Group #45691008 — subgroups

Subgroups — #45691008

Order 33472.

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	#30660419	C₂	1	1	yes	maximal, characteristic, complemented
2	#30660399	C₂	1	3	yes	maximal, complemented
3	#30660398	C₂	1	3	yes	maximal, complemented
4	#30660393	C₂	1	3	yes	maximal, complemented
5	C₂×#22552626	C₂	1	3	yes	maximal, complemented
6	#30660380	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#22552642	—	2	1	no	—
9	#22552641	C₂²	1	3	yes	complemented
10	C₂×D₂₀₉₂	C₂²	1	3	yes	complemented
11	D₄×D₅₂₃	—	2	3	no	—
12	#22552631	C₂²	1	1	yes	characteristic, complemented
13	#22552629	C₂²	1	3	yes	complemented
14	#22552628	C₂²	1	3	yes	complemented
15	#22552626	C₂²	1	3	yes	complemented
16	#22552626	C₂²	1	6	yes	complemented
17	#22552625	C₂²	1	3	yes	complemented
18	C₂×C₅₂₃⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₅₂₃	—	2	1	no	—
20	#41×C₅₂₃	C₂²	1	1	yes	characteristic
21	#41×C₅₂₃	C₂²	1	3	yes	complemented
22	C₂×D₄×C₅₂₃	—	2	3	no	—
23	C₂×D₄×C₅₂₃	C₂²	1	3	yes	complemented
24	C₂²×D₅₂₃	—	2	3	no	—
25	Dic₁₀₄₆	C₂³	1	1	yes	characteristic
26	D₂₀₉₂	—	2	3	no	—
27	D₂₀₉₂	C₂³	1	3	yes	—
28	C₂²⋊D₅₂₃	—	2	3	no	—
29	C₄×D₅₂₃	—	2	3	no	—
30	C₅₂₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₅₂₃⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₅₂₃	—	2	3	no	—
33	Q₈×C₅₂₃	D₄	1	1	yes	characteristic
34	Q₈×C₅₂₃	C₂³	1	1	yes	characteristic
35	D₄×C₅₂₃	—	2	3	no	—
36	D₄×C₅₂₃	—	2	3	no	—
37	D₄×C₅₂₃	D₄	1	3	yes	—
38	D₄×C₅₂₃	C₂³	1	3	yes	—
39	C₂×C₄×C₅₂₃	—	2	3	no	—
40	C₂×C₄×C₅₂₃	C₂³	1	3	yes	—
41	C₂²×C₅₂₃	—	2	3	no	—
42	C₂²×C₅₂₃	—	8	3	no	—
43	C₂²×C₅₂₃	C₂×D₄	1	3	yes	—
44	C₂×D₅₂₃	—	2	3	no	—
45	C₂×D₅₂₃	—	8	3	no	—
46	C₄×C₅₂₃	—	2	1	no	—
47	C₄×C₅₂₃	C₂×D₄	1	3	yes	—
48	C₄×C₅₂₃	C₂⁴	1	1	yes	characteristic
49	Dic₅₂₃	—	2	1	no	—
50	C₂×C₅₂₃	—	2	3	no	—
51	C₂×C₅₂₃	—	4	3	no	—
52	C₂×C₅₂₃	C₂²×D₄	1	1	yes	characteristic
53	D₅₂₃	—	4	3	no	—
54	C₅₂₃	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	523	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	523	1	no	—
57	H(5,𝔽₂)	D₅₂₃	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	523	3	no	—
59	AGL(1,ℤ/8)	—	523	3	no	—
60	#135	—	523	3	no	—
61	C₂×D₈	—	523	3	no	—
62	#131	—	523	1	no	—
63	#41	—	2	1	no	—
64	#41	—	523	3	no	—
65	#41	—	1046	1	no	—
66	#41	C₂×D₅₂₃	1	1	yes	characteristic
67	#41	C₂×D₅₂₃	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	523	3	no	—
70	C₂×D₄	—	1046	3	no	—
71	C₂×D₄	C₂×D₅₂₃	1	3	yes	complemented
72	Q₁₆	—	523	1	no	—
73	SD₁₆	—	523	3	no	—
74	SD₁₆	—	523	3	no	—
75	D₈	—	523	3	no	—
76	D₈	—	523	6	no	—
77	M₁₆	—	523	3	no	—
78	C₂×C₈	—	523	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	1046	3	no	—
81	Q₈	—	523	1	no	—
82	Q₈	C₂²⋊D₅₂₃	1	1	yes	characteristic
83	Q₈	C₂²×D₅₂₃	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	523	3	no	—
87	D₄	—	1046	3	no	—
88	D₄	—	1046	3	no	—
89	D₄	C₂²⋊D₅₂₃	1	3	yes	—
90	D₄	C₂²×D₅₂₃	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	1046	3	no	—
93	C₂×C₄	C₂²×D₅₂₃	1	3	yes	—
94	C₈	—	523	1	no	—
95	C₈	—	523	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	1046	3	no	—
99	C₂²	—	4184	3	no	—
100	C₂²	C₂×C₂²⋊D₅₂₃	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	1046	1	no	—
103	C₄	C₂×C₂²⋊D₅₂₃	1	3	yes	—
104	C₄	C₂³×D₅₂₃	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	2092	3	no	—
108	C₂	C₂²×C₂²⋊D₅₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30660419

