Group #37527255 — subgroups

Subgroups — #37527255

Order 22976.

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	#26040062	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040061	C₂	1	1	yes	maximal, characteristic, complemented
3	#26040048	C₂	1	3	yes	maximal, complemented
4	#26040043	C₂	1	3	yes	maximal, complemented
5	#26040037	C₂	1	3	yes	maximal, complemented
6	C₂×#19153330	C₂	1	3	yes	maximal, complemented
7	#131×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153342	—	2	1	no	—
9	#19153341	C₂²	1	3	yes	—
10	#19153341	C₂²	1	3	yes	complemented
11	#19153340	—	2	3	no	—
12	C₂×Dic₇₁₈	C₂²	1	3	yes	complemented
13	Q₈×D₃₅₉	—	2	1	no	—
14	C₂×D₁₄₃₆	C₂²	1	3	yes	—
15	D₄×D₃₅₉	—	2	3	no	—
16	Dic₁₄₃₆	C₂²	1	3	yes	complemented
17	#19153330	C₂²	1	3	yes	complemented
18	#19153330	C₂²	1	6	yes	complemented
19	#19153330	C₂²	1	1	yes	characteristic, complemented
20	D₂₈₇₂	C₂²	1	3	yes	complemented
21	#41×C₃₅₉	C₂²	1	1	yes	characteristic
22	M₁₆×C₃₅₉	C₂²	1	3	yes	complemented
23	C₂×C₈×C₃₅₉	C₂²	1	3	yes	complemented
24	C₂²×D₃₅₉	—	2	3	no	—
25	Dic₇₁₈	—	2	3	no	—
26	Dic₇₁₈	C₂³	1	3	yes	—
27	Dic₇₁₈	C₂³	1	1	yes	characteristic
28	D₁₄₃₆	—	2	3	no	—
29	D₁₄₃₆	C₂³	1	1	yes	characteristic
30	D₁₄₃₆	C₂³	1	3	yes	—
31	C₂²⋊D₃₅₉	—	2	3	no	—
32	C₂²⋊D₃₅₉	—	2	3	no	—
33	C₂×Dic₃₅₉	—	2	3	no	—
34	C₄×D₃₅₉	—	2	3	no	—
35	C₄×D₃₅₉	—	2	3	no	—
36	Q₈×C₃₅₉	D₄	1	1	yes	characteristic
37	D₄×C₃₅₉	D₄	1	3	yes	—
38	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
39	C₈×C₃₅₉	C₂³	1	1	yes	characteristic
40	C₈×C₃₅₉	C₂³	1	3	yes	—
41	C₂²×C₃₅₉	C₂×D₄	1	3	yes	—
42	C₂×D₃₅₉	—	2	1	no	—
43	C₂×D₃₅₉	—	2	3	no	—
44	C₂×D₃₅₉	—	8	3	no	—
45	C₄×C₃₅₉	C₂×D₄	1	3	yes	—
46	C₄×C₃₅₉	C₂⁴	1	1	yes	characteristic
47	Dic₃₅₉	—	2	1	no	—
48	Dic₃₅₉	—	2	3	no	—
49	C₂×C₃₅₉	—	2	3	no	—
50	C₂×C₃₅₉	C₂²×D₄	1	1	yes	characteristic
51	D₃₅₉	—	4	1	no	—
52	D₃₅₉	—	4	3	no	—
53	C₃₅₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	359	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	359	1	no	—
56	H(5,𝔽₂)	—	359	1	no	—
57	#137	—	359	3	no	—
58	AGL(1,ℤ/8)	—	359	3	no	—
59	#135	—	359	3	no	—
60	C₂×SD₁₆	—	359	3	no	—
61	#131	D₃₅₉	1	1	yes	characteristic, complemented
62	#41	—	359	3	no	—
63	#41	—	359	3	no	—
64	#41	—	718	1	no	—
65	#41	—	718	3	no	—
66	#41	C₂×D₃₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	359	3	no	—
68	C₂×Q₈	—	718	1	no	—
69	C₂×D₄	—	359	3	no	—
70	C₂×D₄	—	718	3	no	—
71	Q₁₆	—	359	3	no	—
72	SD₁₆	—	359	3	no	—
73	SD₁₆	—	359	6	no	—
74	SD₁₆	—	359	1	no	—
75	D₈	—	359	3	no	—
76	M₁₆	C₂×D₃₅₉	1	3	yes	complemented
77	C₂×C₈	C₂×D₃₅₉	1	3	yes	complemented
78	C₂³	—	718	3	no	—
79	Q₈	—	359	3	no	—
80	Q₈	—	359	1	no	—
81	Q₈	—	718	3	no	—
82	Q₈	D₁₄₃₆	1	1	yes	characteristic
83	D₄	—	359	1	no	—
84	D₄	—	359	3	no	—
85	D₄	—	718	3	no	—
86	D₄	—	718	3	no	—
87	D₄	—	718	3	no	—
88	D₄	D₁₄₃₆	1	3	yes	—
89	C₂×C₄	—	718	3	no	—
90	C₂×C₄	—	718	3	no	—
91	C₂×C₄	—	718	3	no	—
92	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
93	C₈	C₂²×D₃₅₉	1	1	yes	characteristic
94	C₈	C₂²×D₃₅₉	1	3	yes	—
95	C₂²	—	2872	3	no	—
96	C₂²	—	718	1	no	—
97	C₂²	—	718	3	no	—
98	C₂²	C₂×D₁₄₃₆	1	3	yes	—
99	C₄	—	718	1	no	—
100	C₄	—	718	3	no	—
101	C₄	C₂×D₁₄₃₆	1	3	yes	—
102	C₄	C₂³×D₃₅₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	1436	3	no	—
105	C₂	—	1436	1	no	—
106	C₂	C₂²×D₁₄₃₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26040062

