Group #45656049 — subgroups

Subgroups — #45656049

Order 32576.

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	#30654262	C₂	1	1	yes	maximal, characteristic, complemented
2	#30654245	C₂	1	3	yes	maximal, complemented
3	#30654241	C₂	1	3	yes	maximal, complemented
4	#30654235	C₂	1	3	yes	maximal, complemented
5	C₂×#22550849	C₂	1	3	yes	maximal, complemented
6	#30654222	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550861	C₂²	1	3	yes	complemented
9	#22550860	—	2	3	no	—
10	C₂×Dic₁₀₁₈	C₂²	1	3	yes	complemented
11	Q₈×D₅₀₉	—	2	1	no	—
12	#22550851	C₂²	1	3	yes	complemented
13	#22550849	C₂²	1	3	yes	complemented
14	#22550849	C₂²	1	6	yes	complemented
15	#22550848	C₂²	1	1	yes	characteristic, complemented
16	#22550846	C₂²	1	3	yes	complemented
17	#22550845	C₂²	1	3	yes	complemented
18	C₂×#16861325	C₂²	1	3	yes	complemented
19	#41×C₅₀₉	—	2	1	no	—
20	#41×C₅₀₉	C₂²	1	3	yes	—
21	#41×C₅₀₉	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₅₀₉	—	2	3	no	—
23	C₂×D₄×C₅₀₉	C₂²	1	3	yes	—
24	Dic₁₀₁₈	—	2	3	no	—
25	Dic₁₀₁₈	C₂³	1	3	yes	—
26	D₂₀₃₆	C₂³	1	1	yes	characteristic
27	C₂²⋊D₅₀₉	—	2	3	no	—
28	C₂×Dic₅₀₉	—	2	3	no	—
29	C₄×D₅₀₉	—	2	3	no	—
30	#16861325	C₂³	1	1	yes	characteristic
31	#16861325	C₂³	1	3	yes	—
32	C₂³×C₅₀₉	—	2	3	no	—
33	Q₈×C₅₀₉	D₄	1	1	yes	characteristic
34	Q₈×C₅₀₉	C₂³	1	1	yes	characteristic
35	D₄×C₅₀₉	—	2	3	no	—
36	D₄×C₅₀₉	—	2	3	no	—
37	D₄×C₅₀₉	D₄	1	3	yes	—
38	D₄×C₅₀₉	C₂³	1	3	yes	—
39	C₂×C₄×C₅₀₉	—	2	3	no	—
40	C₂×C₄×C₅₀₉	C₂³	1	3	yes	—
41	C₂²×C₅₀₉	—	2	3	no	—
42	C₂²×C₅₀₉	—	8	3	no	—
43	C₂²×C₅₀₉	C₂×D₄	1	3	yes	—
44	C₂×D₅₀₉	—	2	1	no	—
45	C₄×C₅₀₉	—	2	1	no	—
46	C₄×C₅₀₉	C₂×D₄	1	3	yes	—
47	C₄×C₅₀₉	C₂⁴	1	1	yes	characteristic
48	Dic₅₀₉	—	2	3	no	—
49	C₂×C₅₀₉	—	2	3	no	—
50	C₂×C₅₀₉	—	4	3	no	—
51	C₂×C₅₀₉	C₂²×D₄	1	1	yes	characteristic
52	D₅₀₉	—	4	1	no	—
53	C₅₀₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	509	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	509	1	no	—
56	H(5,𝔽₂)	D₅₀₉	1	1	yes	characteristic, complemented
57	#137	—	509	3	no	—
58	AGL(1,ℤ/8)	—	509	3	no	—
59	#135	—	509	3	no	—
60	C₂×SD₁₆	—	509	3	no	—
61	#131	—	509	1	no	—
62	#41	—	2	1	no	—
63	#41	—	1018	3	no	—
64	#41	—	509	3	no	—
65	#41	C₂×D₅₀₉	1	3	yes	—
66	#41	C₂×D₅₀₉	1	1	yes	characteristic
67	C₂×Q₈	—	1018	1	no	—
68	C₂×Q₈	—	509	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₅₀₉	1	3	yes	—
71	Q₁₆	—	509	3	no	—
72	SD₁₆	—	509	1	no	—
73	SD₁₆	—	509	3	no	—
74	SD₁₆	—	509	6	no	—
75	D₈	—	509	3	no	—
76	M₁₆	—	509	3	no	—
77	C₂×C₈	—	509	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	1018	3	no	—
80	Q₈	—	509	3	no	—
81	Q₈	C₂²⋊D₅₀₉	1	1	yes	characteristic
82	Q₈	C₂²×D₅₀₉	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	1018	3	no	—
86	D₄	—	509	1	no	—
87	D₄	C₂²⋊D₅₀₉	1	3	yes	—
88	D₄	C₂²×D₅₀₉	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	1018	3	no	—
91	C₂×C₄	—	1018	3	no	—
92	C₂×C₄	C₂²×D₅₀₉	1	3	yes	—
93	C₈	—	509	1	no	—
94	C₈	—	509	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	1018	1	no	—
98	C₂²	C₂×C₂²⋊D₅₀₉	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	1018	3	no	—
101	C₄	C₂×C₂²⋊D₅₀₉	1	3	yes	—
102	C₄	C₂³×D₅₀₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	2036	1	no	—
106	C₂	C₂²×C₂²⋊D₅₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30654262

