Group #37589163 — subgroups

Subgroups — #37589163

Order 24512.

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	#27133888	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133887	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133874	C₂	1	3	yes	maximal, complemented
4	#27133869	C₂	1	3	yes	maximal, complemented
5	#27133863	C₂	1	3	yes	maximal, complemented
6	C₂×#20259446	C₂	1	3	yes	maximal, complemented
7	#131×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20259458	—	2	1	no	—
9	#20259457	C₂²	1	3	yes	—
10	#20259457	C₂²	1	3	yes	complemented
11	#20259456	—	2	3	no	—
12	C₂×Dic₇₆₆	C₂²	1	3	yes	complemented
13	Q₈×D₃₈₃	—	2	1	no	—
14	C₂×D₁₅₃₂	C₂²	1	3	yes	—
15	D₄×D₃₈₃	—	2	3	no	—
16	Dic₁₅₃₂	C₂²	1	3	yes	complemented
17	#20259446	C₂²	1	3	yes	complemented
18	#20259446	C₂²	1	6	yes	complemented
19	#20259446	C₂²	1	1	yes	characteristic, complemented
20	D₃₀₆₄	C₂²	1	3	yes	complemented
21	#41×C₃₈₃	C₂²	1	1	yes	characteristic
22	M₁₆×C₃₈₃	C₂²	1	3	yes	complemented
23	C₂×C₈×C₃₈₃	C₂²	1	3	yes	complemented
24	C₂²×D₃₈₃	—	2	3	no	—
25	Dic₇₆₆	—	2	3	no	—
26	Dic₇₆₆	C₂³	1	3	yes	—
27	Dic₇₆₆	C₂³	1	1	yes	characteristic
28	D₁₅₃₂	—	2	3	no	—
29	D₁₅₃₂	C₂³	1	1	yes	characteristic
30	D₁₅₃₂	C₂³	1	3	yes	—
31	C₂²⋊D₃₈₃	—	2	3	no	—
32	C₂²⋊D₃₈₃	—	2	3	no	—
33	C₂×Dic₃₈₃	—	2	3	no	—
34	C₄×D₃₈₃	—	2	3	no	—
35	C₄×D₃₈₃	—	2	3	no	—
36	Q₈×C₃₈₃	D₄	1	1	yes	characteristic
37	D₄×C₃₈₃	D₄	1	3	yes	—
38	C₂×C₄×C₃₈₃	C₂³	1	3	yes	—
39	C₈×C₃₈₃	C₂³	1	1	yes	characteristic
40	C₈×C₃₈₃	C₂³	1	3	yes	—
41	C₂²×C₃₈₃	C₂×D₄	1	3	yes	—
42	C₂×D₃₈₃	—	2	1	no	—
43	C₂×D₃₈₃	—	2	3	no	—
44	C₂×D₃₈₃	—	8	3	no	—
45	C₄×C₃₈₃	C₂×D₄	1	3	yes	—
46	C₄×C₃₈₃	C₂⁴	1	1	yes	characteristic
47	Dic₃₈₃	—	2	1	no	—
48	Dic₃₈₃	—	2	3	no	—
49	C₂×C₃₈₃	—	2	3	no	—
50	C₂×C₃₈₃	C₂²×D₄	1	1	yes	characteristic
51	D₃₈₃	—	4	1	no	—
52	D₃₈₃	—	4	3	no	—
53	C₃₈₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	383	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	383	1	no	—
56	H(5,𝔽₂)	—	383	1	no	—
57	#137	—	383	3	no	—
58	AGL(1,ℤ/8)	—	383	3	no	—
59	#135	—	383	3	no	—
60	C₂×SD₁₆	—	383	3	no	—
61	#131	D₃₈₃	1	1	yes	characteristic, complemented
62	#41	—	383	3	no	—
63	#41	—	383	3	no	—
64	#41	—	766	1	no	—
65	#41	—	766	3	no	—
66	#41	C₂×D₃₈₃	1	1	yes	characteristic
67	C₂×Q₈	—	383	3	no	—
68	C₂×Q₈	—	766	1	no	—
69	C₂×D₄	—	383	3	no	—
70	C₂×D₄	—	766	3	no	—
71	Q₁₆	—	383	3	no	—
72	SD₁₆	—	383	3	no	—
73	SD₁₆	—	383	6	no	—
74	SD₁₆	—	383	1	no	—
75	D₈	—	383	3	no	—
76	M₁₆	C₂×D₃₈₃	1	3	yes	complemented
77	C₂×C₈	C₂×D₃₈₃	1	3	yes	complemented
78	C₂³	—	766	3	no	—
79	Q₈	—	383	3	no	—
80	Q₈	—	383	1	no	—
81	Q₈	—	766	3	no	—
82	Q₈	D₁₅₃₂	1	1	yes	characteristic
83	D₄	—	383	1	no	—
84	D₄	—	383	3	no	—
85	D₄	—	766	3	no	—
86	D₄	—	766	3	no	—
87	D₄	—	766	3	no	—
88	D₄	D₁₅₃₂	1	3	yes	—
89	C₂×C₄	—	766	3	no	—
90	C₂×C₄	—	766	3	no	—
91	C₂×C₄	—	766	3	no	—
92	C₂×C₄	C₂²×D₃₈₃	1	3	yes	—
93	C₈	C₂²×D₃₈₃	1	1	yes	characteristic
94	C₈	C₂²×D₃₈₃	1	3	yes	—
95	C₂²	—	3064	3	no	—
96	C₂²	—	766	1	no	—
97	C₂²	—	766	3	no	—
98	C₂²	C₂×D₁₅₃₂	1	3	yes	—
99	C₄	—	766	1	no	—
100	C₄	—	766	3	no	—
101	C₄	C₂×D₁₅₃₂	1	3	yes	—
102	C₄	C₂³×D₃₈₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	1532	3	no	—
105	C₂	—	1532	1	no	—
106	C₂	C₂²×D₁₅₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133888

