Group #44417588 — subgroups

Subgroups — #44417588

Order 29248.

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	#28410599	C₂	1	1	yes	maximal, characteristic, complemented
2	#28410582	C₂	1	3	yes	maximal, complemented
3	#28410578	C₂	1	3	yes	maximal, complemented
4	#28410572	C₂	1	3	yes	maximal, complemented
5	C₂×#20300729	C₂	1	3	yes	maximal, complemented
6	#28410559	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20300741	C₂²	1	3	yes	complemented
9	#20300740	—	2	3	no	—
10	C₂×Dic₉₁₄	C₂²	1	3	yes	complemented
11	Q₈×D₄₅₇	—	2	1	no	—
12	#20300731	C₂²	1	3	yes	complemented
13	#20300729	C₂²	1	3	yes	complemented
14	#20300729	C₂²	1	6	yes	complemented
15	#20300728	C₂²	1	1	yes	characteristic, complemented
16	#20300726	C₂²	1	3	yes	complemented
17	#20300725	C₂²	1	3	yes	complemented
18	C₂×#16814532	C₂²	1	3	yes	complemented
19	#41×C₄₅₇	—	2	1	no	—
20	#41×C₄₅₇	C₂²	1	3	yes	—
21	#41×C₄₅₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₄₅₇	—	2	3	no	—
23	C₂×D₄×C₄₅₇	C₂²	1	3	yes	—
24	Dic₉₁₄	—	2	3	no	—
25	Dic₉₁₄	C₂³	1	3	yes	—
26	D₁₈₂₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₄₅₇	—	2	3	no	—
28	C₂×Dic₄₅₇	—	2	3	no	—
29	C₄×D₄₅₇	—	2	3	no	—
30	#16814532	C₂³	1	1	yes	characteristic
31	#16814532	C₂³	1	3	yes	—
32	C₂³×C₄₅₇	—	2	3	no	—
33	Q₈×C₄₅₇	D₄	1	1	yes	characteristic
34	Q₈×C₄₅₇	C₂³	1	1	yes	characteristic
35	D₄×C₄₅₇	—	2	3	no	—
36	D₄×C₄₅₇	—	2	3	no	—
37	D₄×C₄₅₇	D₄	1	3	yes	—
38	D₄×C₄₅₇	C₂³	1	3	yes	—
39	C₂×C₄×C₄₅₇	—	2	3	no	—
40	C₂×C₄×C₄₅₇	C₂³	1	3	yes	—
41	C₂²×C₄₅₇	—	2	3	no	—
42	C₂²×C₄₅₇	—	8	3	no	—
43	C₂²×C₄₅₇	C₂×D₄	1	3	yes	—
44	C₂×D₄₅₇	—	2	1	no	—
45	C₄×C₄₅₇	—	2	1	no	—
46	C₄×C₄₅₇	C₂×D₄	1	3	yes	—
47	C₄×C₄₅₇	C₂⁴	1	1	yes	characteristic
48	Dic₄₅₇	—	2	3	no	—
49	C₂×C₄₅₇	—	2	3	no	—
50	C₂×C₄₅₇	—	4	3	no	—
51	C₂×C₄₅₇	C₂²×D₄	1	1	yes	characteristic
52	D₄₅₇	—	4	1	no	—
53	C₄₅₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	457	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	457	1	no	—
56	H(5,𝔽₂)	D₄₅₇	1	1	yes	characteristic, complemented
57	#137	—	457	3	no	—
58	AGL(1,ℤ/8)	—	457	3	no	—
59	#135	—	457	3	no	—
60	C₂×SD₁₆	—	457	3	no	—
61	#131	—	457	1	no	—
62	#41	—	2	1	no	—
63	#41	—	914	3	no	—
64	#41	—	457	3	no	—
65	#41	C₂×D₄₅₇	1	3	yes	—
66	#41	C₂×D₄₅₇	1	1	yes	characteristic
67	C₂×Q₈	—	914	1	no	—
68	C₂×Q₈	—	457	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₄₅₇	1	3	yes	—
71	Q₁₆	—	457	3	no	—
72	SD₁₆	—	457	1	no	—
73	SD₁₆	—	457	3	no	—
74	SD₁₆	—	457	6	no	—
75	D₈	—	457	3	no	—
76	M₁₆	—	457	3	no	—
77	C₂×C₈	—	457	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	914	3	no	—
80	Q₈	—	457	3	no	—
81	Q₈	C₂²⋊D₄₅₇	1	1	yes	characteristic
82	Q₈	C₂²×D₄₅₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	914	3	no	—
86	D₄	—	457	1	no	—
87	D₄	C₂²⋊D₄₅₇	1	3	yes	—
88	D₄	C₂²×D₄₅₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	914	3	no	—
91	C₂×C₄	—	914	3	no	—
92	C₂×C₄	C₂²×D₄₅₇	1	3	yes	—
93	C₈	—	457	1	no	—
94	C₈	—	457	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	914	1	no	—
98	C₂²	C₂×C₂²⋊D₄₅₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	914	3	no	—
101	C₄	C₂×C₂²⋊D₄₅₇	1	3	yes	—
102	C₄	C₂³×D₄₅₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1828	1	no	—
106	C₂	C₂²×C₂²⋊D₄₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28410599

