Group #45688243 — subgroups

Subgroups — #45688243

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	#30659738	C₂	1	3	yes	maximal, complemented
3	#30659734	C₂	1	3	yes	maximal, complemented
4	#30659728	C₂	1	3	yes	maximal, complemented
5	C₂×#22552352	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	#22552364	C₂²	1	3	yes	complemented
9	#22552363	—	2	3	no	—
10	C₂×Dic₁₀₄₂	C₂²	1	3	yes	complemented
11	Q₈×D₅₂₁	—	2	1	no	—
12	#22552354	C₂²	1	3	yes	complemented
13	#22552352	C₂²	1	3	yes	complemented
14	#22552352	C₂²	1	6	yes	complemented
15	#22552351	C₂²	1	1	yes	characteristic, complemented
16	#22552349	C₂²	1	3	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	3	yes	—
21	#41×C₅₂₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₅₂₁	—	2	3	no	—
23	C₂×D₄×C₅₂₁	C₂²	1	3	yes	—
24	Dic₁₀₄₂	—	2	3	no	—
25	Dic₁₀₄₂	C₂³	1	3	yes	—
26	D₂₀₈₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₅₂₁	—	2	3	no	—
28	C₂×Dic₅₂₁	—	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	1	no	—
45	C₄×C₅₂₁	—	2	1	no	—
46	C₄×C₅₂₁	C₂×D₄	1	3	yes	—
47	C₄×C₅₂₁	C₂⁴	1	1	yes	characteristic
48	Dic₅₂₁	—	2	3	no	—
49	C₂×C₅₂₁	—	2	3	no	—
50	C₂×C₅₂₁	—	4	3	no	—
51	C₂×C₅₂₁	C₂²×D₄	1	1	yes	characteristic
52	D₅₂₁	—	4	1	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)	D₅₂₁	1	1	yes	characteristic, complemented
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	—	521	1	no	—
62	#41	—	2	1	no	—
63	#41	—	521	3	no	—
64	#41	—	1042	3	no	—
65	#41	C₂×D₅₂₁	1	3	yes	—
66	#41	C₂×D₅₂₁	1	1	yes	characteristic
67	C₂×Q₈	—	521	3	no	—
68	C₂×Q₈	—	1042	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₅₂₁	1	3	yes	—
71	Q₁₆	—	521	3	no	—
72	SD₁₆	—	521	1	no	—
73	SD₁₆	—	521	3	no	—
74	SD₁₆	—	521	6	no	—
75	D₈	—	521	3	no	—
76	M₁₆	—	521	3	no	—
77	C₂×C₈	—	521	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	521	3	no	—
80	Q₈	—	1042	3	no	—
81	Q₈	C₂²⋊D₅₂₁	1	1	yes	characteristic
82	Q₈	C₂²×D₅₂₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	521	1	no	—
86	D₄	—	1042	3	no	—
87	D₄	C₂²⋊D₅₂₁	1	3	yes	—
88	D₄	C₂²×D₅₂₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	1042	3	no	—
91	C₂×C₄	—	1042	3	no	—
92	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
93	C₈	—	521	1	no	—
94	C₈	—	521	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	1042	1	no	—
98	C₂²	C₂×C₂²⋊D₅₂₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	1042	3	no	—
101	C₄	C₂×C₂²⋊D₅₂₁	1	3	yes	—
102	C₄	C₂³×D₅₂₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	2084	1	no	—
106	C₂	C₂²×C₂²⋊D₅₂₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659755

C₂

yes

maximal, characteristic, complemented

#30659738

C₂

yes

maximal, complemented

#30659734

C₂

yes

maximal, complemented

#30659728

C₂

yes

maximal, complemented

C₂×#22552352

C₂

yes

maximal, complemented

#30659715

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₂₁

C₂

yes

maximal, characteristic, complemented

#22552364

C₂²

yes

complemented

#22552363

—

C₂×Dic₁₀₄₂

C₂²

yes

complemented

Q₈×D₅₂₁

—

#22552354

C₂²

yes

complemented

#22552352

C₂²

yes

complemented

#22552352

C₂²

yes

complemented

#22552351

C₂²

yes

characteristic, complemented

#22552349

C₂²

yes

complemented

#22552348

C₂²

yes

complemented

C₂×#16861960

C₂²

yes

complemented

#41×C₅₂₁

—

#41×C₅₂₁

C₂²

yes

—

#41×C₅₂₁

C₂²

yes

characteristic

C₂×D₄×C₅₂₁

—

C₂×D₄×C₅₂₁

C₂²

yes

—

Dic₁₀₄₂

—

Dic₁₀₄₂

C₂³

yes

—

D₂₀₈₄

C₂³

yes

characteristic

C₂²⋊D₅₂₁

—

C₂×Dic₅₂₁

—

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

—

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

#577

yes

characteristic, complemented, Sylow

#577

—

521

maximal, Sylow

2₋¹⁺⁴

—

521

—

H(5,𝔽₂)

