Group #45688235 — subgroups

Subgroups — #45688235

Order 33344.

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	#30659754	C₂	1	1	yes	maximal, characteristic, complemented
2	#30659734	C₂	1	3	yes	maximal, complemented
3	#30659733	C₂	1	3	yes	maximal, complemented
4	#30659728	C₂	1	3	yes	maximal, complemented
5	C₂×#22552349	C₂	1	3	yes	maximal, complemented
6	#30659715	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#22552365	—	2	1	no	—
9	#22552364	C₂²	1	3	yes	complemented
10	C₂×D₂₀₈₄	C₂²	1	3	yes	complemented
11	D₄×D₅₂₁	—	2	3	no	—
12	#22552354	C₂²	1	1	yes	characteristic, complemented
13	#22552352	C₂²	1	3	yes	complemented
14	#22552351	C₂²	1	3	yes	complemented
15	#22552349	C₂²	1	3	yes	complemented
16	#22552349	C₂²	1	6	yes	complemented
17	#22552348	C₂²	1	3	yes	complemented
18	C₂×#16861960	C₂²	1	3	yes	complemented
19	#41×C₅₂₁	—	2	1	no	—
20	#41×C₅₂₁	C₂²	1	1	yes	characteristic
21	#41×C₅₂₁	C₂²	1	3	yes	complemented
22	C₂×D₄×C₅₂₁	—	2	3	no	—
23	C₂×D₄×C₅₂₁	C₂²	1	3	yes	complemented
24	C₂²×D₅₂₁	—	2	3	no	—
25	Dic₁₀₄₂	C₂³	1	1	yes	characteristic
26	D₂₀₈₄	—	2	3	no	—
27	D₂₀₈₄	C₂³	1	3	yes	—
28	C₂²⋊D₅₂₁	—	2	3	no	—
29	C₄×D₅₂₁	—	2	3	no	—
30	#16861960	C₂³	1	1	yes	characteristic
31	#16861960	C₂³	1	3	yes	—
32	C₂³×C₅₂₁	—	2	3	no	—
33	Q₈×C₅₂₁	D₄	1	1	yes	characteristic
34	Q₈×C₅₂₁	C₂³	1	1	yes	characteristic
35	D₄×C₅₂₁	—	2	3	no	—
36	D₄×C₅₂₁	—	2	3	no	—
37	D₄×C₅₂₁	D₄	1	3	yes	—
38	D₄×C₅₂₁	C₂³	1	3	yes	—
39	C₂×C₄×C₅₂₁	—	2	3	no	—
40	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
41	C₂²×C₅₂₁	—	2	3	no	—
42	C₂²×C₅₂₁	—	8	3	no	—
43	C₂²×C₅₂₁	C₂×D₄	1	3	yes	—
44	C₂×D₅₂₁	—	2	3	no	—
45	C₂×D₅₂₁	—	8	3	no	—
46	C₄×C₅₂₁	—	2	1	no	—
47	C₄×C₅₂₁	C₂×D₄	1	3	yes	—
48	C₄×C₅₂₁	C₂⁴	1	1	yes	characteristic
49	Dic₅₂₁	—	2	1	no	—
50	C₂×C₅₂₁	—	2	3	no	—
51	C₂×C₅₂₁	—	4	3	no	—
52	C₂×C₅₂₁	C₂²×D₄	1	1	yes	characteristic
53	D₅₂₁	—	4	3	no	—
54	C₅₂₁	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	521	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	521	1	no	—
57	H(5,𝔽₂)	D₅₂₁	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	521	3	no	—
59	AGL(1,ℤ/8)	—	521	3	no	—
60	#135	—	521	3	no	—
61	C₂×D₈	—	521	3	no	—
62	#131	—	521	1	no	—
63	#41	—	2	1	no	—
64	#41	—	521	3	no	—
65	#41	—	1042	1	no	—
66	#41	C₂×D₅₂₁	1	1	yes	characteristic
67	#41	C₂×D₅₂₁	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	521	3	no	—
70	C₂×D₄	—	1042	3	no	—
71	C₂×D₄	C₂×D₅₂₁	1	3	yes	complemented
72	Q₁₆	—	521	1	no	—
73	SD₁₆	—	521	3	no	—
74	SD₁₆	—	521	3	no	—
75	D₈	—	521	3	no	—
76	D₈	—	521	6	no	—
77	M₁₆	—	521	3	no	—
78	C₂×C₈	—	521	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	1042	3	no	—
81	Q₈	—	521	1	no	—
82	Q₈	C₂²⋊D₅₂₁	1	1	yes	characteristic
83	Q₈	C₂²×D₅₂₁	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	521	3	no	—
87	D₄	—	1042	3	no	—
88	D₄	—	1042	3	no	—
89	D₄	C₂²⋊D₅₂₁	1	3	yes	—
90	D₄	C₂²×D₅₂₁	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	1042	3	no	—
93	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
94	C₈	—	521	1	no	—
95	C₈	—	521	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	1042	3	no	—
99	C₂²	—	4168	3	no	—
100	C₂²	C₂×C₂²⋊D₅₂₁	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	1042	1	no	—
103	C₄	C₂×C₂²⋊D₅₂₁	1	3	yes	—
104	C₄	C₂³×D₅₂₁	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	2084	3	no	—
108	C₂	C₂²×C₂²⋊D₅₂₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659754