C₂

yes

maximal, characteristic, complemented

#28410582

C₂

yes

maximal, complemented

#28410578

C₂

yes

maximal, complemented

#28410572

C₂

yes

maximal, complemented

C₂×#20300729

C₂

yes

maximal, complemented

#28410559

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₄₅₇

C₂

yes

maximal, characteristic, complemented

#20300741

C₂²

yes

complemented

#20300740

—

C₂×Dic₉₁₄

C₂²

yes

complemented

Q₈×D₄₅₇

—

#20300731

C₂²

yes

complemented

#20300729

C₂²

yes

complemented

#20300729

C₂²

yes

complemented

#20300728

C₂²

yes

characteristic, complemented

#20300726

C₂²

yes

complemented

#20300725

C₂²

yes

complemented

C₂×#16814532

C₂²

yes

complemented

#41×C₄₅₇

—

#41×C₄₅₇

C₂²

yes

—

#41×C₄₅₇

C₂²

yes

characteristic

C₂×D₄×C₄₅₇

—

C₂×D₄×C₄₅₇

C₂²

yes

—

Dic₉₁₄

—

Dic₉₁₄

C₂³

yes

—

D₁₈₂₈

C₂³

yes

characteristic

C₂²⋊D₄₅₇

—

C₂×Dic₄₅₇

—

C₄×D₄₅₇

—

#16814532

C₂³

yes

characteristic

#16814532

C₂³

yes

—

C₂³×C₄₅₇

—

Q₈×C₄₅₇

D₄

yes

characteristic

Q₈×C₄₅₇

C₂³

yes

characteristic

D₄×C₄₅₇

—

D₄×C₄₅₇

—

D₄×C₄₅₇

D₄

yes

—

D₄×C₄₅₇

C₂³

yes

—

C₂×C₄×C₄₅₇

—

C₂×C₄×C₄₅₇

C₂³

yes

—

C₂²×C₄₅₇

—

C₂²×C₄₅₇

—

C₂²×C₄₅₇

C₂×D₄

yes

—

C₂×D₄₅₇

—

C₄×C₄₅₇

—

C₄×C₄₅₇

C₂×D₄

yes

—

C₄×C₄₅₇

C₂⁴

yes

characteristic

Dic₄₅₇

—

C₂×C₄₅₇

—

C₂×C₄₅₇

—

C₂×C₄₅₇

C₂²×D₄

yes

characteristic

D₄₅₇

—

C₄₅₇

#577

yes

characteristic, complemented, Sylow

#577

—

457

maximal, Sylow

2₋¹⁺⁴

—

457

—

H(5,𝔽₂)

D₄₅₇

yes

characteristic, complemented

#137

—

457

—

AGL(1,ℤ/8)

