Group #14583702 — subgroups

Subgroups — #14583702

Order 2752.

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

normal subgroups only

This group has 268 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	#13063371	C₂	1	1	yes	maximal, characteristic, complemented
2	#13063351	C₂	1	3	yes	maximal, complemented
3	#13063350	C₂	1	3	yes	maximal, complemented
4	#13063345	C₂	1	3	yes	maximal, complemented
5	C₂×#10621882	C₂	1	3	yes	maximal, complemented
6	#13063332	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×C₄₃	—	2	1	no	—
9	#41×C₄₃	C₂²	1	1	yes	characteristic
10	#41×C₄₃	C₂²	1	3	yes	complemented
11	C₂×D₄×C₄₃	—	2	3	no	—
12	C₂×D₄×C₄₃	C₂²	1	3	yes	complemented
13	#10621902	—	2	1	no	—
14	D₄×D₄₃	—	2	3	no	—
15	#10621898	C₂²	1	3	yes	complemented
16	C₂×D₁₇₂	C₂²	1	3	yes	complemented
17	#10621885	C₂²	1	1	yes	characteristic, complemented
18	#10621884	C₂²	1	3	yes	complemented
19	#10621883	C₂²	1	3	yes	complemented
20	#10621882	C₂²	1	3	yes	complemented
21	#10621882	C₂²	1	6	yes	complemented
22	#10621877	C₂²	1	3	yes	complemented
23	C₂×C₄₃⋊C₈	C₂²	1	3	yes	complemented
24	C₂³×C₄₃	—	2	3	no	—
25	C₂²×D₄₃	—	2	3	no	—
26	Q₈×C₄₃	D₄	1	1	yes	characteristic
27	Q₈×C₄₃	C₂³	1	1	yes	characteristic
28	D₄×C₄₃	—	2	3	no	—
29	D₄×C₄₃	—	2	3	no	—
30	D₄×C₄₃	D₄	1	3	yes	—
31	D₄×C₄₃	C₂³	1	3	yes	—
32	C₂×C₄×C₄₃	—	2	3	no	—
33	C₂×C₄×C₄₃	C₂³	1	3	yes	—
34	C₂²⋊D₄₃	—	2	3	no	—
35	D₁₇₂	—	2	3	no	—
36	D₁₇₂	C₂³	1	3	yes	—
37	C₄×D₄₃	—	2	3	no	—
38	Dic₈₆	C₂³	1	1	yes	characteristic
39	C₄₃⋊C₈	C₂³	1	1	yes	characteristic
40	C₄₃⋊C₈	C₂³	1	3	yes	—
41	C₂²×C₄₃	—	2	3	no	—
42	C₂²×C₄₃	—	8	3	no	—
43	C₂²×C₄₃	C₂×D₄	1	3	yes	—
44	C₂×D₄₃	—	2	3	no	—
45	C₂×D₄₃	—	8	3	no	—
46	C₄×C₄₃	—	2	1	no	—
47	C₄×C₄₃	C₂×D₄	1	3	yes	—
48	C₄×C₄₃	C₂⁴	1	1	yes	characteristic
49	Dic₄₃	—	2	1	no	—
50	C₂×C₄₃	—	2	3	no	—
51	C₂×C₄₃	—	4	3	no	—
52	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
53	D₄₃	—	4	3	no	—
54	#576	—	43	1	no	maximal, Sylow
55	C₄₃	#576	1	1	yes	characteristic, complemented, Sylow
56	H(5,𝔽₂)	—	43	1	no	—
57	H(5,𝔽₂)	D₄₃	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	43	3	no	—
59	AGL(1,ℤ/8)	—	43	3	no	—
60	#135	—	43	3	no	—
61	C₂×D₈	—	43	3	no	—
62	#131	—	43	1	no	—
63	#41	—	2	1	no	—
64	#41	—	43	3	no	—
65	#41	—	86	1	no	—
66	#41	C₂×D₄₃	1	1	yes	characteristic
67	#41	C₂×D₄₃	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	43	3	no	—
70	C₂×D₄	—	86	3	no	—
71	C₂×D₄	C₂×D₄₃	1	3	yes	complemented
72	Q₁₆	—	43	1	no	—
73	SD₁₆	—	43	3	no	—
74	SD₁₆	—	43	3	no	—
75	D₈	—	43	3	no	—
76	D₈	—	43	6	no	—
77	M₁₆	—	43	3	no	—
78	C₂×C₈	—	43	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	86	3	no	—
81	Q₈	—	43	1	no	—
82	Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
83	Q₈	C₂²×D₄₃	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	43	3	no	—
87	D₄	—	86	3	no	—
88	D₄	—	86	3	no	—
89	D₄	C₂²⋊D₄₃	1	3	yes	—
90	D₄	C₂²×D₄₃	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	86	3	no	—
93	C₂×C₄	C₂²×D₄₃	1	3	yes	—
94	C₈	—	43	1	no	—
95	C₈	—	43	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	86	3	no	—
99	C₂²	—	344	3	no	—
100	C₂²	C₂×C₂²⋊D₄₃	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	86	1	no	—
103	C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
104	C₄	C₂³×D₄₃	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	172	3	no	—
108	C₂	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063371

