108	C₂²	#13133132	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	1	no	—
113	C₄	—	848	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	8	1	no	—
116	C₂	—	8	1	no	—
117	C₂	—	8	1	no	—
118	C₂	#16810410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810416

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

C₅₃×#453

C₂

yes

maximal, characteristic

C₄⋊#88322

—

#13133132

—

H(5,𝔽₂)×C₅₃

C₄

yes

characteristic, complemented

AGL(1,ℤ/8)×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

#104×C₅₃

—

#100×C₅₃

C₄

yes

characteristic, complemented

#41×C₅₃

—

#41×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

SD₁₆×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#99

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1402

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

AΣL(1,𝔽₁₆)

yes

characteristic

#453

D₅₃

yes

characteristic

#353

—

#331

—

C₅₃

#1402

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

Dic₅₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

#127

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#104

—

#100

Dic₅₃

yes

characteristic, complemented

#99

—

106

—

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

characteristic

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

212

—

C₂²⋊C₄

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

100

C₈

—

101

C₈

—

424

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#13133132

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

848

—

114

C₂

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

#16810410

yes

characteristic

	subgroup	G/H
center	C₂	#16810410
commutator	C₂×D₄×C₅₃	C₂×C₄
Frattini	#127	C₂×D₅₃
Fitting	C₅₃×#453	C₂
socle	C₂×C₅₃	AΣL(1,𝔽₁₆)
hypercenter	#453	D₅₃
hypocenter	C₅₃	#1402
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #453

subgroup

G/H

center

C₂

#16810410

commutator

C₂×D₄×C₅₃

C₂×C₄

Frattini

#127

C₂×D₅₃

Fitting

C₅₃×#453

C₂

socle

C₂×C₅₃

AΣL(1,𝔽₁₆)

D₅₃

C₅₃

G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #453

N	H ≅ G/N	type
1. #16810416	C₂	semidirect
2. #16810367	C₂	semidirect
6. H(5,𝔽₂)×C₅₃	C₄	semidirect
10. #100×C₅₃	C₄	semidirect
63. C₅₃	#1402	semidirect
64. H(5,𝔽₂)	Dic₅₃	semidirect
69. #100	Dic₅₃	semidirect

H ≅ G/N

type

1. #16810416

C₂

semidirect

2. #16810367

C₂

semidirect

6. H(5,𝔽₂)×C₅₃

C₄

semidirect

10. #100×C₅₃

C₄

semidirect

63. C₅₃

#1402

semidirect

64. H(5,𝔽₂)

Dic₅₃

semidirect

69. #100

Dic₅₃

semidirect

Subgroups — #20272860

Conjugacy classes of subgroups

Complements