Group #20256529 — subgroups

Subgroups — #20256529

Order 6016.

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	#15689928	C₂	1	2	yes	maximal, complemented
2	#15689561	C₂	1	6	yes	maximal, complemented
3	#15689555	C₂	1	3	yes	maximal, complemented
4	C₂×D₇₅₂	C₂	1	3	yes	maximal, complemented
5	C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094070	—	2	2	no	—
7	#13094052	—	2	6	no	—
8	#13094046	C₂²	1	6	yes	complemented
9	C₂×D₃₇₆	C₂²	1	6	yes	complemented
10	Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
11	#13093969	C₂²	1	6	yes	complemented
12	D₇₅₂	C₂²	1	3	yes	complemented
13	D₇₅₂	C₂²	1	6	yes	complemented
14	#131×C₄₇	C₂²	1	1	yes	characteristic
15	M₃₂×C₄₇	C₂²	1	3	yes	complemented
16	C₂×C₁₆×C₄₇	C₂²	1	3	yes	complemented
17	#41×C₄₇	D₄	1	1	yes	characteristic
18	#10625354	—	4	2	no	—
19	D₄×D₄₇	—	4	6	no	—
20	#10625350	—	2	6	no	—
21	C₂×D₁₈₈	—	2	6	no	—
22	M₁₆×C₄₇	D₄	1	3	yes	—
23	C₂×C₈×C₄₇	C₂³	1	3	yes	—
24	Dic₁₈₈	C₂³	1	2	yes	—
25	D₃₇₆	—	2	6	no	—
26	D₃₇₆	C₂³	1	6	yes	—
27	#10625325	—	2	6	no	—
28	C₁₆×C₄₇	C₂³	1	1	yes	characteristic
29	C₁₆×C₄₇	C₂³	1	3	yes	—
30	C₂²×D₄₇	—	4	6	no	—
31	Q₈×C₄₇	D₈	1	1	yes	characteristic
32	D₄×C₄₇	D₈	1	3	yes	—
33	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂²⋊D₄₇	—	4	6	no	—
35	D₁₈₈	—	2	6	no	—
36	D₁₈₈	—	4	6	no	—
37	C₄×D₄₇	—	4	6	no	—
38	Dic₉₄	—	2	2	no	—
39	C₈×C₄₇	C₂×D₄	1	3	yes	—
40	C₈×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄₇	C₂×D₈	1	3	yes	—
42	C₂×D₄₇	—	4	6	no	—
43	C₂×D₄₇	—	16	6	no	—
44	C₄×C₄₇	C₂×D₈	1	3	yes	—
45	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
46	Dic₄₇	—	4	2	no	—
47	#3415	—	47	1	no	maximal, Sylow
48	C₂×C₄₇	—	2	3	no	—
49	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
50	D₄₇	—	8	6	no	—
51	#576	—	47	2	no	—
52	#509	—	47	6	no	—
53	#508	—	47	3	no	—
54	C₂×D₁₆	—	47	3	no	—
55	#504	D₄₇	1	1	yes	characteristic, complemented
56	C₄₇	#3415	1	1	yes	characteristic, complemented, Sylow
57	H(5,𝔽₂)	—	94	2	no	—
58	AGL(1,ℤ/8)	—	94	6	no	—
59	#135	—	47	6	no	—
60	C₂×D₈	—	47	6	no	—
61	#131	C₂×D₄₇	1	1	yes	characteristic
62	Q₃₂	—	47	1	no	—
63	SD₃₂	—	47	6	no	—
64	D₁₆	—	47	3	no	—
65	D₁₆	—	47	6	no	—
66	M₃₂	C₂×D₄₇	1	3	yes	complemented
67	C₂×C₁₆	C₂×D₄₇	1	3	yes	complemented
68	#41	—	94	6	no	—
69	#41	—	188	2	no	—
70	#41	D₁₈₈	1	1	yes	characteristic
71	C₂×D₄	—	94	6	no	—
72	C₂×D₄	—	188	6	no	—
73	Q₁₆	—	47	2	no	—
74	SD₁₆	—	94	6	no	—
75	D₈	—	47	6	no	—
76	D₈	—	94	6	no	—
77	M₁₆	D₁₈₈	1	3	yes	—
78	C₂×C₈	C₂²×D₄₇	1	3	yes	—
79	C₁₆	C₂²×D₄₇	1	1	yes	characteristic
80	C₁₆	C₂²×D₄₇	1	3	yes	—
81	C₂³	—	188	6	no	—
82	Q₈	—	94	2	no	—
83	Q₈	D₃₇₆	1	1	yes	characteristic
84	D₄	—	94	6	no	—
85	D₄	—	188	6	no	—
86	D₄	—	188	6	no	—
87	D₄	D₃₇₆	1	3	yes	—
88	C₂×C₄	—	188	6	no	—
89	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
90	C₈	C₂×D₁₈₈	1	3	yes	—
91	C₈	C₂³×D₄₇	1	1	yes	characteristic
92	C₂²	—	188	6	no	—
93	C₂²	—	752	6	no	—
94	C₂²	C₂×D₃₇₆	1	3	yes	—
95	C₄	—	188	2	no	—
96	C₄	C₂×D₃₇₆	1	3	yes	—
97	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
98	C₂	—	2	3	no	—
99	C₂	—	376	6	no	—
100	C₂	C₂²×D₃₇₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689928

