Group #45688244 — subgroups

Subgroups — #45688244

Order 33344.

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	#30659755	C₂	1	1	yes	maximal, characteristic, complemented
2	#30659754	C₂	1	1	yes	maximal, characteristic, complemented
3	#30659741	C₂	1	3	yes	maximal, complemented
4	#30659736	C₂	1	3	yes	maximal, complemented
5	#30659730	C₂	1	3	yes	maximal, complemented
6	C₂×#22552353	C₂	1	3	yes	maximal, complemented
7	#131×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#22552365	—	2	1	no	—
9	#22552364	C₂²	1	3	yes	—
10	#22552364	C₂²	1	3	yes	complemented
11	#22552363	—	2	3	no	—
12	C₂×Dic₁₀₄₂	C₂²	1	3	yes	complemented
13	Q₈×D₅₂₁	—	2	1	no	—
14	C₂×D₂₀₈₄	C₂²	1	3	yes	—
15	D₄×D₅₂₁	—	2	3	no	—
16	Dic₂₀₈₄	C₂²	1	3	yes	complemented
17	#22552353	C₂²	1	3	yes	complemented
18	#22552353	C₂²	1	6	yes	complemented
19	#22552353	C₂²	1	1	yes	characteristic, complemented
20	D₄₁₆₈	C₂²	1	3	yes	complemented
21	#41×C₅₂₁	C₂²	1	1	yes	characteristic
22	M₁₆×C₅₂₁	C₂²	1	3	yes	complemented
23	C₂×C₈×C₅₂₁	C₂²	1	3	yes	complemented
24	C₂²×D₅₂₁	—	2	3	no	—
25	Dic₁₀₄₂	—	2	3	no	—
26	Dic₁₀₄₂	C₂³	1	3	yes	—
27	Dic₁₀₄₂	C₂³	1	1	yes	characteristic
28	D₂₀₈₄	—	2	3	no	—
29	D₂₀₈₄	C₂³	1	1	yes	characteristic
30	D₂₀₈₄	C₂³	1	3	yes	—
31	C₂²⋊D₅₂₁	—	2	3	no	—
32	C₂²⋊D₅₂₁	—	2	3	no	—
33	C₂×Dic₅₂₁	—	2	3	no	—
34	C₄×D₅₂₁	—	2	3	no	—
35	C₄×D₅₂₁	—	2	3	no	—
36	Q₈×C₅₂₁	D₄	1	1	yes	characteristic
37	D₄×C₅₂₁	D₄	1	3	yes	—
38	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
39	C₈×C₅₂₁	C₂³	1	1	yes	characteristic
40	C₈×C₅₂₁	C₂³	1	3	yes	—
41	C₂²×C₅₂₁	C₂×D₄	1	3	yes	—
42	C₂×D₅₂₁	—	2	1	no	—
43	C₂×D₅₂₁	—	2	3	no	—
44	C₂×D₅₂₁	—	8	3	no	—
45	C₄×C₅₂₁	C₂×D₄	1	3	yes	—
46	C₄×C₅₂₁	C₂⁴	1	1	yes	characteristic
47	Dic₅₂₁	—	2	1	no	—
48	Dic₅₂₁	—	2	3	no	—
49	C₂×C₅₂₁	—	2	3	no	—
50	C₂×C₅₂₁	C₂²×D₄	1	1	yes	characteristic
51	D₅₂₁	—	4	1	no	—
52	D₅₂₁	—	4	3	no	—
53	C₅₂₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	521	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	521	1	no	—
56	H(5,𝔽₂)	—	521	1	no	—
57	#137	—	521	3	no	—
58	AGL(1,ℤ/8)	—	521	3	no	—
59	#135	—	521	3	no	—
60	C₂×SD₁₆	—	521	3	no	—
61	#131	D₅₂₁	1	1	yes	characteristic, complemented
62	#41	—	521	3	no	—
63	#41	—	521	3	no	—
64	#41	—	1042	1	no	—
65	#41	—	1042	3	no	—
66	#41	C₂×D₅₂₁	1	1	yes	characteristic
67	C₂×Q₈	—	521	3	no	—
68	C₂×Q₈	—	1042	1	no	—
69	C₂×D₄	—	521	3	no	—
70	C₂×D₄	—	1042	3	no	—
71	Q₁₆	—	521	3	no	—
72	SD₁₆	—	521	3	no	—
73	SD₁₆	—	521	6	no	—
74	SD₁₆	—	521	1	no	—
75	D₈	—	521	3	no	—
76	M₁₆	C₂×D₅₂₁	1	3	yes	complemented
77	C₂×C₈	C₂×D₅₂₁	1	3	yes	complemented
78	C₂³	—	1042	3	no	—
79	Q₈	—	521	3	no	—
80	Q₈	—	521	1	no	—
81	Q₈	—	1042	3	no	—
82	Q₈	D₂₀₈₄	1	1	yes	characteristic
83	D₄	—	521	1	no	—
84	D₄	—	521	3	no	—
85	D₄	—	1042	3	no	—
86	D₄	—	1042	3	no	—
87	D₄	—	1042	3	no	—
88	D₄	D₂₀₈₄	1	3	yes	—
89	C₂×C₄	—	1042	3	no	—
90	C₂×C₄	—	1042	3	no	—
91	C₂×C₄	—	1042	3	no	—
92	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
93	C₈	C₂²×D₅₂₁	1	1	yes	characteristic
94	C₈	C₂²×D₅₂₁	1	3	yes	—
95	C₂²	—	1042	1	no	—
96	C₂²	—	1042	3	no	—
97	C₂²	—	4168	3	no	—
98	C₂²	C₂×D₂₀₈₄	1	3	yes	—
99	C₄	—	1042	1	no	—
100	C₄	—	1042	3	no	—
101	C₄	C₂×D₂₀₈₄	1	3	yes	—
102	C₄	C₂³×D₅₂₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	2084	3	no	—
105	C₂	—	2084	1	no	—
106	C₂	C₂²×D₂₀₈₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659755

