Group #14558336 — subgroups

Subgroups — #14558336

Order 2368.

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	#11924582	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924562	C₂	1	3	yes	maximal, complemented
3	#11924561	C₂	1	3	yes	maximal, complemented
4	#11924556	C₂	1	3	yes	maximal, complemented
5	C₂×#10596939	C₂	1	3	yes	maximal, complemented
6	#11924543	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×C₃₇	—	2	1	no	—
9	#41×C₃₇	C₂²	1	1	yes	characteristic
10	#41×C₃₇	C₂²	1	3	yes	complemented
11	C₂×D₄×C₃₇	—	2	3	no	—
12	C₂×D₄×C₃₇	C₂²	1	3	yes	complemented
13	#10596966	—	2	1	no	—
14	D₄×D₃₇	—	2	3	no	—
15	#10596962	C₂²	1	3	yes	complemented
16	C₂×D₁₄₈	C₂²	1	3	yes	complemented
17	#10596942	C₂²	1	1	yes	characteristic, complemented
18	#10596941	C₂²	1	3	yes	complemented
19	#10596940	C₂²	1	3	yes	complemented
20	#10596939	C₂²	1	3	yes	complemented
21	#10596939	C₂²	1	6	yes	complemented
22	#10596934	C₂²	1	3	yes	complemented
23	C₂×#64436	C₂²	1	3	yes	complemented
24	C₂³×C₃₇	—	2	3	no	—
25	C₂²×D₃₇	—	2	3	no	—
26	Q₈×C₃₇	D₄	1	1	yes	characteristic
27	Q₈×C₃₇	C₂³	1	1	yes	characteristic
28	D₄×C₃₇	—	2	3	no	—
29	D₄×C₃₇	—	2	3	no	—
30	D₄×C₃₇	D₄	1	3	yes	—
31	D₄×C₃₇	C₂³	1	3	yes	—
32	C₂×C₄×C₃₇	—	2	3	no	—
33	C₂×C₄×C₃₇	C₂³	1	3	yes	—
34	C₂²⋊D₃₇	—	2	3	no	—
35	D₁₄₈	—	2	3	no	—
36	D₁₄₈	C₂³	1	3	yes	—
37	C₄×D₃₇	—	2	3	no	—
38	Dic₇₄	C₂³	1	1	yes	characteristic
39	#64436	C₂³	1	1	yes	characteristic
40	#64436	C₂³	1	3	yes	—
41	C₂²×C₃₇	—	2	3	no	—
42	C₂²×C₃₇	—	8	3	no	—
43	C₂²×C₃₇	C₂×D₄	1	3	yes	—
44	C₂×D₃₇	—	2	3	no	—
45	C₂×D₃₇	—	8	3	no	—
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	C₂×D₄	1	3	yes	—
48	C₄×C₃₇	C₂⁴	1	1	yes	characteristic
49	Dic₃₇	—	2	1	no	—
50	C₂×C₃₇	—	2	3	no	—
51	C₂×C₃₇	—	4	3	no	—
52	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
53	D₃₇	—	4	3	no	—
54	#576	—	37	1	no	maximal, Sylow
55	C₃₇	#576	1	1	yes	characteristic, complemented, Sylow
56	H(5,𝔽₂)	—	37	1	no	—
57	H(5,𝔽₂)	D₃₇	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	37	3	no	—
59	AGL(1,ℤ/8)	—	37	3	no	—
60	#135	—	37	3	no	—
61	C₂×D₈	—	37	3	no	—
62	#131	—	37	1	no	—
63	#41	—	2	1	no	—
64	#41	—	37	3	no	—
65	#41	—	74	1	no	—
66	#41	C₂×D₃₇	1	1	yes	characteristic
67	#41	C₂×D₃₇	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	37	3	no	—
70	C₂×D₄	—	74	3	no	—
71	C₂×D₄	C₂×D₃₇	1	3	yes	complemented
72	Q₁₆	—	37	1	no	—
73	SD₁₆	—	37	3	no	—
74	SD₁₆	—	37	3	no	—
75	D₈	—	37	3	no	—
76	D₈	—	37	6	no	—
77	M₁₆	—	37	3	no	—
78	C₂×C₈	—	37	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	74	3	no	—
81	Q₈	—	37	1	no	—
82	Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
83	Q₈	C₂²×D₃₇	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	37	3	no	—
87	D₄	—	74	3	no	—
88	D₄	—	74	3	no	—
89	D₄	C₂²⋊D₃₇	1	3	yes	—
90	D₄	C₂²×D₃₇	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	74	3	no	—
93	C₂×C₄	C₂²×D₃₇	1	3	yes	—
94	C₈	—	37	1	no	—
95	C₈	—	37	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	296	3	no	—
99	C₂²	—	74	3	no	—
100	C₂²	C₂×C₂²⋊D₃₇	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	74	1	no	—
103	C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
104	C₄	C₂³×D₃₇	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	148	3	no	—
108	C₂	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924582

