Group #44567093 — subgroups

Subgroups — #44567093

Order 32192.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#30632911	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632910	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632897	C₂	1	3	yes	maximal, complemented
4	#30632892	C₂	1	3	yes	maximal, complemented
5	#30632886	C₂	1	3	yes	maximal, complemented
6	C₂×#22548872	C₂	1	3	yes	maximal, complemented
7	#131×C₅₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#22548884	—	2	1	no	—
9	#22548883	C₂²	1	3	yes	—
10	#22548883	C₂²	1	3	yes	complemented
11	#22548882	—	2	3	no	—
12	C₂×Dic₁₀₀₆	C₂²	1	3	yes	complemented
13	Q₈×D₅₀₃	—	2	1	no	—
14	C₂×D₂₀₁₂	C₂²	1	3	yes	—
15	D₄×D₅₀₃	—	2	3	no	—
16	Dic₂₀₁₂	C₂²	1	3	yes	complemented
17	#22548872	C₂²	1	3	yes	complemented
18	#22548872	C₂²	1	6	yes	complemented
19	#22548872	C₂²	1	1	yes	characteristic, complemented
20	D₄₀₂₄	C₂²	1	3	yes	complemented
21	#41×C₅₀₃	C₂²	1	1	yes	characteristic
22	M₁₆×C₅₀₃	C₂²	1	3	yes	complemented
23	C₂×C₈×C₅₀₃	C₂²	1	3	yes	complemented
24	C₂²×D₅₀₃	—	2	3	no	—
25	Dic₁₀₀₆	—	2	3	no	—
26	Dic₁₀₀₆	C₂³	1	3	yes	—
27	Dic₁₀₀₆	C₂³	1	1	yes	characteristic
28	D₂₀₁₂	—	2	3	no	—
29	D₂₀₁₂	C₂³	1	1	yes	characteristic
30	D₂₀₁₂	C₂³	1	3	yes	—
31	C₂²⋊D₅₀₃	—	2	3	no	—
32	C₂²⋊D₅₀₃	—	2	3	no	—
33	C₂×Dic₅₀₃	—	2	3	no	—
34	C₄×D₅₀₃	—	2	3	no	—
35	C₄×D₅₀₃	—	2	3	no	—
36	Q₈×C₅₀₃	D₄	1	1	yes	characteristic
37	D₄×C₅₀₃	D₄	1	3	yes	—
38	C₂×C₄×C₅₀₃	C₂³	1	3	yes	—
39	C₈×C₅₀₃	C₂³	1	1	yes	characteristic
40	C₈×C₅₀₃	C₂³	1	3	yes	—
41	C₂²×C₅₀₃	C₂×D₄	1	3	yes	—
42	C₂×D₅₀₃	—	2	1	no	—
43	C₂×D₅₀₃	—	2	3	no	—
44	C₂×D₅₀₃	—	8	3	no	—
45	C₄×C₅₀₃	C₂×D₄	1	3	yes	—
46	C₄×C₅₀₃	C₂⁴	1	1	yes	characteristic
47	Dic₅₀₃	—	2	1	no	—
48	Dic₅₀₃	—	2	3	no	—
49	C₂×C₅₀₃	—	2	3	no	—
50	C₂×C₅₀₃	C₂²×D₄	1	1	yes	characteristic
51	D₅₀₃	—	4	1	no	—
52	D₅₀₃	—	4	3	no	—
53	C₅₀₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	503	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	503	1	no	—
56	H(5,𝔽₂)	—	503	1	no	—
57	#137	—	503	3	no	—
58	AGL(1,ℤ/8)	—	503	3	no	—
59	#135	—	503	3	no	—
60	C₂×SD₁₆	—	503	3	no	—
61	#131	D₅₀₃	1	1	yes	characteristic, complemented
62	#41	—	1006	1	no	—
63	#41	—	1006	3	no	—
64	#41	—	503	3	no	—
65	#41	—	503	3	no	—
66	#41	C₂×D₅₀₃	1	1	yes	characteristic
67	C₂×Q₈	—	1006	1	no	—
68	C₂×Q₈	—	503	3	no	—
69	C₂×D₄	—	1006	3	no	—
70	C₂×D₄	—	503	3	no	—
71	Q₁₆	—	503	3	no	—
72	SD₁₆	—	503	3	no	—
73	SD₁₆	—	503	6	no	—
74	SD₁₆	—	503	1	no	—
75	D₈	—	503	3	no	—
76	M₁₆	C₂×D₅₀₃	1	3	yes	complemented
77	C₂×C₈	C₂×D₅₀₃	1	3	yes	complemented
78	C₂³	—	1006	3	no	—
79	Q₈	—	1006	3	no	—
80	Q₈	—	503	3	no	—
81	Q₈	—	503	1	no	—
82	Q₈	D₂₀₁₂	1	1	yes	characteristic
83	D₄	—	1006	3	no	—
84	D₄	—	1006	3	no	—
85	D₄	—	1006	3	no	—
86	D₄	—	503	1	no	—
87	D₄	—	503	3	no	—
88	D₄	D₂₀₁₂	1	3	yes	—
89	C₂×C₄	—	1006	3	no	—
90	C₂×C₄	—	1006	3	no	—
91	C₂×C₄	—	1006	3	no	—
92	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
93	C₈	C₂²×D₅₀₃	1	1	yes	characteristic
94	C₈	C₂²×D₅₀₃	1	3	yes	—
95	C₂²	—	4024	3	no	—
96	C₂²	—	1006	1	no	—
97	C₂²	—	1006	3	no	—
98	C₂²	C₂×D₂₀₁₂	1	3	yes	—
99	C₄	—	1006	1	no	—
100	C₄	—	1006	3	no	—
101	C₄	C₂×D₂₀₁₂	1	3	yes	—
102	C₄	C₂³×D₅₀₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	2012	3	no	—
105	C₂	—	2012	1	no	—
106	C₂	C₂²×D₂₀₁₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632911

