Group #43248529 — subgroups

Subgroups — #43248529

Order 28352.

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	#28381933	C₂	1	1	yes	maximal, characteristic, complemented
2	#28381932	C₂	1	1	yes	maximal, characteristic, complemented
3	#28381919	C₂	1	3	yes	maximal, complemented
4	#28381914	C₂	1	3	yes	maximal, complemented
5	#28381908	C₂	1	3	yes	maximal, complemented
6	C₂×#20296776	C₂	1	3	yes	maximal, complemented
7	#131×C₄₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296788	—	2	1	no	—
9	#20296787	C₂²	1	3	yes	—
10	#20296787	C₂²	1	3	yes	complemented
11	#20296786	—	2	3	no	—
12	C₂×Dic₈₈₆	C₂²	1	3	yes	complemented
13	Q₈×D₄₄₃	—	2	1	no	—
14	C₂×D₁₇₇₂	C₂²	1	3	yes	—
15	D₄×D₄₄₃	—	2	3	no	—
16	Dic₁₇₇₂	C₂²	1	3	yes	complemented
17	#20296776	C₂²	1	3	yes	complemented
18	#20296776	C₂²	1	6	yes	complemented
19	#20296776	C₂²	1	1	yes	characteristic, complemented
20	D₃₅₄₄	C₂²	1	3	yes	complemented
21	#41×C₄₄₃	C₂²	1	1	yes	characteristic
22	M₁₆×C₄₄₃	C₂²	1	3	yes	complemented
23	C₂×C₈×C₄₄₃	C₂²	1	3	yes	complemented
24	C₂²×D₄₄₃	—	2	3	no	—
25	Dic₈₈₆	—	2	3	no	—
26	Dic₈₈₆	C₂³	1	3	yes	—
27	Dic₈₈₆	C₂³	1	1	yes	characteristic
28	D₁₇₇₂	—	2	3	no	—
29	D₁₇₇₂	C₂³	1	1	yes	characteristic
30	D₁₇₇₂	C₂³	1	3	yes	—
31	C₂²⋊D₄₄₃	—	2	3	no	—
32	C₂²⋊D₄₄₃	—	2	3	no	—
33	C₂×Dic₄₄₃	—	2	3	no	—
34	C₄×D₄₄₃	—	2	3	no	—
35	C₄×D₄₄₃	—	2	3	no	—
36	Q₈×C₄₄₃	D₄	1	1	yes	characteristic
37	D₄×C₄₄₃	D₄	1	3	yes	—
38	C₂×C₄×C₄₄₃	C₂³	1	3	yes	—
39	C₈×C₄₄₃	C₂³	1	1	yes	characteristic
40	C₈×C₄₄₃	C₂³	1	3	yes	—
41	C₂²×C₄₄₃	C₂×D₄	1	3	yes	—
42	C₂×D₄₄₃	—	2	1	no	—
43	C₂×D₄₄₃	—	2	3	no	—
44	C₂×D₄₄₃	—	8	3	no	—
45	C₄×C₄₄₃	C₂×D₄	1	3	yes	—
46	C₄×C₄₄₃	C₂⁴	1	1	yes	characteristic
47	Dic₄₄₃	—	2	1	no	—
48	Dic₄₄₃	—	2	3	no	—
49	C₂×C₄₄₃	—	2	3	no	—
50	C₂×C₄₄₃	C₂²×D₄	1	1	yes	characteristic
51	D₄₄₃	—	4	1	no	—
52	D₄₄₃	—	4	3	no	—
53	C₄₄₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	443	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	443	1	no	—
56	H(5,𝔽₂)	—	443	1	no	—
57	#137	—	443	3	no	—
58	AGL(1,ℤ/8)	—	443	3	no	—
59	#135	—	443	3	no	—
60	C₂×SD₁₆	—	443	3	no	—
61	#131	D₄₄₃	1	1	yes	characteristic, complemented
62	#41	—	886	1	no	—
63	#41	—	886	3	no	—
64	#41	—	443	3	no	—
65	#41	—	443	3	no	—
66	#41	C₂×D₄₄₃	1	1	yes	characteristic
67	C₂×Q₈	—	886	1	no	—
68	C₂×Q₈	—	443	3	no	—
69	C₂×D₄	—	886	3	no	—
70	C₂×D₄	—	443	3	no	—
71	Q₁₆	—	443	3	no	—
72	SD₁₆	—	443	3	no	—
73	SD₁₆	—	443	6	no	—
74	SD₁₆	—	443	1	no	—
75	D₈	—	443	3	no	—
76	M₁₆	C₂×D₄₄₃	1	3	yes	complemented
77	C₂×C₈	C₂×D₄₄₃	1	3	yes	complemented
78	C₂³	—	886	3	no	—
79	Q₈	—	886	3	no	—
80	Q₈	—	443	3	no	—
81	Q₈	—	443	1	no	—
82	Q₈	D₁₇₇₂	1	1	yes	characteristic
83	D₄	—	886	3	no	—
84	D₄	—	886	3	no	—
85	D₄	—	886	3	no	—
86	D₄	—	443	1	no	—
87	D₄	—	443	3	no	—
88	D₄	D₁₇₇₂	1	3	yes	—
89	C₂×C₄	—	886	3	no	—
90	C₂×C₄	—	886	3	no	—
91	C₂×C₄	—	886	3	no	—
92	C₂×C₄	C₂²×D₄₄₃	1	3	yes	—
93	C₈	C₂²×D₄₄₃	1	1	yes	characteristic
94	C₈	C₂²×D₄₄₃	1	3	yes	—
95	C₂²	—	886	1	no	—
96	C₂²	—	886	3	no	—
97	C₂²	—	3544	3	no	—
98	C₂²	C₂×D₁₇₇₂	1	3	yes	—
99	C₄	—	886	1	no	—
100	C₄	—	886	3	no	—
101	C₄	C₂×D₁₇₇₂	1	3	yes	—
102	C₄	C₂³×D₄₄₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	1772	3	no	—
105	C₂	—	1772	1	no	—
106	C₂	C₂²×D₁₇₇₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28381933