C₂

yes

maximal, characteristic, complemented

#11924562

C₂

yes

maximal, complemented

#11924561

C₂

yes

maximal, complemented

#11924556

C₂

yes

maximal, complemented

C₂×#10596939

C₂

yes

maximal, complemented

#11924543

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₇

C₂

yes

maximal, characteristic, complemented

#41×C₃₇

—

#41×C₃₇

C₂²

yes

characteristic

#41×C₃₇

C₂²

yes

complemented

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂²

yes

complemented

#10596966

—

D₄×D₃₇

—

#10596962

C₂²

yes

complemented

C₂×D₁₄₈

C₂²

yes

complemented

#10596942

C₂²

yes

characteristic, complemented

#10596941

C₂²

yes

complemented

#10596940

C₂²

yes

complemented

#10596939

C₂²

yes

complemented

#10596939

C₂²

yes

complemented

#10596934

C₂²

yes

complemented

C₂×#64436

C₂²

yes

complemented

C₂³×C₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

D₄

yes

characteristic

Q₈×C₃₇

C₂³

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

D₄

yes

—

D₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂²⋊D₃₇

—

D₁₄₈

—

D₁₄₈

C₂³

yes

—

C₄×D₃₇

—

Dic₇₄

C₂³

yes

characteristic

#64436

C₂³

yes

characteristic

#64436

C₂³

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×D₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₄

yes

—

C₄×C₃₇

C₂⁴

yes

characteristic

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₄

yes

characteristic

D₃₇

—

#576

—

maximal, Sylow

C₃₇

#576

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

D₃₇

yes

characteristic, complemented

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

#135

—

C₂×D₈

—

—

—

—

—

C₂×D₃₇

yes

characteristic

#41

C₂×D₃₇

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

complemented

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

Q₈

—

Q₈

C₂²⋊D₃₇

yes

characteristic

Q₈

C₂²×D₃₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²⋊D₃₇

yes

—

D₄

C₂²×D₃₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₃₇

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

296

—

C₂²

—

100

C₂²

C₂×C₂²⋊D₃₇

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₂×C₂²⋊D₃₇

yes

—

104

C₄

C₂³×D₃₇

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

148

—

108

C₂

C₂²×C₂²⋊D₃₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₄×C₃₇	C₂⁴
Frattini	C₄	C₂³×D₃₇
Fitting	H(5,𝔽₂)×C₃₇	C₂
socle	C₂×C₃₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₇
hypocenter	C₃₇	#576
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₄×C₃₇

C₂⁴

Frattini

C₄

C₂³×D₃₇

Fitting

H(5,𝔽₂)×C₃₇

C₂

socle

C₂×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #11924582	C₂	semidirect
2. #11924562	C₂	semidirect
3. #11924561	C₂	semidirect
4. #11924556	C₂	semidirect
5. C₂×#10596939	C₂	semidirect
6. #11924543	C₂	semidirect
7. H(5,𝔽₂)×C₃₇	C₂	semidirect
10. #41×C₃₇	C₂²	semidirect
12. C₂×D₄×C₃₇	C₂²	semidirect
15. #10596962	C₂²	semidirect
16. C₂×D₁₄₈	C₂²	semidirect
17. #10596942	C₂²	semidirect
18. #10596941	C₂²	semidirect
19. #10596940	C₂²	semidirect
20. #10596939	C₂²	semidirect
21. #10596939	C₂²	semidirect
22. #10596934	C₂²	semidirect
23. C₂×#64436	C₂²	semidirect
55. C₃₇	#576	semidirect
57. H(5,𝔽₂)	D₃₇	semidirect
67. #41	C₂×D₃₇	semidirect
71. C₂×D₄	C₂×D₃₇	semidirect

H ≅ G/N

type

1. #11924582

C₂

semidirect

2. #11924562

C₂

semidirect

3. #11924561

C₂

semidirect

4. #11924556

C₂

semidirect

5. C₂×#10596939

C₂

semidirect

6. #11924543

C₂

semidirect

7. H(5,𝔽₂)×C₃₇

C₂

semidirect

10. #41×C₃₇

C₂²

semidirect

12. C₂×D₄×C₃₇

C₂²

semidirect

15. #10596962

C₂²

semidirect

16. C₂×D₁₄₈

C₂²

semidirect

17. #10596942

C₂²

semidirect

18. #10596941

C₂²

semidirect

19. #10596940

C₂²

semidirect

20. #10596939

C₂²

semidirect

21. #10596939

C₂²

semidirect

22. #10596934

C₂²

semidirect

23. C₂×#64436

C₂²

semidirect

55. C₃₇

#576

semidirect

57. H(5,𝔽₂)

D₃₇

semidirect

67. #41

C₂×D₃₇

semidirect

71. C₂×D₄

C₂×D₃₇

semidirect

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

subgroup	index
#11924582	2
#11924562	2
#11924561	2
#11924556	2
C₂×#10596939	2
#11924543	2
H(5,𝔽₂)×C₃₇	2
#576	37

subgroup

index

H(5,𝔽₂)×C₃₇

Subgroups — #14558336

Conjugacy classes of subgroups

Complements

Maximal subgroups