C₂

yes

maximal, characteristic, complemented

#13063351

C₂

yes

maximal, complemented

#13063350

C₂

yes

maximal, complemented

#13063345

C₂

yes

maximal, complemented

C₂×#10621882

C₂

yes

maximal, complemented

#13063332

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₄₃

C₂

yes

maximal, characteristic, complemented

#41×C₄₃

—

#41×C₄₃

C₂²

yes

characteristic

#41×C₄₃

C₂²

yes

complemented

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂²

yes

complemented

#10621902

—

D₄×D₄₃

—

#10621898

C₂²

yes

complemented

C₂×D₁₇₂

C₂²

yes

complemented

#10621885

C₂²

yes

characteristic, complemented

#10621884

C₂²

yes

complemented

#10621883

C₂²

yes

complemented

#10621882

C₂²

yes

complemented

#10621882

C₂²

yes

complemented

#10621877

C₂²

yes

complemented

C₂×C₄₃⋊C₈

C₂²

yes

complemented

C₂³×C₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

D₄

yes

characteristic

Q₈×C₄₃

C₂³

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

D₄

yes

—

D₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂²⋊D₄₃

—

D₁₇₂

—

D₁₇₂

C₂³

yes

—

C₄×D₄₃

—

Dic₈₆

C₂³

yes

characteristic

C₄₃⋊C₈

C₂³

yes

characteristic

C₄₃⋊C₈

C₂³

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×D₄

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₄

yes

—

C₄×C₄₃

C₂⁴

yes

characteristic

Dic₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₄

yes

characteristic

D₄₃

—

#576

—

maximal, Sylow

C₄₃

#576

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

D₄₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

#135

—

C₂×D₈

—

—

—

—

—

C₂×D₄₃

yes

characteristic

#41

C₂×D₄₃

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₃

yes

complemented

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

Q₈

—

Q₈

C₂²⋊D₄₃

yes

characteristic

Q₈

C₂²×D₄₃

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²⋊D₄₃

yes

—

D₄

C₂²×D₄₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₄₃

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

344

—

100

C₂²

C₂×C₂²⋊D₄₃

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₂×C₂²⋊D₄₃

yes

—

104

C₄

C₂³×D₄₃

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

172

—

108

C₂

C₂²×C₂²⋊D₄₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₄×C₄₃	C₂⁴
Frattini	C₄	C₂³×D₄₃
Fitting	H(5,𝔽₂)×C₄₃	C₂
socle	C₂×C₄₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₄₃
hypocenter	C₄₃	#576
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₄₃

commutator, cosocle

C₄×C₄₃

C₂⁴

Frattini

C₄

C₂³×D₄₃

Fitting

H(5,𝔽₂)×C₄₃

C₂

socle

C₂×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #13063371	C₂	semidirect
2. #13063351	C₂	semidirect
3. #13063350	C₂	semidirect
4. #13063345	C₂	semidirect
5. C₂×#10621882	C₂	semidirect
6. #13063332	C₂	semidirect
7. H(5,𝔽₂)×C₄₃	C₂	semidirect
10. #41×C₄₃	C₂²	semidirect
12. C₂×D₄×C₄₃	C₂²	semidirect
15. #10621898	C₂²	semidirect
16. C₂×D₁₇₂	C₂²	semidirect
17. #10621885	C₂²	semidirect
18. #10621884	C₂²	semidirect
19. #10621883	C₂²	semidirect
20. #10621882	C₂²	semidirect
21. #10621882	C₂²	semidirect
22. #10621877	C₂²	semidirect
23. C₂×C₄₃⋊C₈	C₂²	semidirect
55. C₄₃	#576	semidirect
57. H(5,𝔽₂)	D₄₃	semidirect
67. #41	C₂×D₄₃	semidirect
71. C₂×D₄	C₂×D₄₃	semidirect

H ≅ G/N

type

1. #13063371

C₂

semidirect

2. #13063351

C₂

semidirect

3. #13063350

C₂

semidirect

4. #13063345

C₂

semidirect

5. C₂×#10621882

C₂

semidirect

6. #13063332

C₂

semidirect

7. H(5,𝔽₂)×C₄₃

C₂

semidirect

10. #41×C₄₃

C₂²

semidirect

12. C₂×D₄×C₄₃

C₂²

semidirect

15. #10621898

C₂²

semidirect

16. C₂×D₁₇₂

C₂²

semidirect

17. #10621885

C₂²

semidirect

18. #10621884

C₂²

semidirect

19. #10621883

C₂²

semidirect

20. #10621882

C₂²

semidirect

21. #10621882

C₂²

semidirect

22. #10621877

C₂²

semidirect

23. C₂×C₄₃⋊C₈

C₂²

semidirect

55. C₄₃

#576

semidirect

57. H(5,𝔽₂)

D₄₃

semidirect

67. #41

C₂×D₄₃

semidirect

71. C₂×D₄

C₂×D₄₃

semidirect

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

subgroup	index
#13063371	2
#13063351	2
#13063350	2
#13063345	2
C₂×#10621882	2
#13063332	2
H(5,𝔽₂)×C₄₃	2
#576	43

subgroup

index

H(5,𝔽₂)×C₄₃

Subgroups — #14583702

Conjugacy classes of subgroups

Complements

Maximal subgroups