C₂

yes

maximal, characteristic, complemented

#28381932

C₂

yes

maximal, characteristic, complemented

#28381919

C₂

yes

maximal, complemented

#28381914

C₂

yes

maximal, complemented

#28381908

C₂

yes

maximal, complemented

C₂×#20296776

C₂

yes

maximal, complemented

#131×C₄₄₃

C₂

yes

maximal, characteristic, complemented

#20296788

—

#20296787

C₂²

yes

—

#20296787

C₂²

yes

complemented

#20296786

—

C₂×Dic₈₈₆

C₂²

yes

complemented

Q₈×D₄₄₃

—

C₂×D₁₇₇₂

C₂²

yes

—

D₄×D₄₄₃

—

Dic₁₇₇₂

C₂²

yes

complemented

#20296776

C₂²

yes

complemented

#20296776

C₂²

yes

complemented

#20296776

C₂²

yes

characteristic, complemented

D₃₅₄₄

C₂²

yes

complemented

#41×C₄₄₃

C₂²

yes

characteristic

M₁₆×C₄₄₃

C₂²

yes

complemented

C₂×C₈×C₄₄₃

C₂²

yes

complemented

C₂²×D₄₄₃

—

Dic₈₈₆

—

Dic₈₈₆

C₂³

yes

—

Dic₈₈₆

C₂³

yes

characteristic

D₁₇₇₂

—

D₁₇₇₂

C₂³

yes

characteristic

D₁₇₇₂

C₂³

yes

—

C₂²⋊D₄₄₃

—

C₂²⋊D₄₄₃

—

C₂×Dic₄₄₃

—

C₄×D₄₄₃

—

C₄×D₄₄₃

—

Q₈×C₄₄₃

D₄

yes

characteristic

D₄×C₄₄₃

D₄

yes

—

C₂×C₄×C₄₄₃

C₂³

yes

—

C₈×C₄₄₃

C₂³

yes

characteristic

C₈×C₄₄₃

C₂³

yes

—

C₂²×C₄₄₃

C₂×D₄

yes

—

C₂×D₄₄₃

—

C₂×D₄₄₃

—

C₂×D₄₄₃

—

C₄×C₄₄₃

C₂×D₄

yes

—

C₄×C₄₄₃

C₂⁴

yes

characteristic

Dic₄₄₃

—

Dic₄₄₃

—

C₂×C₄₄₃

—

C₂×C₄₄₃

C₂²×D₄

yes

characteristic

D₄₄₃

—

D₄₄₃

—

C₄₄₃

#577

yes

characteristic, complemented, Sylow

#577

—

443

maximal, Sylow

2₋¹⁺⁴

—

443

—

H(5,𝔽₂)

—

443

—

#137

—

443

—

AGL(1,ℤ/8)

