Group #16838205 — subgroups

Subgroups — #16838205

Order 3776.

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	#14273773	C₂	1	1	yes	maximal, characteristic, complemented
2	#14273753	C₂	1	3	yes	maximal, complemented
3	#14273752	C₂	1	3	yes	maximal, complemented
4	#14273747	C₂	1	3	yes	maximal, complemented
5	C₂×#11745669	C₂	1	3	yes	maximal, complemented
6	#14273734	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×C₅₉	—	2	1	no	—
9	#41×C₅₉	C₂²	1	1	yes	characteristic
10	#41×C₅₉	C₂²	1	3	yes	complemented
11	C₂×D₄×C₅₉	—	2	3	no	—
12	C₂×D₄×C₅₉	C₂²	1	3	yes	complemented
13	#11745689	—	2	1	no	—
14	D₄×D₅₉	—	2	3	no	—
15	#11745685	C₂²	1	3	yes	complemented
16	C₂×D₂₃₆	C₂²	1	3	yes	complemented
17	#11745672	C₂²	1	1	yes	characteristic, complemented
18	#11745671	C₂²	1	3	yes	complemented
19	#11745670	C₂²	1	3	yes	complemented
20	#11745669	C₂²	1	3	yes	complemented
21	#11745669	C₂²	1	6	yes	complemented
22	#11745664	C₂²	1	3	yes	complemented
23	C₂×C₅₉⋊C₈	C₂²	1	3	yes	complemented
24	C₂³×C₅₉	—	2	3	no	—
25	C₂²×D₅₉	—	2	3	no	—
26	Q₈×C₅₉	D₄	1	1	yes	characteristic
27	Q₈×C₅₉	C₂³	1	1	yes	characteristic
28	D₄×C₅₉	—	2	3	no	—
29	D₄×C₅₉	—	2	3	no	—
30	D₄×C₅₉	D₄	1	3	yes	—
31	D₄×C₅₉	C₂³	1	3	yes	—
32	C₂×C₄×C₅₉	—	2	3	no	—
33	C₂×C₄×C₅₉	C₂³	1	3	yes	—
34	C₂²⋊D₅₉	—	2	3	no	—
35	D₂₃₆	—	2	3	no	—
36	D₂₃₆	C₂³	1	3	yes	—
37	C₄×D₅₉	—	2	3	no	—
38	Dic₁₁₈	C₂³	1	1	yes	characteristic
39	C₅₉⋊C₈	C₂³	1	1	yes	characteristic
40	C₅₉⋊C₈	C₂³	1	3	yes	—
41	C₂²×C₅₉	—	2	3	no	—
42	C₂²×C₅₉	—	8	3	no	—
43	C₂²×C₅₉	C₂×D₄	1	3	yes	—
44	C₂×D₅₉	—	2	3	no	—
45	C₂×D₅₉	—	8	3	no	—
46	C₄×C₅₉	—	2	1	no	—
47	C₄×C₅₉	C₂×D₄	1	3	yes	—
48	C₄×C₅₉	C₂⁴	1	1	yes	characteristic
49	Dic₅₉	—	2	1	no	—
50	C₂×C₅₉	—	2	3	no	—
51	C₂×C₅₉	—	4	3	no	—
52	C₂×C₅₉	C₂²×D₄	1	1	yes	characteristic
53	D₅₉	—	4	3	no	—
54	#576	—	59	1	no	maximal, Sylow
55	C₅₉	#576	1	1	yes	characteristic, complemented, Sylow
56	H(5,𝔽₂)	—	59	1	no	—
57	H(5,𝔽₂)	D₅₉	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	59	3	no	—
59	AGL(1,ℤ/8)	—	59	3	no	—
60	#135	—	59	3	no	—
61	C₂×D₈	—	59	3	no	—
62	#131	—	59	1	no	—
63	#41	—	2	1	no	—
64	#41	—	59	3	no	—
65	#41	—	118	1	no	—
66	#41	C₂×D₅₉	1	1	yes	characteristic
67	#41	C₂×D₅₉	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	59	3	no	—
70	C₂×D₄	—	118	3	no	—
71	C₂×D₄	C₂×D₅₉	1	3	yes	complemented
72	Q₁₆	—	59	1	no	—
73	SD₁₆	—	59	3	no	—
74	SD₁₆	—	59	3	no	—
75	D₈	—	59	3	no	—
76	D₈	—	59	6	no	—
77	M₁₆	—	59	3	no	—
78	C₂×C₈	—	59	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	118	3	no	—
81	Q₈	—	59	1	no	—
82	Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
83	Q₈	C₂²×D₅₉	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	59	3	no	—
87	D₄	—	118	3	no	—
88	D₄	—	118	3	no	—
89	D₄	C₂²⋊D₅₉	1	3	yes	—
90	D₄	C₂²×D₅₉	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	118	3	no	—
93	C₂×C₄	C₂²×D₅₉	1	3	yes	—
94	C₈	—	59	1	no	—
95	C₈	—	59	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	118	3	no	—
99	C₂²	—	472	3	no	—
100	C₂²	C₂×C₂²⋊D₅₉	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	118	1	no	—
103	C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
104	C₄	C₂³×D₅₉	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	236	3	no	—
108	C₂	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14273773