—

457

—

#135

—

457

—

C₂×SD₁₆

—

457

—

#131

—

457

—

#41

—

#41

—

914

—

#41

—

457

—

#41

C₂×D₄₅₇

yes

—

#41

C₂×D₄₅₇

yes

characteristic

C₂×Q₈

—

914

—

C₂×Q₈

—

457

—

C₂×D₄

—

C₂×D₄

C₂×D₄₅₇

yes

—

Q₁₆

—

457

—

SD₁₆

—

457

—

SD₁₆

—

457

—

SD₁₆

—

457

—

D₈

—

457

—

M₁₆

—

457

—

C₂×C₈

—

457

—

C₂³

—

Q₈

—

914

—

Q₈

—

457

—

Q₈

C₂²⋊D₄₅₇

yes

characteristic

Q₈

C₂²×D₄₅₇

yes

characteristic

D₄

—

D₄

—

D₄

—

914

—

D₄

—

457

—

D₄

C₂²⋊D₄₅₇

yes

—

D₄

C₂²×D₄₅₇

yes

—

C₂×C₄

—

C₂×C₄

—

914

—

C₂×C₄

—

914

—

C₂×C₄

C₂²×D₄₅₇

yes

—

C₈

—

457

—

C₈

—

457

—

C₂²

—

C₂²

—

C₂²

—

914

—

C₂²

C₂×C₂²⋊D₄₅₇

yes

—

C₄

—

100

C₄

—

914

—

101

C₄

C₂×C₂²⋊D₄₅₇

yes

—

102

C₄

C₂³×D₄₅₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1828

—

106

C₂

C₂²×C₂²⋊D₄₅₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₄₅₇
commutator, cosocle	C₄×C₄₅₇	C₂⁴
Frattini	C₄	C₂³×D₄₅₇
Fitting	H(5,𝔽₂)×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₄₅₇
hypocenter	C₄₅₇	#577
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₄₅₇

commutator, cosocle

C₄×C₄₅₇

C₂⁴

Frattini

C₄

C₂³×D₄₅₇

Fitting

H(5,𝔽₂)×C₄₅₇

C₂

socle

C₂×C₄₅₇

C₂²×D₄

D₄₅₇

C₄₅₇

G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #28410599	C₂	semidirect
2. #28410582	C₂	semidirect
3. #28410578	C₂	semidirect
4. #28410572	C₂	semidirect
5. C₂×#20300729	C₂	semidirect
6. #28410559	C₂	semidirect
7. H(5,𝔽₂)×C₄₅₇	C₂	semidirect
8. #20300741	C₂²	semidirect
10. C₂×Dic₉₁₄	C₂²	semidirect
12. #20300731	C₂²	semidirect
13. #20300729	C₂²	semidirect
14. #20300729	C₂²	semidirect
15. #20300728	C₂²	semidirect
16. #20300726	C₂²	semidirect
17. #20300725	C₂²	semidirect
18. C₂×#16814532	C₂²	semidirect
53. C₄₅₇	#577	semidirect
56. H(5,𝔽₂)	D₄₅₇	semidirect

H ≅ G/N

type

1. #28410599

C₂

semidirect

2. #28410582

C₂

semidirect

3. #28410578

C₂

semidirect

4. #28410572

C₂

semidirect

5. C₂×#20300729

C₂

semidirect

6. #28410559

C₂

semidirect

7. H(5,𝔽₂)×C₄₅₇

C₂

semidirect

8. #20300741

C₂²

semidirect

10. C₂×Dic₉₁₄

C₂²

semidirect

12. #20300731

C₂²

semidirect

13. #20300729

C₂²

semidirect

14. #20300729

C₂²

semidirect

15. #20300728

C₂²

semidirect

16. #20300726

C₂²

semidirect

17. #20300725

C₂²

semidirect

18. C₂×#16814532

C₂²

semidirect

53. C₄₅₇

#577

semidirect

56. H(5,𝔽₂)

D₄₅₇

semidirect

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

subgroup	index
#28410599	2
#28410582	2
#28410578	2
#28410572	2
C₂×#20300729	2
#28410559	2
H(5,𝔽₂)×C₄₅₇	2
#577	457

subgroup

index

H(5,𝔽₂)×C₄₅₇

457

Subgroups — #44417588

Conjugacy classes of subgroups

Complements

Maximal subgroups