Group #37527246 — subgroups

Subgroups — #37527246

Order 22976.

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	#26040061	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040041	C₂	1	3	yes	maximal, complemented
3	#26040040	C₂	1	3	yes	maximal, complemented
4	#26040035	C₂	1	3	yes	maximal, complemented
5	C₂×#19153326	C₂	1	3	yes	maximal, complemented
6	#26040022	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153342	—	2	1	no	—
9	#19153341	C₂²	1	3	yes	complemented
10	C₂×D₁₄₃₆	C₂²	1	3	yes	complemented
11	D₄×D₃₅₉	—	2	3	no	—
12	#19153331	C₂²	1	1	yes	characteristic, complemented
13	#19153329	C₂²	1	3	yes	complemented
14	#19153328	C₂²	1	3	yes	complemented
15	#19153326	C₂²	1	3	yes	complemented
16	#19153326	C₂²	1	6	yes	complemented
17	#19153325	C₂²	1	3	yes	complemented
18	C₂×C₃₅₉⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₃₅₉	—	2	1	no	—
20	#41×C₃₅₉	C₂²	1	1	yes	characteristic
21	#41×C₃₅₉	C₂²	1	3	yes	complemented
22	C₂×D₄×C₃₅₉	—	2	3	no	—
23	C₂×D₄×C₃₅₉	C₂²	1	3	yes	complemented
24	C₂²×D₃₅₉	—	2	3	no	—
25	Dic₇₁₈	C₂³	1	1	yes	characteristic
26	D₁₄₃₆	—	2	3	no	—
27	D₁₄₃₆	C₂³	1	3	yes	—
28	C₂²⋊D₃₅₉	—	2	3	no	—
29	C₄×D₃₅₉	—	2	3	no	—
30	C₃₅₉⋊C₈	C₂³	1	1	yes	characteristic
31	C₃₅₉⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₃₅₉	—	2	3	no	—
33	Q₈×C₃₅₉	D₄	1	1	yes	characteristic
34	Q₈×C₃₅₉	C₂³	1	1	yes	characteristic
35	D₄×C₃₅₉	—	2	3	no	—
36	D₄×C₃₅₉	—	2	3	no	—
37	D₄×C₃₅₉	D₄	1	3	yes	—
38	D₄×C₃₅₉	C₂³	1	3	yes	—
39	C₂×C₄×C₃₅₉	—	2	3	no	—
40	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
41	C₂²×C₃₅₉	—	2	3	no	—
42	C₂²×C₃₅₉	—	8	3	no	—
43	C₂²×C₃₅₉	C₂×D₄	1	3	yes	—
44	C₂×D₃₅₉	—	2	3	no	—
45	C₂×D₃₅₉	—	8	3	no	—
46	C₄×C₃₅₉	—	2	1	no	—
47	C₄×C₃₅₉	C₂×D₄	1	3	yes	—
48	C₄×C₃₅₉	C₂⁴	1	1	yes	characteristic
49	Dic₃₅₉	—	2	1	no	—
50	C₂×C₃₅₉	—	2	3	no	—
51	C₂×C₃₅₉	—	4	3	no	—
52	C₂×C₃₅₉	C₂²×D₄	1	1	yes	characteristic
53	D₃₅₉	—	4	3	no	—
54	C₃₅₉	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	359	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	359	1	no	—
57	H(5,𝔽₂)	D₃₅₉	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	359	3	no	—
59	AGL(1,ℤ/8)	—	359	3	no	—
60	#135	—	359	3	no	—
61	C₂×D₈	—	359	3	no	—
62	#131	—	359	1	no	—
63	#41	—	2	1	no	—
64	#41	—	359	3	no	—
65	#41	—	718	1	no	—
66	#41	C₂×D₃₅₉	1	1	yes	characteristic
67	#41	C₂×D₃₅₉	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	359	3	no	—
70	C₂×D₄	—	718	3	no	—
71	C₂×D₄	C₂×D₃₅₉	1	3	yes	complemented
72	Q₁₆	—	359	1	no	—
73	SD₁₆	—	359	3	no	—
74	SD₁₆	—	359	3	no	—
75	D₈	—	359	3	no	—
76	D₈	—	359	6	no	—
77	M₁₆	—	359	3	no	—
78	C₂×C₈	—	359	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	718	3	no	—
81	Q₈	—	359	1	no	—
82	Q₈	C₂²⋊D₃₅₉	1	1	yes	characteristic
83	Q₈	C₂²×D₃₅₉	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	359	3	no	—
87	D₄	—	718	3	no	—
88	D₄	—	718	3	no	—
89	D₄	C₂²⋊D₃₅₉	1	3	yes	—
90	D₄	C₂²×D₃₅₉	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	718	3	no	—
93	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
94	C₈	—	359	1	no	—
95	C₈	—	359	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	2872	3	no	—
99	C₂²	—	718	3	no	—
100	C₂²	C₂×C₂²⋊D₃₅₉	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	718	1	no	—
103	C₄	C₂×C₂²⋊D₃₅₉	1	3	yes	—
104	C₄	C₂³×D₃₅₉	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	1436	3	no	—
108	C₂	C₂²×C₂²⋊D₃₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26040061

