Group #44567092 — subgroups

Subgroups — #44567092

Order 32192.

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	#30632911	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632894	C₂	1	3	yes	maximal, complemented
3	#30632890	C₂	1	3	yes	maximal, complemented
4	#30632884	C₂	1	3	yes	maximal, complemented
5	C₂×#22548871	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	#22548883	C₂²	1	3	yes	complemented
9	#22548882	—	2	3	no	—
10	C₂×Dic₁₀₀₆	C₂²	1	3	yes	complemented
11	Q₈×D₅₀₃	—	2	1	no	—
12	#22548873	C₂²	1	3	yes	complemented
13	#22548871	C₂²	1	3	yes	complemented
14	#22548871	C₂²	1	6	yes	complemented
15	#22548870	C₂²	1	1	yes	characteristic, complemented
16	#22548868	C₂²	1	3	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	3	yes	—
21	#41×C₅₀₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₅₀₃	—	2	3	no	—
23	C₂×D₄×C₅₀₃	C₂²	1	3	yes	—
24	Dic₁₀₀₆	—	2	3	no	—
25	Dic₁₀₀₆	C₂³	1	3	yes	—
26	D₂₀₁₂	C₂³	1	1	yes	characteristic
27	C₂²⋊D₅₀₃	—	2	3	no	—
28	C₂×Dic₅₀₃	—	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	1	no	—
45	C₄×C₅₀₃	—	2	1	no	—
46	C₄×C₅₀₃	C₂×D₄	1	3	yes	—
47	C₄×C₅₀₃	C₂⁴	1	1	yes	characteristic
48	Dic₅₀₃	—	2	3	no	—
49	C₂×C₅₀₃	—	2	3	no	—
50	C₂×C₅₀₃	—	4	3	no	—
51	C₂×C₅₀₃	C₂²×D₄	1	1	yes	characteristic
52	D₅₀₃	—	4	1	no	—
53	C₅₀₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	503	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	503	1	no	—
56	H(5,𝔽₂)	D₅₀₃	1	1	yes	characteristic, complemented
57	#137	—	503	3	no	—
58	AGL(1,ℤ/8)	—	503	3	no	—
59	#135	—	503	3	no	—
60	C₂×SD₁₆	—	503	3	no	—
61	#131	—	503	1	no	—
62	#41	—	2	1	no	—
63	#41	—	1006	3	no	—
64	#41	—	503	3	no	—
65	#41	C₂×D₅₀₃	1	3	yes	—
66	#41	C₂×D₅₀₃	1	1	yes	characteristic
67	C₂×Q₈	—	1006	1	no	—
68	C₂×Q₈	—	503	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₅₀₃	1	3	yes	—
71	Q₁₆	—	503	3	no	—
72	SD₁₆	—	503	1	no	—
73	SD₁₆	—	503	3	no	—
74	SD₁₆	—	503	6	no	—
75	D₈	—	503	3	no	—
76	M₁₆	—	503	3	no	—
77	C₂×C₈	—	503	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	1006	3	no	—
80	Q₈	—	503	3	no	—
81	Q₈	C₂²⋊D₅₀₃	1	1	yes	characteristic
82	Q₈	C₂²×D₅₀₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	1006	3	no	—
86	D₄	—	503	1	no	—
87	D₄	C₂²⋊D₅₀₃	1	3	yes	—
88	D₄	C₂²×D₅₀₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	1006	3	no	—
91	C₂×C₄	—	1006	3	no	—
92	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
93	C₈	—	503	1	no	—
94	C₈	—	503	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	1006	1	no	—
98	C₂²	C₂×C₂²⋊D₅₀₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	1006	3	no	—
101	C₄	C₂×C₂²⋊D₅₀₃	1	3	yes	—
102	C₄	C₂³×D₅₀₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	2012	1	no	—
106	C₂	C₂²×C₂²⋊D₅₀₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632911