C₂

yes

maximal, characteristic, complemented

#30659754

C₂

yes

maximal, characteristic, complemented

#30659741

C₂

yes

maximal, complemented

#30659736

C₂

yes

maximal, complemented

#30659730

C₂

yes

maximal, complemented

C₂×#22552353

C₂

yes

maximal, complemented

#131×C₅₂₁

C₂

yes

maximal, characteristic, complemented

#22552365

—

#22552364

C₂²

yes

—

#22552364

C₂²

yes

complemented

#22552363

—

C₂×Dic₁₀₄₂

C₂²

yes

complemented

Q₈×D₅₂₁

—

C₂×D₂₀₈₄

C₂²

yes

—

D₄×D₅₂₁

—

Dic₂₀₈₄

C₂²

yes

complemented

#22552353

C₂²

yes

complemented

#22552353

C₂²

yes

complemented

#22552353

C₂²

yes

characteristic, complemented

D₄₁₆₈

C₂²

yes

complemented

#41×C₅₂₁

C₂²

yes

characteristic

M₁₆×C₅₂₁

C₂²

yes

complemented

C₂×C₈×C₅₂₁

C₂²

yes

complemented

C₂²×D₅₂₁

—

Dic₁₀₄₂

—

Dic₁₀₄₂

C₂³

yes

—

Dic₁₀₄₂

C₂³

yes

characteristic

D₂₀₈₄

—

D₂₀₈₄

C₂³

yes

characteristic

D₂₀₈₄

C₂³

yes

—

C₂²⋊D₅₂₁

—

C₂²⋊D₅₂₁

—

C₂×Dic₅₂₁

—

C₄×D₅₂₁

—

C₄×D₅₂₁

—

Q₈×C₅₂₁

D₄

yes

characteristic

D₄×C₅₂₁

D₄

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₈×C₅₂₁

C₂³

yes

characteristic

C₈×C₅₂₁

C₂³

yes

—

C₂²×C₅₂₁

C₂×D₄

yes

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₄×C₅₂₁

C₂×D₄

yes

—

C₄×C₅₂₁

C₂⁴

yes

characteristic

Dic₅₂₁

—

Dic₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

C₂²×D₄

yes

characteristic

D₅₂₁

—

D₅₂₁

—

C₅₂₁

#577

yes

characteristic, complemented, Sylow

#577

—

521

maximal, Sylow

2₋¹⁺⁴

—

521

—

H(5,𝔽₂)

—

521

—

#137

—

521

—

AGL(1,ℤ/8)

