Group #44567084 — subgroups

Subgroups — #44567084

Order 32192.

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	#30632910	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632890	C₂	1	3	yes	maximal, complemented
3	#30632889	C₂	1	3	yes	maximal, complemented
4	#30632884	C₂	1	3	yes	maximal, complemented
5	C₂×#22548868	C₂	1	3	yes	maximal, complemented
6	#30632871	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₅₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#22548884	—	2	1	no	—
9	#22548883	C₂²	1	3	yes	complemented
10	C₂×D₂₀₁₂	C₂²	1	3	yes	complemented
11	D₄×D₅₀₃	—	2	3	no	—
12	#22548873	C₂²	1	1	yes	characteristic, complemented
13	#22548871	C₂²	1	3	yes	complemented
14	#22548870	C₂²	1	3	yes	complemented
15	#22548868	C₂²	1	3	yes	complemented
16	#22548868	C₂²	1	6	yes	complemented
17	#22548867	C₂²	1	3	yes	complemented
18	C₂×C₅₀₃⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₅₀₃	—	2	1	no	—
20	#41×C₅₀₃	C₂²	1	1	yes	characteristic
21	#41×C₅₀₃	C₂²	1	3	yes	complemented
22	C₂×D₄×C₅₀₃	—	2	3	no	—
23	C₂×D₄×C₅₀₃	C₂²	1	3	yes	complemented
24	C₂²×D₅₀₃	—	2	3	no	—
25	Dic₁₀₀₆	C₂³	1	1	yes	characteristic
26	D₂₀₁₂	—	2	3	no	—
27	D₂₀₁₂	C₂³	1	3	yes	—
28	C₂²⋊D₅₀₃	—	2	3	no	—
29	C₄×D₅₀₃	—	2	3	no	—
30	C₅₀₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₅₀₃⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₅₀₃	—	2	3	no	—
33	Q₈×C₅₀₃	D₄	1	1	yes	characteristic
34	Q₈×C₅₀₃	C₂³	1	1	yes	characteristic
35	D₄×C₅₀₃	—	2	3	no	—
36	D₄×C₅₀₃	—	2	3	no	—
37	D₄×C₅₀₃	D₄	1	3	yes	—
38	D₄×C₅₀₃	C₂³	1	3	yes	—
39	C₂×C₄×C₅₀₃	—	2	3	no	—
40	C₂×C₄×C₅₀₃	C₂³	1	3	yes	—
41	C₂²×C₅₀₃	—	2	3	no	—
42	C₂²×C₅₀₃	—	8	3	no	—
43	C₂²×C₅₀₃	C₂×D₄	1	3	yes	—
44	C₂×D₅₀₃	—	2	3	no	—
45	C₂×D₅₀₃	—	8	3	no	—
46	C₄×C₅₀₃	—	2	1	no	—
47	C₄×C₅₀₃	C₂×D₄	1	3	yes	—
48	C₄×C₅₀₃	C₂⁴	1	1	yes	characteristic
49	Dic₅₀₃	—	2	1	no	—
50	C₂×C₅₀₃	—	2	3	no	—
51	C₂×C₅₀₃	—	4	3	no	—
52	C₂×C₅₀₃	C₂²×D₄	1	1	yes	characteristic
53	D₅₀₃	—	4	3	no	—
54	C₅₀₃	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	503	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	503	1	no	—
57	H(5,𝔽₂)	D₅₀₃	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	503	3	no	—
59	AGL(1,ℤ/8)	—	503	3	no	—
60	#135	—	503	3	no	—
61	C₂×D₈	—	503	3	no	—
62	#131	—	503	1	no	—
63	#41	—	2	1	no	—
64	#41	—	1006	1	no	—
65	#41	—	503	3	no	—
66	#41	C₂×D₅₀₃	1	1	yes	characteristic
67	#41	C₂×D₅₀₃	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	1006	3	no	—
70	C₂×D₄	—	503	3	no	—
71	C₂×D₄	C₂×D₅₀₃	1	3	yes	complemented
72	Q₁₆	—	503	1	no	—
73	SD₁₆	—	503	3	no	—
74	SD₁₆	—	503	3	no	—
75	D₈	—	503	3	no	—
76	D₈	—	503	6	no	—
77	M₁₆	—	503	3	no	—
78	C₂×C₈	—	503	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	1006	3	no	—
81	Q₈	—	503	1	no	—
82	Q₈	C₂²⋊D₅₀₃	1	1	yes	characteristic
83	Q₈	C₂²×D₅₀₃	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	1006	3	no	—
87	D₄	—	1006	3	no	—
88	D₄	—	503	3	no	—
89	D₄	C₂²⋊D₅₀₃	1	3	yes	—
90	D₄	C₂²×D₅₀₃	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	1006	3	no	—
93	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
94	C₈	—	503	1	no	—
95	C₈	—	503	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	4024	3	no	—
99	C₂²	—	1006	3	no	—
100	C₂²	C₂×C₂²⋊D₅₀₃	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	1006	1	no	—
103	C₄	C₂×C₂²⋊D₅₀₃	1	3	yes	—
104	C₄	C₂³×D₅₀₃	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	2012	3	no	—
108	C₂	C₂²×C₂²⋊D₅₀₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632910