C₂

yes

maximal, characteristic, complemented

#30632894

C₂

yes

maximal, complemented

#30632890

C₂

yes

maximal, complemented

#30632884

C₂

yes

maximal, complemented

C₂×#22548871

C₂

yes

maximal, complemented

#30632871

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₀₃

C₂

yes

maximal, characteristic, complemented

#22548883

C₂²

yes

complemented

#22548882

—

C₂×Dic₁₀₀₆

C₂²

yes

complemented

Q₈×D₅₀₃

—

#22548873

C₂²

yes

complemented

#22548871

C₂²

yes

complemented

#22548871

C₂²

yes

complemented

#22548870

C₂²

yes

characteristic, complemented

#22548868

C₂²

yes

complemented

#22548867

C₂²

yes

complemented

C₂×C₅₀₃⋊C₈

C₂²

yes

complemented

#41×C₅₀₃

—

#41×C₅₀₃

C₂²

yes

—

#41×C₅₀₃

C₂²

yes

characteristic

C₂×D₄×C₅₀₃

—

C₂×D₄×C₅₀₃

C₂²

yes

—

Dic₁₀₀₆

—

Dic₁₀₀₆

C₂³

yes

—

D₂₀₁₂

C₂³

yes

characteristic

C₂²⋊D₅₀₃

—

C₂×Dic₅₀₃

—

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

—

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

#577

yes

characteristic, complemented, Sylow

#577

—

503

maximal, Sylow

2₋¹⁺⁴

—

503

—

H(5,𝔽₂)

D₅₀₃

yes

characteristic, complemented

#137

—

503

—

AGL(1,ℤ/8)

—

503

—

#135

—

503

—

C₂×SD₁₆

—

503

—

#131

—

503

—

#41

—

#41

—

1006

—

#41

—

503

—

#41

C₂×D₅₀₃

yes

—

#41

C₂×D₅₀₃

yes

characteristic

C₂×Q₈

—

1006

—

C₂×Q₈

—

503

—

C₂×D₄

—

C₂×D₄

C₂×D₅₀₃

yes

—

Q₁₆

—

503

—

SD₁₆

—

503

—

SD₁₆

—

503

—

SD₁₆

—

503

—

D₈

—

503

—

M₁₆

—

503

—

C₂×C₈

—

503

—

C₂³

—

Q₈

—

1006

—

Q₈

—

503

—

Q₈

C₂²⋊D₅₀₃

yes

characteristic

Q₈

C₂²×D₅₀₃

yes

characteristic

D₄

—

D₄

—

D₄

—

1006

—

D₄

—

503

—

D₄

C₂²⋊D₅₀₃

yes

—

D₄

C₂²×D₅₀₃

yes

—

C₂×C₄

—

C₂×C₄

—

1006

—

C₂×C₄

—

1006

—

C₂×C₄

C₂²×D₅₀₃

yes

—

C₈

—

503

—

C₈

—

503

—

C₂²

—

C₂²

—

C₂²

—

1006

—

C₂²

C₂×C₂²⋊D₅₀₃

yes

—

C₄

—

100

C₄

—

1006

—

101

C₄

C₂×C₂²⋊D₅₀₃

yes

—

102

C₄

C₂³×D₅₀₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

2012

—

106

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	#577
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. #30632911	C₂	semidirect
2. #30632894	C₂	semidirect
3. #30632890	C₂	semidirect
4. #30632884	C₂	semidirect
5. C₂×#22548871	C₂	semidirect
6. #30632871	C₂	semidirect
7. H(5,𝔽₂)×C₅₀₃	C₂	semidirect
8. #22548883	C₂²	semidirect
10. C₂×Dic₁₀₀₆	C₂²	semidirect
12. #22548873	C₂²	semidirect
13. #22548871	C₂²	semidirect
14. #22548871	C₂²	semidirect
15. #22548870	C₂²	semidirect
16. #22548868	C₂²	semidirect
17. #22548867	C₂²	semidirect
18. C₂×C₅₀₃⋊C₈	C₂²	semidirect
53. C₅₀₃	#577	semidirect
56. H(5,𝔽₂)	D₅₀₃	semidirect

H ≅ G/N

type

1. #30632911

C₂

semidirect

2. #30632894

C₂

semidirect

3. #30632890

C₂

semidirect

4. #30632884

C₂

semidirect

5. C₂×#22548871

C₂

semidirect

6. #30632871

C₂

semidirect

7. H(5,𝔽₂)×C₅₀₃

C₂

semidirect

8. #22548883

C₂²

semidirect

10. C₂×Dic₁₀₀₆

C₂²

semidirect

12. #22548873

C₂²

semidirect

13. #22548871

C₂²

semidirect

14. #22548871

C₂²

semidirect

15. #22548870

C₂²

semidirect

16. #22548868

C₂²

semidirect

17. #22548867

C₂²

semidirect

18. C₂×C₅₀₃⋊C₈

C₂²

semidirect

53. C₅₀₃

#577

semidirect

56. H(5,𝔽₂)

D₅₀₃

semidirect

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

subgroup	index
#30632911	2
#30632894	2
#30632890	2
#30632884	2
C₂×#22548871	2
#30632871	2
H(5,𝔽₂)×C₅₀₃	2
#577	503

subgroup

index

H(5,𝔽₂)×C₅₀₃

503

Subgroups — #44567092

Conjugacy classes of subgroups

Complements

Maximal subgroups