C₂

yes

maximal, characteristic, complemented

#30659734

C₂

yes

maximal, complemented

#30659733

C₂

yes

maximal, complemented

#30659728

C₂

yes

maximal, complemented

C₂×#22552349

C₂

yes

maximal, complemented

#30659715

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₂₁

C₂

yes

maximal, characteristic, complemented

#22552365

—

#22552364

C₂²

yes

complemented

C₂×D₂₀₈₄

C₂²

yes

complemented

D₄×D₅₂₁

—

#22552354

C₂²

yes

characteristic, complemented

#22552352

C₂²

yes

complemented

#22552351

C₂²

yes

complemented

#22552349

C₂²

yes

complemented

#22552349

C₂²

yes

complemented

#22552348

C₂²

yes

complemented

C₂×#16861960

C₂²

yes

complemented

#41×C₅₂₁

—

#41×C₅₂₁

C₂²

yes

characteristic

#41×C₅₂₁

C₂²

yes

complemented

C₂×D₄×C₅₂₁

—

C₂×D₄×C₅₂₁

C₂²

yes

complemented

C₂²×D₅₂₁

—

Dic₁₀₄₂

C₂³

yes

characteristic

D₂₀₈₄

—

D₂₀₈₄

C₂³

yes

—

C₂²⋊D₅₂₁

—

C₄×D₅₂₁

—

#16861960

C₂³

yes

characteristic

#16861960

C₂³

yes

—

C₂³×C₅₂₁

—

Q₈×C₅₂₁

D₄

yes

characteristic

Q₈×C₅₂₁

C₂³

yes

characteristic

D₄×C₅₂₁

—

D₄×C₅₂₁

—

D₄×C₅₂₁

D₄

yes

—

D₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂×D₄

yes

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

C₂×D₄

yes

—

C₄×C₅₂₁

C₂⁴

yes

characteristic

Dic₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

C₂²×D₄

yes

characteristic

D₅₂₁

—

C₅₂₁

#576

yes

characteristic, complemented, Sylow

#576

—

521

maximal, Sylow

H(5,𝔽₂)

—

521

—

H(5,𝔽₂)

D₅₂₁

yes

characteristic, complemented

AGL(1,ℤ/8)

—

521

—

AGL(1,ℤ/8)