C₂

yes

maximal, characteristic, complemented

#30632890

C₂

yes

maximal, complemented

#30632889

C₂

yes

maximal, complemented

#30632884

C₂

yes

maximal, complemented

C₂×#22548868

C₂

yes

maximal, complemented

#30632871

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₀₃

C₂

yes

maximal, characteristic, complemented

#22548884

—

#22548883

C₂²

yes

complemented

C₂×D₂₀₁₂

C₂²

yes

complemented

D₄×D₅₀₃

—

#22548873

C₂²

yes

characteristic, complemented

#22548871

C₂²

yes

complemented

#22548870

C₂²

yes

complemented

#22548868

C₂²

yes

complemented

#22548868

C₂²

yes

complemented

#22548867

C₂²

yes

complemented

C₂×C₅₀₃⋊C₈

C₂²

yes

complemented

#41×C₅₀₃

—

#41×C₅₀₃

C₂²

yes

characteristic

#41×C₅₀₃

C₂²

yes

complemented

C₂×D₄×C₅₀₃

—

C₂×D₄×C₅₀₃

C₂²

yes

complemented

C₂²×D₅₀₃

—

Dic₁₀₀₆

C₂³

yes

characteristic

D₂₀₁₂

—

D₂₀₁₂

C₂³

yes

—

C₂²⋊D₅₀₃

—

C₄×D₅₀₃

—

C₅₀₃⋊C₈

C₂³

yes

characteristic

C₅₀₃⋊C₈

C₂³

yes

—

C₂³×C₅₀₃

—

Q₈×C₅₀₃

D₄

yes

characteristic

Q₈×C₅₀₃

C₂³

yes

characteristic

D₄×C₅₀₃

—

D₄×C₅₀₃

—

D₄×C₅₀₃

D₄

yes

—

D₄×C₅₀₃

C₂³

yes

—

C₂×C₄×C₅₀₃

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

C₂×D₄

yes

—

C₂×D₅₀₃

—

C₂×D₅₀₃

—

C₄×C₅₀₃

—

C₄×C₅₀₃

C₂×D₄

yes

—

C₄×C₅₀₃

C₂⁴

yes

characteristic

Dic₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

C₂²×D₄

yes

characteristic

D₅₀₃

—

C₅₀₃

#576

yes

characteristic, complemented, Sylow

#576

—

503

maximal, Sylow

H(5,𝔽₂)

—

503

—

H(5,𝔽₂)

D₅₀₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

503

—

AGL(1,ℤ/8)

