Group #20287856 — subgroups

Subgroups — #20287856

Order 6784.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16811428	C₂	1	2	yes	maximal, complemented
2	#16811061	C₂	1	6	yes	maximal, complemented
3	#16811055	C₂	1	3	yes	maximal, complemented
4	C₂×D₈₄₈	C₂	1	3	yes	maximal, complemented
5	C₅₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133263	—	2	2	no	—
7	#13133245	—	2	6	no	—
8	#13133239	C₂²	1	6	yes	complemented
9	C₂×D₄₂₄	C₂²	1	6	yes	complemented
10	Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
11	#13133162	C₂²	1	6	yes	complemented
12	D₈₄₈	C₂²	1	3	yes	complemented
13	D₈₄₈	C₂²	1	6	yes	complemented
14	#131×C₅₃	C₂²	1	1	yes	characteristic
15	M₃₂×C₅₃	C₂²	1	3	yes	complemented
16	C₂×C₁₆×C₅₃	C₂²	1	3	yes	complemented
17	#41×C₅₃	D₄	1	1	yes	characteristic
18	#11719949	—	4	2	no	—
19	D₄×D₅₃	—	4	6	no	—
20	#11719945	—	2	6	no	—
21	C₂×D₂₁₂	—	2	6	no	—
22	M₁₆×C₅₃	D₄	1	3	yes	—
23	C₂×C₈×C₅₃	C₂³	1	3	yes	—
24	Dic₂₁₂	C₂³	1	2	yes	—
25	D₄₂₄	—	2	6	no	—
26	D₄₂₄	C₂³	1	6	yes	—
27	#11719913	—	2	6	no	—
28	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
29	C₁₆×C₅₃	C₂³	1	3	yes	—
30	C₂²×D₅₃	—	4	6	no	—
31	Q₈×C₅₃	D₈	1	1	yes	characteristic
32	D₄×C₅₃	D₈	1	3	yes	—
33	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
34	C₂²⋊D₅₃	—	4	6	no	—
35	D₂₁₂	—	2	6	no	—
36	D₂₁₂	—	4	6	no	—
37	C₄×D₅₃	—	4	6	no	—
38	Dic₁₀₆	—	2	2	no	—
39	C₈×C₅₃	C₂×D₄	1	3	yes	—
40	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂²×C₅₃	C₂×D₈	1	3	yes	—
42	C₂×D₅₃	—	4	6	no	—
43	C₂×D₅₃	—	16	6	no	—
44	C₄×C₅₃	C₂×D₈	1	3	yes	—
45	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
46	Dic₅₃	—	4	2	no	—
47	#3415	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	3	no	—
49	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
50	D₅₃	—	8	6	no	—
51	#576	—	53	2	no	—
52	#509	—	53	6	no	—
53	#508	—	53	3	no	—
54	C₂×D₁₆	—	53	3	no	—
55	#504	D₅₃	1	1	yes	characteristic, complemented
56	C₅₃	#3415	1	1	yes	characteristic, complemented, Sylow
57	H(5,𝔽₂)	—	106	2	no	—
58	AGL(1,ℤ/8)	—	106	6	no	—
59	#135	—	53	6	no	—
60	C₂×D₈	—	53	6	no	—
61	#131	C₂×D₅₃	1	1	yes	characteristic
62	Q₃₂	—	53	1	no	—
63	SD₃₂	—	53	6	no	—
64	D₁₆	—	53	3	no	—
65	D₁₆	—	53	6	no	—
66	M₃₂	C₂×D₅₃	1	3	yes	complemented
67	C₂×C₁₆	C₂×D₅₃	1	3	yes	complemented
68	#41	—	106	6	no	—
69	#41	—	212	2	no	—
70	#41	D₂₁₂	1	1	yes	characteristic
71	C₂×D₄	—	106	6	no	—
72	C₂×D₄	—	212	6	no	—
73	Q₁₆	—	53	2	no	—
74	SD₁₆	—	106	6	no	—
75	D₈	—	53	6	no	—
76	D₈	—	106	6	no	—
77	M₁₆	D₂₁₂	1	3	yes	—
78	C₂×C₈	C₂²×D₅₃	1	3	yes	—
79	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
80	C₁₆	C₂²×D₅₃	1	3	yes	—
81	C₂³	—	212	6	no	—
82	Q₈	—	106	2	no	—
83	Q₈	D₄₂₄	1	1	yes	characteristic
84	D₄	—	106	6	no	—
85	D₄	—	212	6	no	—
86	D₄	—	212	6	no	—
87	D₄	D₄₂₄	1	3	yes	—
88	C₂×C₄	—	212	6	no	—
89	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
90	C₈	C₂×D₂₁₂	1	3	yes	—
91	C₈	C₂³×D₅₃	1	1	yes	characteristic
92	C₂²	—	848	6	no	—
93	C₂²	—	212	6	no	—
94	C₂²	C₂×D₄₂₄	1	3	yes	—
95	C₄	—	212	2	no	—
96	C₄	C₂×D₄₂₄	1	3	yes	—
97	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
98	C₂	—	2	3	no	—
99	C₂	—	424	6	no	—
100	C₂	C₂²×D₄₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811428