C₂

yes

maximal, characteristic, complemented

#30632910

C₂

yes

maximal, characteristic, complemented

#30632897

C₂

yes

maximal, complemented

#30632892

C₂

yes

maximal, complemented

#30632886

C₂

yes

maximal, complemented

C₂×#22548872

C₂

yes

maximal, complemented

#131×C₅₀₃

C₂

yes

maximal, characteristic, complemented

#22548884

—

#22548883

C₂²

yes

—

#22548883

C₂²

yes

complemented

#22548882

—

C₂×Dic₁₀₀₆

C₂²

yes

complemented

Q₈×D₅₀₃

—

C₂×D₂₀₁₂

C₂²

yes

—

D₄×D₅₀₃

—

Dic₂₀₁₂

C₂²

yes

complemented

#22548872

C₂²

yes

complemented

#22548872

C₂²

yes

complemented

#22548872

C₂²

yes

characteristic, complemented

D₄₀₂₄

C₂²

yes

complemented

#41×C₅₀₃

C₂²

yes

characteristic

M₁₆×C₅₀₃

C₂²

yes

complemented

C₂×C₈×C₅₀₃

C₂²

yes

complemented

C₂²×D₅₀₃

—

Dic₁₀₀₆

—

Dic₁₀₀₆

C₂³

yes

—

Dic₁₀₀₆

C₂³

yes

characteristic

D₂₀₁₂

—

D₂₀₁₂

C₂³

yes

characteristic

D₂₀₁₂

C₂³

yes

—

C₂²⋊D₅₀₃

—

C₂²⋊D₅₀₃

—

C₂×Dic₅₀₃

—

C₄×D₅₀₃

—

C₄×D₅₀₃

—

Q₈×C₅₀₃

D₄

yes

characteristic

D₄×C₅₀₃

D₄

yes

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₈×C₅₀₃

C₂³

yes

characteristic

C₈×C₅₀₃

C₂³

yes

—

C₂²×C₅₀₃

C₂×D₄

yes

—

C₂×D₅₀₃

—

C₂×D₅₀₃

—

C₂×D₅₀₃

—

C₄×C₅₀₃

C₂×D₄

yes

—

C₄×C₅₀₃

C₂⁴

yes

characteristic

Dic₅₀₃

—

Dic₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

C₂²×D₄

yes

characteristic

D₅₀₃

—

D₅₀₃

—

C₅₀₃

#577

yes

characteristic, complemented, Sylow

#577

—

503

maximal, Sylow

2₋¹⁺⁴

—

503

—

H(5,𝔽₂)

—

503

—

#137

—

503

—

AGL(1,ℤ/8)