C₂

yes

maximal, characteristic, complemented

#26040061

C₂

yes

maximal, characteristic, complemented

#26040048

C₂

yes

maximal, complemented

#26040043

C₂

yes

maximal, complemented

#26040037

C₂

yes

maximal, complemented

C₂×#19153330

C₂

yes

maximal, complemented

#131×C₃₅₉

C₂

yes

maximal, characteristic, complemented

#19153342

—

#19153341

C₂²

yes

—

#19153341

C₂²

yes

complemented

#19153340

—

C₂×Dic₇₁₈

C₂²

yes

complemented

Q₈×D₃₅₉

—

C₂×D₁₄₃₆

C₂²

yes

—

D₄×D₃₅₉

—

Dic₁₄₃₆

C₂²

yes

complemented

#19153330

C₂²

yes

complemented

#19153330

C₂²

yes

complemented

#19153330

C₂²

yes

characteristic, complemented

D₂₈₇₂

C₂²

yes

complemented

#41×C₃₅₉

C₂²

yes

characteristic

M₁₆×C₃₅₉

C₂²

yes

complemented

C₂×C₈×C₃₅₉

C₂²

yes

complemented

C₂²×D₃₅₉

—

Dic₇₁₈

—

Dic₇₁₈

C₂³

yes

—

Dic₇₁₈

C₂³

yes

characteristic

D₁₄₃₆

—

D₁₄₃₆

C₂³

yes

characteristic

D₁₄₃₆

C₂³

yes

—

C₂²⋊D₃₅₉

—

C₂²⋊D₃₅₉

—

C₂×Dic₃₅₉

—

C₄×D₃₅₉

—

C₄×D₃₅₉

—

Q₈×C₃₅₉

D₄

yes

characteristic

D₄×C₃₅₉

D₄

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₈×C₃₅₉

C₂³

yes

characteristic

C₈×C₃₅₉

C₂³

yes

—

C₂²×C₃₅₉

C₂×D₄

yes

—

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₄×C₃₅₉

C₂×D₄

yes

—

C₄×C₃₅₉

C₂⁴

yes

characteristic

Dic₃₅₉

—

Dic₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

C₂²×D₄

yes

characteristic

D₃₅₉

—

D₃₅₉

—

C₃₅₉

#577

yes

characteristic, complemented, Sylow

#577

—

359

maximal, Sylow

2₋¹⁺⁴

—

359

—

H(5,𝔽₂)

—

359

—

#137

—

359

—

AGL(1,ℤ/8)