C₂

yes

maximal, complemented

#16811061

C₂

yes

maximal, complemented

#16811055

C₂

yes

maximal, complemented

C₂×D₈₄₈

C₂

yes

maximal, complemented

C₅₃×#504

C₂

yes

maximal, characteristic, complemented

#13133263

—

#13133245

—

#13133239

C₂²

yes

complemented

C₂×D₄₂₄

C₂²

yes

complemented

Dic₄₂₄

C₂²

yes

characteristic, complemented

#13133162

C₂²

yes

complemented

D₈₄₈

C₂²

yes

complemented

D₈₄₈

C₂²

yes

complemented

#131×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

complemented

C₂×C₁₆×C₅₃

C₂²

yes

complemented

#41×C₅₃

D₄

yes

characteristic

#11719949

—

D₄×D₅₃

—

#11719945

—

C₂×D₂₁₂

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

Dic₂₁₂

C₂³

yes

—

D₄₂₄

—

D₄₂₄

C₂³

yes

—

#11719913

—

C₁₆×C₅₃

C₂³

yes

characteristic

C₁₆×C₅₃

C₂³

yes

—

C₂²×D₅₃

—

Q₈×C₅₃

D₈

yes

characteristic

D₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂⁴

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

—

#3415

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×D₈

yes

characteristic

D₅₃

—

#576

—

#509

—

#508

—

C₂×D₁₆

—

#504

D₅₃

yes

characteristic, complemented

C₅₃

#3415

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

106

—

AGL(1,ℤ/8)

—

106

—

#135

—

C₂×D₈

—

#131

C₂×D₅₃

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

C₂×D₅₃

yes

complemented

C₂×C₁₆

C₂×D₅₃

yes

complemented

#41

—

106

—

#41

—

212

—

#41

D₂₁₂

yes

characteristic

C₂×D₄

—

106

—

C₂×D₄

—

212

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

D₈

—

106

—

M₁₆

D₂₁₂

yes

—

C₂×C₈

C₂²×D₅₃

yes

—

C₁₆

C₂²×D₅₃

yes

characteristic

C₁₆

C₂²×D₅₃

yes

—

C₂³

—

212

—

Q₈

—

106

—

Q₈

D₄₂₄

yes

characteristic

D₄

—

106

—

D₄

—

212

—

D₄

—

212

—

D₄

D₄₂₄

yes

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₈

C₂×D₂₁₂

yes

—

C₈

C₂³×D₅₃

yes

characteristic

C₂²

—

848

—

C₂²

—

212

—

C₂²

C₂×D₄₂₄

yes

—

C₄

—

212

—

C₄

C₂×D₄₂₄

yes

—

C₄

C₂²×D₂₁₂

yes

characteristic

C₂

—

C₂

—

424

—

100

C₂

C₂²×D₄₂₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₂₄
commutator	C₈×C₅₃	C₂⁴
Frattini	C₈	C₂³×D₅₃
Fitting	C₅₃×#504	C₂
socle	C₂×C₅₃	C₂²×D₈
hypercenter	#504	D₅₃
hypocenter	C₅₃	#3415
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

subgroup

G/H

center

C₂

C₂²×D₄₂₄

commutator

C₈×C₅₃

C₂⁴

Frattini

C₈

C₂³×D₅₃

Fitting

C₅₃×#504

C₂

socle

C₂×C₅₃

C₂²×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

N	H ≅ G/N	type
1. #16811428	C₂	semidirect
2. #16811061	C₂	semidirect
3. #16811055	C₂	semidirect
4. C₂×D₈₄₈	C₂	semidirect
5. C₅₃×#504	C₂	semidirect
8. #13133239	C₂²	semidirect
9. C₂×D₄₂₄	C₂²	semidirect
10. Dic₄₂₄	C₂²	semidirect
11. #13133162	C₂²	semidirect
12. D₈₄₈	C₂²	semidirect
13. D₈₄₈	C₂²	semidirect
15. M₃₂×C₅₃	C₂²	semidirect
16. C₂×C₁₆×C₅₃	C₂²	semidirect
55. #504	D₅₃	semidirect
56. C₅₃	#3415	semidirect
66. M₃₂	C₂×D₅₃	semidirect
67. C₂×C₁₆	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16811428

C₂

semidirect

2. #16811061

C₂

semidirect

3. #16811055

C₂

semidirect

4. C₂×D₈₄₈

C₂

semidirect

5. C₅₃×#504

C₂

semidirect

8. #13133239

C₂²

semidirect

9. C₂×D₄₂₄

C₂²

semidirect

10. Dic₄₂₄

C₂²

semidirect

11. #13133162

C₂²

semidirect

12. D₈₄₈

C₂²

semidirect

13. D₈₄₈

C₂²

semidirect

15. M₃₂×C₅₃

C₂²

semidirect

16. C₂×C₁₆×C₅₃

C₂²

semidirect

55. #504

D₅₃

semidirect

56. C₅₃

#3415

semidirect

66. M₃₂

C₂×D₅₃

semidirect

67. C₂×C₁₆

C₂×D₅₃

semidirect

Subgroups — #20287856

Conjugacy classes of subgroups

Complements