—

521

—

#135

—

521

—

C₂×SD₁₆

—

521

—

#131

D₅₂₁

yes

characteristic, complemented

#41

—

521

—

#41

—

521

—

#41

—

1042

—

#41

—

1042

—

#41

C₂×D₅₂₁

yes

characteristic

C₂×Q₈

—

521

—

C₂×Q₈

—

1042

—

C₂×D₄

—

521

—

C₂×D₄

—

1042

—

Q₁₆

—

521

—

SD₁₆

—

521

—

SD₁₆

—

521

—

SD₁₆

—

521

—

D₈

—

521

—

M₁₆

C₂×D₅₂₁

yes

complemented

C₂×C₈

C₂×D₅₂₁

yes

complemented

C₂³

—

1042

—

Q₈

—

521

—

Q₈

—

521

—

Q₈

—

1042

—

Q₈

D₂₀₈₄

yes

characteristic

D₄

—

521

—

D₄

—

521

—

D₄

—

1042

—

D₄

—

1042

—

D₄

—

1042

—

D₄

D₂₀₈₄

yes

—

C₂×C₄

—

1042

—

C₂×C₄

—

1042

—

C₂×C₄

—

1042

—

C₂×C₄

C₂²×D₅₂₁

yes

—

C₈

C₂²×D₅₂₁

yes

characteristic

C₈

C₂²×D₅₂₁

yes

—

C₂²

—

1042

—

C₂²

—

1042

—

C₂²

—

4168

—

C₂²

C₂×D₂₀₈₄

yes

—

C₄

—

1042

—

100

C₄

—

1042

—

101

C₄

C₂×D₂₀₈₄

yes

—

102

C₄

C₂³×D₅₂₁

yes

characteristic

103

C₂

—

104

C₂

—

2084

—

105

C₂

—

2084

—

106

C₂

C₂²×D₂₀₈₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₀₈₄
commutator, cosocle	C₄×C₅₂₁	C₂⁴
Frattini	C₄	C₂³×D₅₂₁
Fitting	#131×C₅₂₁	C₂
socle	C₂×C₅₂₁	C₂²×D₄
hypercenter	#131	D₅₂₁
hypocenter	C₅₂₁	#577
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₂₀₈₄

commutator, cosocle

C₄×C₅₂₁

C₂⁴

Frattini

C₄

C₂³×D₅₂₁

Fitting

#131×C₅₂₁

C₂

socle

C₂×C₅₂₁

C₂²×D₄

D₅₂₁

C₅₂₁

G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #30659755	C₂	semidirect
2. #30659754	C₂	semidirect
3. #30659741	C₂	semidirect
4. #30659736	C₂	semidirect
5. #30659730	C₂	semidirect
6. C₂×#22552353	C₂	semidirect
7. #131×C₅₂₁	C₂	semidirect
10. #22552364	C₂²	semidirect
12. C₂×Dic₁₀₄₂	C₂²	semidirect
16. Dic₂₀₈₄	C₂²	semidirect
17. #22552353	C₂²	semidirect
18. #22552353	C₂²	semidirect
19. #22552353	C₂²	semidirect
20. D₄₁₆₈	C₂²	semidirect
22. M₁₆×C₅₂₁	C₂²	semidirect
23. C₂×C₈×C₅₂₁	C₂²	semidirect
53. C₅₂₁	#577	semidirect
61. #131	D₅₂₁	semidirect
76. M₁₆	C₂×D₅₂₁	semidirect
77. C₂×C₈	C₂×D₅₂₁	semidirect

H ≅ G/N

type

1. #30659755

C₂

semidirect

2. #30659754

C₂

semidirect

3. #30659741

C₂

semidirect

4. #30659736

C₂

semidirect

5. #30659730

C₂

semidirect

6. C₂×#22552353

C₂

semidirect

7. #131×C₅₂₁

C₂

semidirect

10. #22552364

C₂²

semidirect

12. C₂×Dic₁₀₄₂

C₂²

semidirect

16. Dic₂₀₈₄

C₂²

semidirect

17. #22552353

C₂²

semidirect

18. #22552353

C₂²

semidirect

19. #22552353

C₂²

semidirect

20. D₄₁₆₈

C₂²

semidirect

22. M₁₆×C₅₂₁

C₂²

semidirect

23. C₂×C₈×C₅₂₁

C₂²

semidirect

53. C₅₂₁

#577

semidirect

61. #131

D₅₂₁

semidirect

76. M₁₆

C₂×D₅₂₁

semidirect

77. C₂×C₈

C₂×D₅₂₁

semidirect

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

subgroup	index
#30659755	2
#30659754	2
#30659741	2
#30659736	2
#30659730	2
C₂×#22552353	2
#131×C₅₂₁	2
#577	521

subgroup

index

#131×C₅₂₁

521

Subgroups — #45688244

Conjugacy classes of subgroups

Complements

Maximal subgroups