—

503

—

#135

—

503

—

C₂×SD₁₆

—

503

—

#131

D₅₀₃

yes

characteristic, complemented

#41

—

1006

—

#41

—

1006

—

#41

—

503

—

#41

—

503

—

#41

C₂×D₅₀₃

yes

characteristic

C₂×Q₈

—

1006

—

C₂×Q₈

—

503

—

C₂×D₄

—

1006

—

C₂×D₄

—

503

—

Q₁₆

—

503

—

SD₁₆

—

503

—

SD₁₆

—

503

—

SD₁₆

—

503

—

D₈

—

503

—

M₁₆

C₂×D₅₀₃

yes

complemented

C₂×C₈

C₂×D₅₀₃

yes

complemented

C₂³

—

1006

—

Q₈

—

1006

—

Q₈

—

503

—

Q₈

—

503

—

Q₈

D₂₀₁₂

yes

characteristic

D₄

—

1006

—

D₄

—

1006

—

D₄

—

1006

—

D₄

—

503

—

D₄

—

503

—

D₄

D₂₀₁₂

yes

—

C₂×C₄

—

1006

—

C₂×C₄

—

1006

—

C₂×C₄

—

1006

—

C₂×C₄

C₂²×D₅₀₃

yes

—

C₈

C₂²×D₅₀₃

yes

characteristic

C₈

C₂²×D₅₀₃

yes

—

C₂²

—

4024

—

C₂²

—

1006

—

C₂²

—

1006

—

C₂²

C₂×D₂₀₁₂

yes

—

C₄

—

1006

—

100

C₄

—

1006

—

101

C₄

C₂×D₂₀₁₂

yes

—

102

C₄

C₂³×D₅₀₃

yes

characteristic

103

C₂

—

104

C₂

—

2012

—

105

C₂

—

2012

—

106

C₂

C₂²×D₂₀₁₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₀₁₂
commutator, cosocle	C₄×C₅₀₃	C₂⁴
Frattini	C₄	C₂³×D₅₀₃
Fitting	#131×C₅₀₃	C₂
socle	C₂×C₅₀₃	C₂²×D₄
hypercenter	#131	D₅₀₃
hypocenter	C₅₀₃	#577
LCS	G ▷ C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₂₀₁₂

commutator, cosocle

C₄×C₅₀₃

C₂⁴

Frattini

C₄

C₂³×D₅₀₃

Fitting

#131×C₅₀₃

C₂

socle

C₂×C₅₀₃

C₂²×D₄

D₅₀₃

C₅₀₃

G ▷ C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #30632911	C₂	semidirect
2. #30632910	C₂	semidirect
3. #30632897	C₂	semidirect
4. #30632892	C₂	semidirect
5. #30632886	C₂	semidirect
6. C₂×#22548872	C₂	semidirect
7. #131×C₅₀₃	C₂	semidirect
10. #22548883	C₂²	semidirect
12. C₂×Dic₁₀₀₆	C₂²	semidirect
16. Dic₂₀₁₂	C₂²	semidirect
17. #22548872	C₂²	semidirect
18. #22548872	C₂²	semidirect
19. #22548872	C₂²	semidirect
20. D₄₀₂₄	C₂²	semidirect
22. M₁₆×C₅₀₃	C₂²	semidirect
23. C₂×C₈×C₅₀₃	C₂²	semidirect
53. C₅₀₃	#577	semidirect
61. #131	D₅₀₃	semidirect
76. M₁₆	C₂×D₅₀₃	semidirect
77. C₂×C₈	C₂×D₅₀₃	semidirect

H ≅ G/N

type

1. #30632911

C₂

semidirect

2. #30632910

C₂

semidirect

3. #30632897

C₂

semidirect

4. #30632892

C₂

semidirect

5. #30632886

C₂

semidirect

6. C₂×#22548872

C₂

semidirect

7. #131×C₅₀₃

C₂

semidirect

10. #22548883

C₂²

semidirect

12. C₂×Dic₁₀₀₆

C₂²

semidirect

16. Dic₂₀₁₂

C₂²

semidirect

17. #22548872

C₂²

semidirect

18. #22548872

C₂²

semidirect

19. #22548872

C₂²

semidirect

20. D₄₀₂₄

C₂²

semidirect

22. M₁₆×C₅₀₃

C₂²

semidirect

23. C₂×C₈×C₅₀₃

C₂²

semidirect

53. C₅₀₃

#577

semidirect

61. #131

D₅₀₃

semidirect

76. M₁₆

C₂×D₅₀₃

semidirect

77. C₂×C₈

C₂×D₅₀₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#30632911	2
#30632910	2
#30632897	2
#30632892	2
#30632886	2
C₂×#22548872	2
#131×C₅₀₃	2
#577	503

subgroup

index

#131×C₅₀₃

503

Subgroups — #44567093

Conjugacy classes of subgroups

Complements

Maximal subgroups