—

443

—

#135

—

443

—

C₂×SD₁₆

—

443

—

#131

D₄₄₃

yes

characteristic, complemented

#41

—

886

—

#41

—

886

—

#41

—

443

—

#41

—

443

—

#41

C₂×D₄₄₃

yes

characteristic

C₂×Q₈

—

886

—

C₂×Q₈

—

443

—

C₂×D₄

—

886

—

C₂×D₄

—

443

—

Q₁₆

—

443

—

SD₁₆

—

443

—

SD₁₆

—

443

—

SD₁₆

—

443

—

D₈

—

443

—

M₁₆

C₂×D₄₄₃

yes

complemented

C₂×C₈

C₂×D₄₄₃

yes

complemented

C₂³

—

886

—

Q₈

—

886

—

Q₈

—

443

—

Q₈

—

443

—

Q₈

D₁₇₇₂

yes

characteristic

D₄

—

886

—

D₄

—

886

—

D₄

—

886

—

D₄

—

443

—

D₄

—

443

—

D₄

D₁₇₇₂

yes

—

C₂×C₄

—

886

—

C₂×C₄

—

886

—

C₂×C₄

—

886

—

C₂×C₄

C₂²×D₄₄₃

yes

—

C₈

C₂²×D₄₄₃

yes

characteristic

C₈

C₂²×D₄₄₃

yes

—

C₂²

—

886

—

C₂²

—

886

—

C₂²

—

3544

—

C₂²

C₂×D₁₇₇₂

yes

—

C₄

—

886

—

100

C₄

—

886

—

101

C₄

C₂×D₁₇₇₂

yes

—

102

C₄

C₂³×D₄₄₃

yes

characteristic

103

C₂

—

104

C₂

—

1772

—

105

C₂

—

1772

—

106

C₂

C₂²×D₁₇₇₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₇₇₂
commutator, cosocle	C₄×C₄₄₃	C₂⁴
Frattini	C₄	C₂³×D₄₄₃
Fitting	#131×C₄₄₃	C₂
socle	C₂×C₄₄₃	C₂²×D₄
hypercenter	#131	D₄₄₃
hypocenter	C₄₄₃	#577
LCS	G ▷ C₄×C₄₄₃ ▷ C₂×C₄₄₃ ▷ C₄₄₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₁₇₇₂

commutator, cosocle

C₄×C₄₄₃

C₂⁴

Frattini

C₄

C₂³×D₄₄₃

Fitting

#131×C₄₄₃

C₂

socle

C₂×C₄₄₃

C₂²×D₄

D₄₄₃

C₄₄₃

G ▷ C₄×C₄₄₃ ▷ C₂×C₄₄₃ ▷ C₄₄₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #28381933	C₂	semidirect
2. #28381932	C₂	semidirect
3. #28381919	C₂	semidirect
4. #28381914	C₂	semidirect
5. #28381908	C₂	semidirect
6. C₂×#20296776	C₂	semidirect
7. #131×C₄₄₃	C₂	semidirect
10. #20296787	C₂²	semidirect
12. C₂×Dic₈₈₆	C₂²	semidirect
16. Dic₁₇₇₂	C₂²	semidirect
17. #20296776	C₂²	semidirect
18. #20296776	C₂²	semidirect
19. #20296776	C₂²	semidirect
20. D₃₅₄₄	C₂²	semidirect
22. M₁₆×C₄₄₃	C₂²	semidirect
23. C₂×C₈×C₄₄₃	C₂²	semidirect
53. C₄₄₃	#577	semidirect
61. #131	D₄₄₃	semidirect
76. M₁₆	C₂×D₄₄₃	semidirect
77. C₂×C₈	C₂×D₄₄₃	semidirect

H ≅ G/N

type

1. #28381933

C₂

semidirect

2. #28381932

C₂

semidirect

3. #28381919

C₂

semidirect

4. #28381914

C₂

semidirect

5. #28381908

C₂

semidirect

6. C₂×#20296776

C₂

semidirect

7. #131×C₄₄₃

C₂

semidirect

10. #20296787

C₂²

semidirect

12. C₂×Dic₈₈₆

C₂²

semidirect

16. Dic₁₇₇₂

C₂²

semidirect

17. #20296776

C₂²

semidirect

18. #20296776

C₂²

semidirect

19. #20296776

C₂²

semidirect

20. D₃₅₄₄

C₂²

semidirect

22. M₁₆×C₄₄₃

C₂²

semidirect

23. C₂×C₈×C₄₄₃

C₂²

semidirect

53. C₄₄₃

#577

semidirect

61. #131

D₄₄₃

semidirect

76. M₁₆

C₂×D₄₄₃

semidirect

77. C₂×C₈

C₂×D₄₄₃

semidirect

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

subgroup	index
#28381933	2
#28381932	2
#28381919	2
#28381914	2
#28381908	2
C₂×#20296776	2
#131×C₄₄₃	2
#577	443

subgroup

index

#131×C₄₄₃

443

Subgroups — #43248529

Conjugacy classes of subgroups

Complements

Maximal subgroups