—

359

—

#135

—

359

—

C₂×SD₁₆

—

359

—

#131

D₃₅₉

yes

characteristic, complemented

#41

—

359

—

#41

—

359

—

#41

—

718

—

#41

—

718

—

#41

C₂×D₃₅₉

yes

characteristic

C₂×Q₈

—

359

—

C₂×Q₈

—

718

—

C₂×D₄

—

359

—

C₂×D₄

—

718

—

Q₁₆

—

359

—

SD₁₆

—

359

—

SD₁₆

—

359

—

SD₁₆

—

359

—

D₈

—

359

—

M₁₆

C₂×D₃₅₉

yes

complemented

C₂×C₈

C₂×D₃₅₉

yes

complemented

C₂³

—

718

—

Q₈

—

359

—

Q₈

—

359

—

Q₈

—

718

—

Q₈

D₁₄₃₆

yes

characteristic

D₄

—

359

—

D₄

—

359

—

D₄

—

718

—

D₄

—

718

—

D₄

—

718

—

D₄

D₁₄₃₆

yes

—

C₂×C₄

—

718

—

C₂×C₄

—

718

—

C₂×C₄

—

718

—

C₂×C₄

C₂²×D₃₅₉

yes

—

C₈

C₂²×D₃₅₉

yes

characteristic

C₈

C₂²×D₃₅₉

yes

—

C₂²

—

2872

—

C₂²

—

718

—

C₂²

—

718

—

C₂²

C₂×D₁₄₃₆

yes

—

C₄

—

718

—

100

C₄

—

718

—

101

C₄

C₂×D₁₄₃₆

yes

—

102

C₄

C₂³×D₃₅₉

yes

characteristic

103

C₂

—

104

C₂

—

1436

—

105

C₂

—

1436

—

106

C₂

C₂²×D₁₄₃₆

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₄₃₆
commutator, cosocle	C₄×C₃₅₉	C₂⁴
Frattini	C₄	C₂³×D₃₅₉
Fitting	#131×C₃₅₉	C₂
socle	C₂×C₃₅₉	C₂²×D₄
hypercenter	#131	D₃₅₉
hypocenter	C₃₅₉	#577
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₁₄₃₆

commutator, cosocle

C₄×C₃₅₉

C₂⁴

Frattini

C₄

C₂³×D₃₅₉

Fitting

#131×C₃₅₉

C₂

socle

C₂×C₃₅₉

C₂²×D₄

D₃₅₉

C₃₅₉

G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #26040062	C₂	semidirect
2. #26040061	C₂	semidirect
3. #26040048	C₂	semidirect
4. #26040043	C₂	semidirect
5. #26040037	C₂	semidirect
6. C₂×#19153330	C₂	semidirect
7. #131×C₃₅₉	C₂	semidirect
10. #19153341	C₂²	semidirect
12. C₂×Dic₇₁₈	C₂²	semidirect
16. Dic₁₄₃₆	C₂²	semidirect
17. #19153330	C₂²	semidirect
18. #19153330	C₂²	semidirect
19. #19153330	C₂²	semidirect
20. D₂₈₇₂	C₂²	semidirect
22. M₁₆×C₃₅₉	C₂²	semidirect
23. C₂×C₈×C₃₅₉	C₂²	semidirect
53. C₃₅₉	#577	semidirect
61. #131	D₃₅₉	semidirect
76. M₁₆	C₂×D₃₅₉	semidirect
77. C₂×C₈	C₂×D₃₅₉	semidirect

H ≅ G/N

type

1. #26040062

C₂

semidirect

2. #26040061

C₂

semidirect

3. #26040048

C₂

semidirect

4. #26040043

C₂

semidirect

5. #26040037

C₂

semidirect

6. C₂×#19153330

C₂

semidirect

7. #131×C₃₅₉

C₂

semidirect

10. #19153341

C₂²

semidirect

12. C₂×Dic₇₁₈

C₂²

semidirect

16. Dic₁₄₃₆

C₂²

semidirect

17. #19153330

C₂²

semidirect

18. #19153330

C₂²

semidirect

19. #19153330

C₂²

semidirect

20. D₂₈₇₂

C₂²

semidirect

22. M₁₆×C₃₅₉

C₂²

semidirect

23. C₂×C₈×C₃₅₉

C₂²

semidirect

53. C₃₅₉

#577

semidirect

61. #131

D₃₅₉

semidirect

76. M₁₆

C₂×D₃₅₉

semidirect

77. C₂×C₈

C₂×D₃₅₉

semidirect

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

subgroup	index
#26040062	2
#26040061	2
#26040048	2
#26040043	2
#26040037	2
C₂×#19153330	2
#131×C₃₅₉	2
#577	359

subgroup

index

#131×C₃₅₉

359

Subgroups — #37527255

Conjugacy classes of subgroups

Complements

Maximal subgroups