C₂

yes

maximal, characteristic, complemented

#26040041

C₂

yes

maximal, complemented

#26040040

C₂

yes

maximal, complemented

#26040035

C₂

yes

maximal, complemented

C₂×#19153326

C₂

yes

maximal, complemented

#26040022

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₅₉

C₂

yes

maximal, characteristic, complemented

#19153342

—

#19153341

C₂²

yes

complemented

C₂×D₁₄₃₆

C₂²

yes

complemented

D₄×D₃₅₉

—

#19153331

C₂²

yes

characteristic, complemented

#19153329

C₂²

yes

complemented

#19153328

C₂²

yes

complemented

#19153326

C₂²

yes

complemented

#19153326

C₂²

yes

complemented

#19153325

C₂²

yes

complemented

C₂×C₃₅₉⋊C₈

C₂²

yes

complemented

#41×C₃₅₉

—

#41×C₃₅₉

C₂²

yes

characteristic

#41×C₃₅₉

C₂²

yes

complemented

C₂×D₄×C₃₅₉

—

C₂×D₄×C₃₅₉

C₂²

yes

complemented

C₂²×D₃₅₉

—

Dic₇₁₈

C₂³

yes

characteristic

D₁₄₃₆

—

D₁₄₃₆

C₂³

yes

—

C₂²⋊D₃₅₉

—

C₄×D₃₅₉

—

C₃₅₉⋊C₈

C₂³

yes

characteristic

C₃₅₉⋊C₈

C₂³

yes

—

C₂³×C₃₅₉

—

Q₈×C₃₅₉

D₄

yes

characteristic

Q₈×C₃₅₉

C₂³

yes

characteristic

D₄×C₃₅₉

—

D₄×C₃₅₉

—

D₄×C₃₅₉

D₄

yes

—

D₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

C₂×D₄

yes

—

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

C₂×D₄

yes

—

C₄×C₃₅₉

C₂⁴

yes

characteristic

Dic₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

C₂²×D₄

yes

characteristic

D₃₅₉

—

C₃₅₉

#576

yes

characteristic, complemented, Sylow

#576

—

359

maximal, Sylow

H(5,𝔽₂)

—

359

—

H(5,𝔽₂)

D₃₅₉

yes

characteristic, complemented

AGL(1,ℤ/8)

—

359

—

AGL(1,ℤ/8)