C₂

yes

maximal, characteristic, complemented

#30654245

C₂

yes

maximal, complemented

#30654241

C₂

yes

maximal, complemented

#30654235

C₂

yes

maximal, complemented

C₂×#22550849

C₂

yes

maximal, complemented

#30654222

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₀₉

C₂

yes

maximal, characteristic, complemented

#22550861

C₂²

yes

complemented

#22550860

—

C₂×Dic₁₀₁₈

C₂²

yes

complemented

Q₈×D₅₀₉

—

#22550851

C₂²

yes

complemented

#22550849

C₂²

yes

complemented

#22550849

C₂²

yes

complemented

#22550848

C₂²

yes

characteristic, complemented

#22550846

C₂²

yes

complemented

#22550845

C₂²

yes

complemented

C₂×#16861325

C₂²

yes

complemented

#41×C₅₀₉

—

#41×C₅₀₉

C₂²

yes

—

#41×C₅₀₉

C₂²

yes

characteristic

C₂×D₄×C₅₀₉

—

C₂×D₄×C₅₀₉

C₂²

yes

—

Dic₁₀₁₈

—

Dic₁₀₁₈

C₂³

yes

—

D₂₀₃₆

C₂³

yes

characteristic

C₂²⋊D₅₀₉

—

C₂×Dic₅₀₉

—

C₄×D₅₀₉

—

#16861325

C₂³

yes

characteristic

#16861325

C₂³

yes

—

C₂³×C₅₀₉

—

Q₈×C₅₀₉

D₄

yes

characteristic

Q₈×C₅₀₉

C₂³

yes

characteristic

D₄×C₅₀₉

—

D₄×C₅₀₉

—

D₄×C₅₀₉

D₄

yes

—

D₄×C₅₀₉

C₂³

yes

—

C₂×C₄×C₅₀₉

—

C₂×C₄×C₅₀₉

C₂³

yes

—

C₂²×C₅₀₉

—

C₂²×C₅₀₉

—

C₂²×C₅₀₉

C₂×D₄

yes

—

C₂×D₅₀₉

—

C₄×C₅₀₉

—

C₄×C₅₀₉

C₂×D₄

yes

—

C₄×C₅₀₉

C₂⁴

yes

characteristic

Dic₅₀₉

—

C₂×C₅₀₉

—

C₂×C₅₀₉

—

C₂×C₅₀₉

C₂²×D₄

yes

characteristic

D₅₀₉

—

C₅₀₉

#577

yes

characteristic, complemented, Sylow

#577

—

509

maximal, Sylow

2₋¹⁺⁴

—

509

—

H(5,𝔽₂)

D₅₀₉

yes

characteristic, complemented

#137

—

509

—

AGL(1,ℤ/8)

