Group #20276886 — subgroups

Subgroups — #20276886

Order 6784.

126 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810447	C₂	1	2	yes	maximal, complemented
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810433	C₂	1	2	yes	maximal
5	C₅₃×#443	C₂	1	1	yes	maximal, characteristic
6	#13133217	—	2	1	no	—
7	#13133157	C₂²	1	2	yes	—
8	#13133149	—	2	1	no	—
9	#13133149	C₂²	1	2	yes	—
10	#135×C₅₃	C₂²	1	1	yes	characteristic
11	#131×C₅₃	—	2	1	no	—
12	#108×C₅₃	C₂²	1	1	yes	characteristic
13	#104×C₅₃	—	2	1	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#41×C₅₃	—	2	1	no	—
16	C₂×Dic₁₀₆	—	4	1	no	—
17	Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
18	SD₁₆×C₅₃	—	2	1	no	—
19	D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
20	M₁₆×C₅₃	—	2	1	no	—
21	M₁₆×C₅₃	—	4	1	no	—
22	C₂×C₈×C₅₃	—	4	1	no	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	4	1	no	—
28	#11719908	—	2	2	no	—
29	Q₈×C₅₃	—	2	1	no	—
30	D₄×C₅₃	—	2	1	no	—
31	D₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	2	no	—
36	Dic₁₀₆	—	8	2	no	—
37	C₈×C₅₃	—	4	1	no	—
38	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
39	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₅₃	#41	1	1	yes	characteristic
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	4	1	no	—
46	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₅₃	—	8	2	no	—
49	#2230	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	8	1	no	—
52	C₂×C₅₃	#476	1	1	yes	characteristic
53	#498	—	53	1	no	—
54	#443	D₅₃	1	1	yes	characteristic
55	#365	—	53	2	no	—
56	#364	—	53	1	no	—
57	#360	—	53	2	no	—
58	C₅₃	#2230	1	1	yes	characteristic, complemented, Sylow
59	#135	C₂×D₅₃	1	1	yes	characteristic
60	#131	—	2	1	no	—
61	C₄⋊Q₈	—	106	1	no	—
62	M₃₂	—	53	2	no	—
63	#108	C₂×D₅₃	1	1	yes	characteristic
64	#106	—	53	2	no	—
65	#106	—	106	1	no	—
66	#104	—	2	1	no	—
67	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
68	#41	—	2	1	no	—
69	C₂×Q₈	—	212	1	no	—
70	Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
71	SD₁₆	—	2	1	no	—
72	D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
73	M₁₆	—	2	1	no	—
74	M₁₆	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
78	C₄⋊C₄	—	106	2	no	—
79	C₄⋊C₄	—	212	1	no	—
80	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₁₆	—	106	2	no	—
82	Q₈	—	2	1	no	—
83	Q₈	—	424	2	no	—
84	D₄	—	2	1	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	212	2	no	—
89	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	#11719923	1	1	yes	characteristic
92	C₈	#11719923	1	1	yes	characteristic
93	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
94	C₈	#11719945	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#13133242	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	4	1	no	—
99	C₄	—	424	2	no	—
100	C₄	#13133196	1	1	yes	characteristic
101	C₄	C₂×#11719923	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	8	1	no	—
104	C₂	#16810913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811021

C₂

yes

maximal, characteristic, complemented

#16810447

C₂

yes

maximal, complemented

#16810444

C₂

yes

maximal, characteristic, complemented

#16810433

C₂

yes

maximal

C₅₃×#443

C₂

yes

maximal, characteristic

#13133217

—

#13133157

C₂²

yes

—

#13133149

—

#13133149

C₂²

yes

—

#135×C₅₃

C₂²

yes

characteristic

#131×C₅₃

—

#108×C₅₃

C₂²

yes

characteristic

#104×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

Q₈

yes

characteristic, complemented

SD₁₆×C₅₃

—

D₈×C₅₃

Q₈

yes

characteristic, complemented

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2230

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#2230

yes

characteristic, complemented, Sylow

#135

C₂×D₅₃

yes

characteristic

#131

—

C₄⋊Q₈

—

106

—

M₃₂

—

#108

C₂×D₅₃

yes

characteristic

#106

—

#106

—

106

—

#104

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

C₂×Q₈

—

212

—

Q₁₆

Dic₁₀₆

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₀₆

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

424

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

#11719945

yes

characteristic

C₂²

—

C₂²

#13133242

yes

characteristic

C₄

—

C₄

—

C₄

—

424

—

100

C₄

#13133196

yes

characteristic

101

C₄

C₂×#11719923

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#16810913

yes

characteristic

	subgroup	G/H
center	C₂	#16810913
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#443	C₂
socle	C₂×C₅₃	#476
hypercenter	#443	D₅₃
hypocenter	C₅₃	#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16810913

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#443

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #16811021	C₂	semidirect
2. #16810447	C₂	semidirect
3. #16810444	C₂	semidirect
17. Q₁₆×C₅₃	Q₈	semidirect
19. D₈×C₅₃	Q₈	semidirect
58. C₅₃	#2230	semidirect
70. Q₁₆	Dic₁₀₆	semidirect
72. D₈	Dic₁₀₆	semidirect

H ≅ G/N

type

1. #16811021

C₂

semidirect

2. #16810447

C₂

semidirect

3. #16810444

C₂

semidirect

17. Q₁₆×C₅₃

Q₈

semidirect

19. D₈×C₅₃

Q₈

semidirect

58. C₅₃

#2230

semidirect

70. Q₁₆

Dic₁₀₆

semidirect

72. D₈

Dic₁₀₆

semidirect

Subgroups — #20276886

Conjugacy classes of subgroups

Complements