—

359

—

#135

—

359

—

C₂×D₈

—

359

—

#131

—

359

—

#41

—

#41

—

359

—

#41

—

718

—

#41

C₂×D₃₅₉

yes

characteristic

#41

C₂×D₃₅₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

359

—

C₂×D₄

—

718

—

C₂×D₄

C₂×D₃₅₉

yes

complemented

Q₁₆

—

359

—

SD₁₆

—

359

—

SD₁₆

—

359

—

D₈

—

359

—

D₈

—

359

—

M₁₆

—

359

—

C₂×C₈

—

359

—

C₂³

—

C₂³

—

718

—

Q₈

—

359

—

Q₈

C₂²⋊D₃₅₉

yes

characteristic

Q₈

C₂²×D₃₅₉

yes

characteristic

D₄

—

D₄

—

D₄

—

359

—

D₄

—

718

—

D₄

—

718

—

D₄

C₂²⋊D₃₅₉

yes

—

D₄

C₂²×D₃₅₉

yes

—

C₂×C₄

—

C₂×C₄

—

718

—

C₂×C₄

C₂²×D₃₅₉

yes

—

C₈

—

359

—

C₈

—

359

—

C₂²

—

C₂²

—

C₂²

—

2872

—

C₂²

—

718

—

100

C₂²

C₂×C₂²⋊D₃₅₉

yes

—

101

C₄

—

102

C₄

—

718

—

103

C₄

C₂×C₂²⋊D₃₅₉

yes

—

104

C₄

C₂³×D₃₅₉

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

1436

—

108

C₂

C₂²×C₂²⋊D₃₅₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₅₉
commutator, cosocle	C₄×C₃₅₉	C₂⁴
Frattini	C₄	C₂³×D₃₅₉
Fitting	H(5,𝔽₂)×C₃₅₉	C₂
socle	C₂×C₃₅₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₅₉
hypocenter	C₃₅₉	#576
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₅₉

commutator, cosocle

C₄×C₃₅₉

C₂⁴

Frattini

C₄

C₂³×D₃₅₉

Fitting

H(5,𝔽₂)×C₃₅₉

C₂

socle

C₂×C₃₅₉

C₂²×D₄

D₃₅₉

C₃₅₉

G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #26040061	C₂	semidirect
2. #26040041	C₂	semidirect
3. #26040040	C₂	semidirect
4. #26040035	C₂	semidirect
5. C₂×#19153326	C₂	semidirect
6. #26040022	C₂	semidirect
7. H(5,𝔽₂)×C₃₅₉	C₂	semidirect
9. #19153341	C₂²	semidirect
10. C₂×D₁₄₃₆	C₂²	semidirect
12. #19153331	C₂²	semidirect
13. #19153329	C₂²	semidirect
14. #19153328	C₂²	semidirect
15. #19153326	C₂²	semidirect
16. #19153326	C₂²	semidirect
17. #19153325	C₂²	semidirect
18. C₂×C₃₅₉⋊C₈	C₂²	semidirect
21. #41×C₃₅₉	C₂²	semidirect
23. C₂×D₄×C₃₅₉	C₂²	semidirect
54. C₃₅₉	#576	semidirect
57. H(5,𝔽₂)	D₃₅₉	semidirect
67. #41	C₂×D₃₅₉	semidirect
71. C₂×D₄	C₂×D₃₅₉	semidirect

H ≅ G/N

type

1. #26040061

C₂

semidirect

2. #26040041

C₂

semidirect

3. #26040040

C₂

semidirect

4. #26040035

C₂

semidirect

5. C₂×#19153326

C₂

semidirect

6. #26040022

C₂

semidirect

7. H(5,𝔽₂)×C₃₅₉

C₂

semidirect

9. #19153341

C₂²

semidirect

10. C₂×D₁₄₃₆

C₂²

semidirect

12. #19153331

C₂²

semidirect

13. #19153329

C₂²

semidirect

14. #19153328

C₂²

semidirect

15. #19153326

C₂²

semidirect

16. #19153326

C₂²

semidirect

17. #19153325

C₂²

semidirect

18. C₂×C₃₅₉⋊C₈

C₂²

semidirect

21. #41×C₃₅₉

C₂²

semidirect

23. C₂×D₄×C₃₅₉

C₂²

semidirect

54. C₃₅₉

#576

semidirect

57. H(5,𝔽₂)

D₃₅₉

semidirect

67. #41

C₂×D₃₅₉

semidirect

71. C₂×D₄

C₂×D₃₅₉

semidirect

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

subgroup	index
#26040061	2
#26040041	2
#26040040	2
#26040035	2
C₂×#19153326	2
#26040022	2
H(5,𝔽₂)×C₃₅₉	2
#576	359

subgroup

index

H(5,𝔽₂)×C₃₅₉

359

Subgroups — #37527246

Conjugacy classes of subgroups

Complements

Maximal subgroups