C₂

yes

maximal, characteristic, complemented

#14273753

C₂

yes

maximal, complemented

#14273752

C₂

yes

maximal, complemented

#14273747

C₂

yes

maximal, complemented

C₂×#11745669

C₂

yes

maximal, complemented

#14273734

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₅₉

C₂

yes

maximal, characteristic, complemented

#41×C₅₉

—

#41×C₅₉

C₂²

yes

characteristic

#41×C₅₉

C₂²

yes

complemented

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂²

yes

complemented

#11745689

—

D₄×D₅₉

—

#11745685

C₂²

yes

complemented

C₂×D₂₃₆

C₂²

yes

complemented

#11745672

C₂²

yes

characteristic, complemented

#11745671

C₂²

yes

complemented

#11745670

C₂²

yes

complemented

#11745669

C₂²

yes

complemented

#11745669

C₂²

yes

complemented

#11745664

C₂²

yes

complemented

C₂×C₅₉⋊C₈

C₂²

yes

complemented

C₂³×C₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

D₄

yes

characteristic

Q₈×C₅₉

C₂³

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

D₄

yes

—

D₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂²⋊D₅₉

—

D₂₃₆

—

D₂₃₆

C₂³

yes

—

C₄×D₅₉

—

Dic₁₁₈

C₂³

yes

characteristic

C₅₉⋊C₈

C₂³

yes

characteristic

C₅₉⋊C₈

C₂³

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₄

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₄

yes

—

C₄×C₅₉

C₂⁴

yes

characteristic

Dic₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×D₄

yes

characteristic

D₅₉

—

#576

—

maximal, Sylow

C₅₉

#576

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

D₅₉

yes

characteristic, complemented

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

#135

—

C₂×D₈

—

—

—

—

—

118

—

#41

C₂×D₅₉

yes

characteristic

#41

C₂×D₅₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

C₂×D₅₉

yes

complemented

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

118

—

Q₈

—

Q₈

C₂²⋊D₅₉

yes

characteristic

Q₈

C₂²×D₅₉

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

118

—

D₄

—

118

—

D₄

C₂²⋊D₅₉

yes

—

D₄

C₂²×D₅₉

yes

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

C₂²×D₅₉

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

118

—

C₂²

—

472

—

100

C₂²

C₂×C₂²⋊D₅₉

yes

—

101

C₄

—

102

C₄

—

118

—

103

C₄

C₂×C₂²⋊D₅₉

yes

—

104

C₄

C₂³×D₅₉

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

236

—

108

C₂

C₂²×C₂²⋊D₅₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₅₉
commutator, cosocle	C₄×C₅₉	C₂⁴
Frattini	C₄	C₂³×D₅₉
Fitting	H(5,𝔽₂)×C₅₉	C₂
socle	C₂×C₅₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₅₉
hypocenter	C₅₉	#576
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₅₉

commutator, cosocle

C₄×C₅₉

C₂⁴

Frattini

C₄

C₂³×D₅₉

Fitting

H(5,𝔽₂)×C₅₉

C₂

socle

C₂×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #14273773	C₂	semidirect
2. #14273753	C₂	semidirect
3. #14273752	C₂	semidirect
4. #14273747	C₂	semidirect
5. C₂×#11745669	C₂	semidirect
6. #14273734	C₂	semidirect
7. H(5,𝔽₂)×C₅₉	C₂	semidirect
10. #41×C₅₉	C₂²	semidirect
12. C₂×D₄×C₅₉	C₂²	semidirect
15. #11745685	C₂²	semidirect
16. C₂×D₂₃₆	C₂²	semidirect
17. #11745672	C₂²	semidirect
18. #11745671	C₂²	semidirect
19. #11745670	C₂²	semidirect
20. #11745669	C₂²	semidirect
21. #11745669	C₂²	semidirect
22. #11745664	C₂²	semidirect
23. C₂×C₅₉⋊C₈	C₂²	semidirect
55. C₅₉	#576	semidirect
57. H(5,𝔽₂)	D₅₉	semidirect
67. #41	C₂×D₅₉	semidirect
71. C₂×D₄	C₂×D₅₉	semidirect

H ≅ G/N

type

1. #14273773

C₂

semidirect

2. #14273753

C₂

semidirect

3. #14273752

C₂

semidirect

4. #14273747

C₂

semidirect

5. C₂×#11745669

C₂

semidirect

6. #14273734

C₂

semidirect

7. H(5,𝔽₂)×C₅₉

C₂

semidirect

10. #41×C₅₉

C₂²

semidirect

12. C₂×D₄×C₅₉

C₂²

semidirect

15. #11745685

C₂²

semidirect

16. C₂×D₂₃₆

C₂²

semidirect

17. #11745672

C₂²

semidirect

18. #11745671

C₂²

semidirect

19. #11745670

C₂²

semidirect

20. #11745669

C₂²

semidirect

21. #11745669

C₂²

semidirect

22. #11745664

C₂²

semidirect

23. C₂×C₅₉⋊C₈

C₂²

semidirect

55. C₅₉

#576

semidirect

57. H(5,𝔽₂)

D₅₉

semidirect

67. #41

C₂×D₅₉

semidirect

71. C₂×D₄

C₂×D₅₉

semidirect

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

subgroup	index
#14273773	2
#14273753	2
#14273752	2
#14273747	2
C₂×#11745669	2
#14273734	2
H(5,𝔽₂)×C₅₉	2
#576	59

subgroup

index

H(5,𝔽₂)×C₅₉

Subgroups — #16838205

Conjugacy classes of subgroups

Complements

Maximal subgroups