C₂

yes

maximal, characteristic, complemented

#30660399

C₂

yes

maximal, complemented

#30660398

C₂

yes

maximal, complemented

#30660393

C₂

yes

maximal, complemented

C₂×#22552626

C₂

yes

maximal, complemented

#30660380

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₂₃

C₂

yes

maximal, characteristic, complemented

#22552642

—

#22552641

C₂²

yes

complemented

C₂×D₂₀₉₂

C₂²

yes

complemented

D₄×D₅₂₃

—

#22552631

C₂²

yes

characteristic, complemented

#22552629

C₂²

yes

complemented

#22552628

C₂²

yes

complemented

#22552626

C₂²

yes

complemented

#22552626

C₂²

yes

complemented

#22552625

C₂²

yes

complemented

C₂×C₅₂₃⋊C₈

C₂²

yes

complemented

#41×C₅₂₃

—

#41×C₅₂₃

C₂²

yes

characteristic

#41×C₅₂₃

C₂²

yes

complemented

C₂×D₄×C₅₂₃

—

C₂×D₄×C₅₂₃

C₂²

yes

complemented

C₂²×D₅₂₃

—

Dic₁₀₄₆

C₂³

yes

characteristic

D₂₀₉₂

—

D₂₀₉₂

C₂³

yes

—

C₂²⋊D₅₂₃

—

C₄×D₅₂₃

—

C₅₂₃⋊C₈

C₂³

yes

characteristic

C₅₂₃⋊C₈

C₂³

yes

—

C₂³×C₅₂₃

—

Q₈×C₅₂₃

D₄

yes

characteristic

Q₈×C₅₂₃

C₂³

yes

characteristic

D₄×C₅₂₃

—

D₄×C₅₂₃

—

D₄×C₅₂₃

D₄

yes

—

D₄×C₅₂₃

C₂³

yes

—

C₂×C₄×C₅₂₃

—

C₂×C₄×C₅₂₃

C₂³

yes

—

C₂²×C₅₂₃

—

C₂²×C₅₂₃

—

C₂²×C₅₂₃

C₂×D₄

yes

—

C₂×D₅₂₃

—

C₂×D₅₂₃

—

C₄×C₅₂₃

—

C₄×C₅₂₃

C₂×D₄

yes

—

C₄×C₅₂₃

C₂⁴

yes

characteristic

Dic₅₂₃

—

C₂×C₅₂₃

—

C₂×C₅₂₃

—

C₂×C₅₂₃

C₂²×D₄

yes

characteristic

D₅₂₃

—

C₅₂₃

#576

yes

characteristic, complemented, Sylow

#576

—

523

maximal, Sylow

H(5,𝔽₂)

—

523

—

H(5,𝔽₂)

D₅₂₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

523

—

AGL(1,ℤ/8)