—

509

—

#135

—

509

—

C₂×SD₁₆

—

509

—

#131

—

509

—

#41

—

#41

—

1018

—

#41

—

509

—

#41

C₂×D₅₀₉

yes

—

#41

C₂×D₅₀₉

yes

characteristic

C₂×Q₈

—

1018

—

C₂×Q₈

—

509

—

C₂×D₄

—

C₂×D₄

C₂×D₅₀₉

yes

—

Q₁₆

—

509

—

SD₁₆

—

509

—

SD₁₆

—

509

—

SD₁₆

—

509

—

D₈

—

509

—

M₁₆

—

509

—

C₂×C₈

—

509

—

C₂³

—

Q₈

—

1018

—

Q₈

—

509

—

Q₈

C₂²⋊D₅₀₉

yes

characteristic

Q₈

C₂²×D₅₀₉

yes

characteristic

D₄

—

D₄

—

D₄

—

1018

—

D₄

—

509

—

D₄

C₂²⋊D₅₀₉

yes

—

D₄

C₂²×D₅₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

1018

—

C₂×C₄

—

1018

—

C₂×C₄

C₂²×D₅₀₉

yes

—

C₈

—

509

—

C₈

—

509

—

C₂²

—

C₂²

—

C₂²

—

1018

—

C₂²

C₂×C₂²⋊D₅₀₉

yes

—

C₄

—

100

C₄

—

1018

—

101

C₄

C₂×C₂²⋊D₅₀₉

yes

—

102

C₄

C₂³×D₅₀₉

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

2036

—

106

C₂

C₂²×C₂²⋊D₅₀₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₅₀₉
commutator, cosocle	C₄×C₅₀₉	C₂⁴
Frattini	C₄	C₂³×D₅₀₉
Fitting	H(5,𝔽₂)×C₅₀₉	C₂
socle	C₂×C₅₀₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₅₀₉
hypocenter	C₅₀₉	#577
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₅₀₉

commutator, cosocle

C₄×C₅₀₉

C₂⁴

Frattini

C₄

C₂³×D₅₀₉

Fitting

H(5,𝔽₂)×C₅₀₉

C₂

socle

C₂×C₅₀₉

C₂²×D₄

D₅₀₉

C₅₀₉

G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #30654262	C₂	semidirect
2. #30654245	C₂	semidirect
3. #30654241	C₂	semidirect
4. #30654235	C₂	semidirect
5. C₂×#22550849	C₂	semidirect
6. #30654222	C₂	semidirect
7. H(5,𝔽₂)×C₅₀₉	C₂	semidirect
8. #22550861	C₂²	semidirect
10. C₂×Dic₁₀₁₈	C₂²	semidirect
12. #22550851	C₂²	semidirect
13. #22550849	C₂²	semidirect
14. #22550849	C₂²	semidirect
15. #22550848	C₂²	semidirect
16. #22550846	C₂²	semidirect
17. #22550845	C₂²	semidirect
18. C₂×#16861325	C₂²	semidirect
53. C₅₀₉	#577	semidirect
56. H(5,𝔽₂)	D₅₀₉	semidirect

H ≅ G/N

type

1. #30654262

C₂

semidirect

2. #30654245

C₂

semidirect

3. #30654241

C₂

semidirect

4. #30654235

C₂

semidirect

5. C₂×#22550849

C₂

semidirect

6. #30654222

C₂

semidirect

7. H(5,𝔽₂)×C₅₀₉

C₂

semidirect

8. #22550861

C₂²

semidirect

10. C₂×Dic₁₀₁₈

C₂²

semidirect

12. #22550851

C₂²

semidirect

13. #22550849

C₂²

semidirect

14. #22550849

C₂²

semidirect

15. #22550848

C₂²

semidirect

16. #22550846

C₂²

semidirect

17. #22550845

C₂²

semidirect

18. C₂×#16861325

C₂²

semidirect

53. C₅₀₉

#577

semidirect

56. H(5,𝔽₂)

D₅₀₉

semidirect

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

subgroup	index
#30654262	2
#30654245	2
#30654241	2
#30654235	2
C₂×#22550849	2
#30654222	2
H(5,𝔽₂)×C₅₀₉	2
#577	509

subgroup

index

H(5,𝔽₂)×C₅₀₉

509

Subgroups — #45656049

Conjugacy classes of subgroups

Complements

Maximal subgroups