D₅₂₁

yes

characteristic, complemented

#137

—

521

—

AGL(1,ℤ/8)

—

521

—

#135

—

521

—

C₂×SD₁₆

—

521

—

#131

—

521

—

#41

—

#41

—

521

—

#41

—

1042

—

#41

C₂×D₅₂₁

yes

—

#41

C₂×D₅₂₁

yes

characteristic

C₂×Q₈

—

521

—

C₂×Q₈

—

1042

—

C₂×D₄

—

C₂×D₄

C₂×D₅₂₁

yes

—

Q₁₆

—

521

—

SD₁₆

—

521

—

SD₁₆

—

521

—

SD₁₆

—

521

—

D₈

—

521

—

M₁₆

—

521

—

C₂×C₈

—

521

—

C₂³

—

Q₈

—

521

—

Q₈

—

1042

—

Q₈

C₂²⋊D₅₂₁

yes

characteristic

Q₈

C₂²×D₅₂₁

yes

characteristic

D₄

—

D₄

—

D₄

—

521

—

D₄

—

1042

—

D₄

C₂²⋊D₅₂₁

yes

—

D₄

C₂²×D₅₂₁

yes

—

C₂×C₄

—

C₂×C₄

—

1042

—

C₂×C₄

—

1042

—

C₂×C₄

C₂²×D₅₂₁

yes

—

C₈

—

521

—

C₈

—

521

—

C₂²

—

C₂²

—

C₂²

—

1042

—

C₂²

C₂×C₂²⋊D₅₂₁

yes

—

C₄

—

100

C₄

—

1042

—

101

C₄

C₂×C₂²⋊D₅₂₁

yes

—

102

C₄

C₂³×D₅₂₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

2084

—

106

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	#577
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. #30659755	C₂	semidirect
2. #30659738	C₂	semidirect
3. #30659734	C₂	semidirect
4. #30659728	C₂	semidirect
5. C₂×#22552352	C₂	semidirect
6. #30659715	C₂	semidirect
7. H(5,𝔽₂)×C₅₂₁	C₂	semidirect
8. #22552364	C₂²	semidirect
10. C₂×Dic₁₀₄₂	C₂²	semidirect
12. #22552354	C₂²	semidirect
13. #22552352	C₂²	semidirect
14. #22552352	C₂²	semidirect
15. #22552351	C₂²	semidirect
16. #22552349	C₂²	semidirect
17. #22552348	C₂²	semidirect
18. C₂×#16861960	C₂²	semidirect
53. C₅₂₁	#577	semidirect
56. H(5,𝔽₂)	D₅₂₁	semidirect

H ≅ G/N

type

1. #30659755

C₂

semidirect

2. #30659738

C₂

semidirect

3. #30659734

C₂

semidirect

4. #30659728

C₂

semidirect

5. C₂×#22552352

C₂

semidirect

6. #30659715

C₂

semidirect

7. H(5,𝔽₂)×C₅₂₁

C₂

semidirect

8. #22552364

C₂²

semidirect

10. C₂×Dic₁₀₄₂

C₂²

semidirect

12. #22552354

C₂²

semidirect

13. #22552352

C₂²

semidirect

14. #22552352

C₂²

semidirect

15. #22552351

C₂²

semidirect

16. #22552349

C₂²

semidirect

17. #22552348

C₂²

semidirect

18. C₂×#16861960

C₂²

semidirect

53. C₅₂₁

#577

semidirect

56. H(5,𝔽₂)

D₅₂₁

semidirect

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

subgroup	index
#30659755	2
#30659738	2
#30659734	2
#30659728	2
C₂×#22552352	2
#30659715	2
H(5,𝔽₂)×C₅₂₁	2
#577	521

subgroup

index

H(5,𝔽₂)×C₅₂₁

521

Subgroups — #45688243

Conjugacy classes of subgroups

Complements

Maximal subgroups