C₂

yes

maximal, complemented

#15689561

C₂

yes

maximal, complemented

#15689555

C₂

yes

maximal, complemented

C₂×D₇₅₂

C₂

yes

maximal, complemented

C₄₇×#504

C₂

yes

maximal, characteristic, complemented

#13094070

—

#13094052

—

#13094046

C₂²

yes

complemented

C₂×D₃₇₆

C₂²

yes

complemented

Dic₃₇₆

C₂²

yes

characteristic, complemented

#13093969

C₂²

yes

complemented

D₇₅₂

C₂²

yes

complemented

D₇₅₂

C₂²

yes

complemented

#131×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

C₂²

yes

complemented

C₂×C₁₆×C₄₇

C₂²

yes

complemented

#41×C₄₇

D₄

yes

characteristic

#10625354

—

D₄×D₄₇

—

#10625350

—

C₂×D₁₈₈

—

M₁₆×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

Dic₁₈₈

C₂³

yes

—

D₃₇₆

—

D₃₇₆

C₂³

yes

—

#10625325

—

C₁₆×C₄₇

C₂³

yes

characteristic

C₁₆×C₄₇

C₂³

yes

—

C₂²×D₄₇

—

Q₈×C₄₇

D₈

yes

characteristic

D₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂²⋊D₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

—

C₈×C₄₇

C₂⁴

yes

characteristic

C₂²×C₄₇

C₂×D₈

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

#3415

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₈

yes

characteristic

D₄₇

—

#576

—

#509

—

#508

—

C₂×D₁₆

—

#504

D₄₇

yes

characteristic, complemented

C₄₇

#3415

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

AGL(1,ℤ/8)

—

#135

—

C₂×D₈

—

#131

C₂×D₄₇

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

C₂×D₄₇

yes

complemented

C₂×C₁₆

C₂×D₄₇

yes

complemented

#41

—

#41

—

188

—

#41

D₁₈₈

yes

characteristic

C₂×D₄

—

C₂×D₄

—

188

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

D₁₈₈

yes

—

C₂×C₈

C₂²×D₄₇

yes

—

C₁₆

C₂²×D₄₇

yes

characteristic

C₁₆

C₂²×D₄₇

yes

—

C₂³

—

188

—

Q₈

—

Q₈

D₃₇₆

yes

characteristic

D₄

—

D₄

—

188

—

D₄

—

188

—

D₄

D₃₇₆

yes

—

C₂×C₄

—

188

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₈

C₂×D₁₈₈

yes

—

C₈

C₂³×D₄₇

yes

characteristic

C₂²

—

188

—

C₂²

—

752

—

C₂²

C₂×D₃₇₆

yes

—

C₄

—

188

—

C₄

C₂×D₃₇₆

yes

—

C₄

C₂²×D₁₈₈

yes

characteristic

C₂

—

C₂

—

376

—

100

C₂

C₂²×D₃₇₆

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₃₇₆
commutator	C₈×C₄₇	C₂⁴
Frattini	C₈	C₂³×D₄₇
Fitting	C₄₇×#504	C₂
socle	C₂×C₄₇	C₂²×D₈
hypercenter	#504	D₄₇
hypocenter	C₄₇	#3415
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

subgroup

G/H

center

C₂

C₂²×D₃₇₆

commutator

C₈×C₄₇

C₂⁴

Frattini

C₈

C₂³×D₄₇

Fitting

C₄₇×#504

C₂

socle

C₂×C₄₇

C₂²×D₈

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

N	H ≅ G/N	type
1. #15689928	C₂	semidirect
2. #15689561	C₂	semidirect
3. #15689555	C₂	semidirect
4. C₂×D₇₅₂	C₂	semidirect
5. C₄₇×#504	C₂	semidirect
8. #13094046	C₂²	semidirect
9. C₂×D₃₇₆	C₂²	semidirect
10. Dic₃₇₆	C₂²	semidirect
11. #13093969	C₂²	semidirect
12. D₇₅₂	C₂²	semidirect
13. D₇₅₂	C₂²	semidirect
15. M₃₂×C₄₇	C₂²	semidirect
16. C₂×C₁₆×C₄₇	C₂²	semidirect
55. #504	D₄₇	semidirect
56. C₄₇	#3415	semidirect
66. M₃₂	C₂×D₄₇	semidirect
67. C₂×C₁₆	C₂×D₄₇	semidirect

H ≅ G/N

type

1. #15689928

C₂

semidirect

2. #15689561

C₂

semidirect

3. #15689555

C₂

semidirect

4. C₂×D₇₅₂

C₂

semidirect

5. C₄₇×#504

C₂

semidirect

8. #13094046

C₂²

semidirect

9. C₂×D₃₇₆

C₂²

semidirect

10. Dic₃₇₆

C₂²

semidirect

11. #13093969

C₂²

semidirect

12. D₇₅₂

C₂²

semidirect

13. D₇₅₂

C₂²

semidirect

15. M₃₂×C₄₇

C₂²

semidirect

16. C₂×C₁₆×C₄₇

C₂²

semidirect

55. #504

D₄₇

semidirect

56. C₄₇

#3415

semidirect

66. M₃₂

C₂×D₄₇

semidirect

67. C₂×C₁₆

C₂×D₄₇

semidirect

Subgroups — #20256529

Conjugacy classes of subgroups

Complements