—

521

—

#135

—

521

—

C₂×D₈

—

521

—

#131

—

521

—

#41

—

#41

—

521

—

#41

—

1042

—

#41

C₂×D₅₂₁

yes

characteristic

#41

C₂×D₅₂₁

yes

complemented

C₂×D₄

—

C₂×D₄

—

521

—

C₂×D₄

—

1042

—

C₂×D₄

C₂×D₅₂₁

yes

complemented

Q₁₆

—

521

—

SD₁₆

—

521

—

SD₁₆

—

521

—

D₈

—

521

—

D₈

—

521

—

M₁₆

—

521

—

C₂×C₈

—

521

—

C₂³

—

C₂³

—

1042

—

Q₈

—

521

—

Q₈

C₂²⋊D₅₂₁

yes

characteristic

Q₈

C₂²×D₅₂₁

yes

characteristic

D₄

—

D₄

—

D₄

—

521

—

D₄

—

1042

—

D₄

—

1042

—

D₄

C₂²⋊D₅₂₁

yes

—

D₄

C₂²×D₅₂₁

yes

—

C₂×C₄

—

C₂×C₄

—

1042

—

C₂×C₄

C₂²×D₅₂₁

yes

—

C₈

—

521

—

C₈

—

521

—

C₂²

—

C₂²

—

C₂²

—

1042

—

C₂²

—

4168

—

100

C₂²

C₂×C₂²⋊D₅₂₁

yes

—

101

C₄

—

102

C₄

—

1042

—

103

C₄

C₂×C₂²⋊D₅₂₁

yes

—

104

C₄

C₂³×D₅₂₁

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

2084

—

108

C₂

C₂²×C₂²⋊D₅₂₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₅₂₁
commutator, cosocle	C₄×C₅₂₁	C₂⁴
Frattini	C₄	C₂³×D₅₂₁
Fitting	H(5,𝔽₂)×C₅₂₁	C₂
socle	C₂×C₅₂₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₅₂₁
hypocenter	C₅₂₁	#576
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₅₂₁

commutator, cosocle

C₄×C₅₂₁

C₂⁴

Frattini

C₄

C₂³×D₅₂₁

Fitting

H(5,𝔽₂)×C₅₂₁

C₂

socle

C₂×C₅₂₁

C₂²×D₄

D₅₂₁

C₅₂₁

G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #30659754	C₂	semidirect
2. #30659734	C₂	semidirect
3. #30659733	C₂	semidirect
4. #30659728	C₂	semidirect
5. C₂×#22552349	C₂	semidirect
6. #30659715	C₂	semidirect
7. H(5,𝔽₂)×C₅₂₁	C₂	semidirect
9. #22552364	C₂²	semidirect
10. C₂×D₂₀₈₄	C₂²	semidirect
12. #22552354	C₂²	semidirect
13. #22552352	C₂²	semidirect
14. #22552351	C₂²	semidirect
15. #22552349	C₂²	semidirect
16. #22552349	C₂²	semidirect
17. #22552348	C₂²	semidirect
18. C₂×#16861960	C₂²	semidirect
21. #41×C₅₂₁	C₂²	semidirect
23. C₂×D₄×C₅₂₁	C₂²	semidirect
54. C₅₂₁	#576	semidirect
57. H(5,𝔽₂)	D₅₂₁	semidirect
67. #41	C₂×D₅₂₁	semidirect
71. C₂×D₄	C₂×D₅₂₁	semidirect

H ≅ G/N

type

1. #30659754

C₂

semidirect

2. #30659734

C₂

semidirect

3. #30659733

C₂

semidirect

4. #30659728

C₂

semidirect

5. C₂×#22552349

C₂

semidirect

6. #30659715

C₂

semidirect

7. H(5,𝔽₂)×C₅₂₁

C₂

semidirect

9. #22552364

C₂²

semidirect

10. C₂×D₂₀₈₄

C₂²

semidirect

12. #22552354

C₂²

semidirect

13. #22552352

C₂²

semidirect

14. #22552351

C₂²

semidirect

15. #22552349

C₂²

semidirect

16. #22552349

C₂²

semidirect

17. #22552348

C₂²

semidirect

18. C₂×#16861960

C₂²

semidirect

21. #41×C₅₂₁

C₂²

semidirect

23. C₂×D₄×C₅₂₁

C₂²

semidirect

54. C₅₂₁

#576

semidirect

57. H(5,𝔽₂)

D₅₂₁

semidirect

67. #41

C₂×D₅₂₁

semidirect

71. C₂×D₄

C₂×D₅₂₁

semidirect

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

subgroup	index
#30659754	2
#30659734	2
#30659733	2
#30659728	2
C₂×#22552349	2
#30659715	2
H(5,𝔽₂)×C₅₂₁	2
#576	521

subgroup

index

H(5,𝔽₂)×C₅₂₁

521

Subgroups — #45688235

Conjugacy classes of subgroups

Complements

Maximal subgroups