—

503

—

#135

—

503

—

C₂×D₈

—

503

—

#131

—

503

—

#41

—

#41

—

1006

—

#41

—

503

—

#41

C₂×D₅₀₃

yes

characteristic

#41

C₂×D₅₀₃

yes

complemented

C₂×D₄

—

C₂×D₄

—

1006

—

C₂×D₄

—

503

—

C₂×D₄

C₂×D₅₀₃

yes

complemented

Q₁₆

—

503

—

SD₁₆

—

503

—

SD₁₆

—

503

—

D₈

—

503

—

D₈

—

503

—

M₁₆

—

503

—

C₂×C₈

—

503

—

C₂³

—

C₂³

—

1006

—

Q₈

—

503

—

Q₈

C₂²⋊D₅₀₃

yes

characteristic

Q₈

C₂²×D₅₀₃

yes

characteristic

D₄

—

D₄

—

D₄

—

1006

—

D₄

—

1006

—

D₄

—

503

—

D₄

C₂²⋊D₅₀₃

yes

—

D₄

C₂²×D₅₀₃

yes

—

C₂×C₄

—

C₂×C₄

—

1006

—

C₂×C₄

C₂²×D₅₀₃

yes

—

C₈

—

503

—

C₈

—

503

—

C₂²

—

C₂²

—

C₂²

—

4024

—

C₂²

—

1006

—

100

C₂²

C₂×C₂²⋊D₅₀₃

yes

—

101

C₄

—

102

C₄

—

1006

—

103

C₄

C₂×C₂²⋊D₅₀₃

yes

—

104

C₄

C₂³×D₅₀₃

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

2012

—

108

C₂

C₂²×C₂²⋊D₅₀₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₅₀₃
commutator, cosocle	C₄×C₅₀₃	C₂⁴
Frattini	C₄	C₂³×D₅₀₃
Fitting	H(5,𝔽₂)×C₅₀₃	C₂
socle	C₂×C₅₀₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₅₀₃
hypocenter	C₅₀₃	#576
LCS	G ▷ C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₅₀₃

commutator, cosocle

C₄×C₅₀₃

C₂⁴

Frattini

C₄

C₂³×D₅₀₃

Fitting

H(5,𝔽₂)×C₅₀₃

C₂

socle

C₂×C₅₀₃

C₂²×D₄

D₅₀₃

C₅₀₃

G ▷ C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #30632910	C₂	semidirect
2. #30632890	C₂	semidirect
3. #30632889	C₂	semidirect
4. #30632884	C₂	semidirect
5. C₂×#22548868	C₂	semidirect
6. #30632871	C₂	semidirect
7. H(5,𝔽₂)×C₅₀₃	C₂	semidirect
9. #22548883	C₂²	semidirect
10. C₂×D₂₀₁₂	C₂²	semidirect
12. #22548873	C₂²	semidirect
13. #22548871	C₂²	semidirect
14. #22548870	C₂²	semidirect
15. #22548868	C₂²	semidirect
16. #22548868	C₂²	semidirect
17. #22548867	C₂²	semidirect
18. C₂×C₅₀₃⋊C₈	C₂²	semidirect
21. #41×C₅₀₃	C₂²	semidirect
23. C₂×D₄×C₅₀₃	C₂²	semidirect
54. C₅₀₃	#576	semidirect
57. H(5,𝔽₂)	D₅₀₃	semidirect
67. #41	C₂×D₅₀₃	semidirect
71. C₂×D₄	C₂×D₅₀₃	semidirect

H ≅ G/N

type

1. #30632910

C₂

semidirect

2. #30632890

C₂

semidirect

3. #30632889

C₂

semidirect

4. #30632884

C₂

semidirect

5. C₂×#22548868

C₂

semidirect

6. #30632871

C₂

semidirect

7. H(5,𝔽₂)×C₅₀₃

C₂

semidirect

9. #22548883

C₂²

semidirect

10. C₂×D₂₀₁₂

C₂²

semidirect

12. #22548873

C₂²

semidirect

13. #22548871

C₂²

semidirect

14. #22548870

C₂²

semidirect

15. #22548868

C₂²

semidirect

16. #22548868

C₂²

semidirect

17. #22548867

C₂²

semidirect

18. C₂×C₅₀₃⋊C₈

C₂²

semidirect

21. #41×C₅₀₃

C₂²

semidirect

23. C₂×D₄×C₅₀₃

C₂²

semidirect

54. C₅₀₃

#576

semidirect

57. H(5,𝔽₂)

D₅₀₃

semidirect

67. #41

C₂×D₅₀₃

semidirect

71. C₂×D₄

C₂×D₅₀₃

semidirect

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

subgroup	index
#30632910	2
#30632890	2
#30632889	2
#30632884	2
C₂×#22548868	2
#30632871	2
H(5,𝔽₂)×C₅₀₃	2
#576	503

subgroup

index

H(5,𝔽₂)×C₅₀₃

503

Subgroups — #44567084

Conjugacy classes of subgroups

Complements

Maximal subgroups