—

523

—

#135

—

523

—

C₂×D₈

—

523

—

#131

—

523

—

#41

—

#41

—

523

—

#41

—

1046

—

#41

C₂×D₅₂₃

yes

characteristic

#41

C₂×D₅₂₃

yes

complemented

C₂×D₄

—

C₂×D₄

—

523

—

C₂×D₄

—

1046

—

C₂×D₄

C₂×D₅₂₃

yes

complemented

Q₁₆

—

523

—

SD₁₆

—

523

—

SD₁₆

—

523

—

D₈

—

523

—

D₈

—

523

—

M₁₆

—

523

—

C₂×C₈

—

523

—

C₂³

—

C₂³

—

1046

—

Q₈

—

523

—

Q₈

C₂²⋊D₅₂₃

yes

characteristic

Q₈

C₂²×D₅₂₃

yes

characteristic

D₄

—

D₄

—

D₄

—

523

—

D₄

—

1046

—

D₄

—

1046

—

D₄

C₂²⋊D₅₂₃

yes

—

D₄

C₂²×D₅₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

1046

—

C₂×C₄

C₂²×D₅₂₃

yes

—

C₈

—

523

—

C₈

—

523

—

C₂²

—

C₂²

—

C₂²

—

1046

—

C₂²

—

4184

—

100

C₂²

C₂×C₂²⋊D₅₂₃

yes

—

101

C₄

—

102

C₄

—

1046

—

103

C₄

C₂×C₂²⋊D₅₂₃

yes

—

104

C₄

C₂³×D₅₂₃

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

2092

—

108

C₂

C₂²×C₂²⋊D₅₂₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₅₂₃
commutator, cosocle	C₄×C₅₂₃	C₂⁴
Frattini	C₄	C₂³×D₅₂₃
Fitting	H(5,𝔽₂)×C₅₂₃	C₂
socle	C₂×C₅₂₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₅₂₃
hypocenter	C₅₂₃	#576
LCS	G ▷ C₄×C₅₂₃ ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₅₂₃

commutator, cosocle

C₄×C₅₂₃

C₂⁴

Frattini

C₄

C₂³×D₅₂₃

Fitting

H(5,𝔽₂)×C₅₂₃

C₂

socle

C₂×C₅₂₃

C₂²×D₄

D₅₂₃

C₅₂₃

G ▷ C₄×C₅₂₃ ▷ C₂×C₅₂₃ ▷ C₅₂₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #30660419	C₂	semidirect
2. #30660399	C₂	semidirect
3. #30660398	C₂	semidirect
4. #30660393	C₂	semidirect
5. C₂×#22552626	C₂	semidirect
6. #30660380	C₂	semidirect
7. H(5,𝔽₂)×C₅₂₃	C₂	semidirect
9. #22552641	C₂²	semidirect
10. C₂×D₂₀₉₂	C₂²	semidirect
12. #22552631	C₂²	semidirect
13. #22552629	C₂²	semidirect
14. #22552628	C₂²	semidirect
15. #22552626	C₂²	semidirect
16. #22552626	C₂²	semidirect
17. #22552625	C₂²	semidirect
18. C₂×C₅₂₃⋊C₈	C₂²	semidirect
21. #41×C₅₂₃	C₂²	semidirect
23. C₂×D₄×C₅₂₃	C₂²	semidirect
54. C₅₂₃	#576	semidirect
57. H(5,𝔽₂)	D₅₂₃	semidirect
67. #41	C₂×D₅₂₃	semidirect
71. C₂×D₄	C₂×D₅₂₃	semidirect

H ≅ G/N

type

1. #30660419

C₂

semidirect

2. #30660399

C₂

semidirect

3. #30660398

C₂

semidirect

4. #30660393

C₂

semidirect

5. C₂×#22552626

C₂

semidirect

6. #30660380

C₂

semidirect

7. H(5,𝔽₂)×C₅₂₃

C₂

semidirect

9. #22552641

C₂²

semidirect

10. C₂×D₂₀₉₂

C₂²

semidirect

12. #22552631

C₂²

semidirect

13. #22552629

C₂²

semidirect

14. #22552628

C₂²

semidirect

15. #22552626

C₂²

semidirect

16. #22552626

C₂²

semidirect

17. #22552625

C₂²

semidirect

18. C₂×C₅₂₃⋊C₈

C₂²

semidirect

21. #41×C₅₂₃

C₂²

semidirect

23. C₂×D₄×C₅₂₃

C₂²

semidirect

54. C₅₂₃

#576

semidirect

57. H(5,𝔽₂)

D₅₂₃

semidirect

67. #41

C₂×D₅₂₃

semidirect

71. C₂×D₄

C₂×D₅₂₃

semidirect

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

subgroup	index
#30660419	2
#30660399	2
#30660398	2
#30660393	2
C₂×#22552626	2
#30660380	2
H(5,𝔽₂)×C₅₂₃	2
#576	523

subgroup

index

H(5,𝔽₂)×C₅₂₃

523

Subgroups — #45691008

Conjugacy classes of subgroups

Complements

Maximal subgroups