C₂

yes

maximal, characteristic, complemented

#27133887

C₂

yes

maximal, characteristic, complemented

#27133874

C₂

yes

maximal, complemented

#27133869

C₂

yes

maximal, complemented

#27133863

C₂

yes

maximal, complemented

C₂×#20259446

C₂

yes

maximal, complemented

#131×C₃₈₃

C₂

yes

maximal, characteristic, complemented

#20259458

—

#20259457

C₂²

yes

—

#20259457

C₂²

yes

complemented

#20259456

—

C₂×Dic₇₆₆

C₂²

yes

complemented

Q₈×D₃₈₃

—

C₂×D₁₅₃₂

C₂²

yes

—

D₄×D₃₈₃

—

Dic₁₅₃₂

C₂²

yes

complemented

#20259446

C₂²

yes

complemented

#20259446

C₂²

yes

complemented

#20259446

C₂²

yes

characteristic, complemented

D₃₀₆₄

C₂²

yes

complemented

#41×C₃₈₃

C₂²

yes

characteristic

M₁₆×C₃₈₃

C₂²

yes

complemented

C₂×C₈×C₃₈₃

C₂²

yes

complemented

C₂²×D₃₈₃

—

Dic₇₆₆

—

Dic₇₆₆

C₂³

yes

—

Dic₇₆₆

C₂³

yes

characteristic

D₁₅₃₂

—

D₁₅₃₂

C₂³

yes

characteristic

D₁₅₃₂

C₂³

yes

—

C₂²⋊D₃₈₃

—

C₂²⋊D₃₈₃

—

C₂×Dic₃₈₃

—

C₄×D₃₈₃

—

C₄×D₃₈₃

—

Q₈×C₃₈₃

D₄

yes

characteristic

D₄×C₃₈₃

D₄

yes

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₈×C₃₈₃

C₂³

yes

characteristic

C₈×C₃₈₃

C₂³

yes

—

C₂²×C₃₈₃

C₂×D₄

yes

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₄×C₃₈₃

C₂×D₄

yes

—

C₄×C₃₈₃

C₂⁴

yes

characteristic

Dic₃₈₃

—

Dic₃₈₃

—

C₂×C₃₈₃

—

C₂×C₃₈₃

C₂²×D₄

yes

characteristic

D₃₈₃

—

D₃₈₃

—

C₃₈₃

#577

yes

characteristic, complemented, Sylow

#577

—

383

maximal, Sylow

2₋¹⁺⁴

—

383

—

H(5,𝔽₂)

—

383

—

#137

—

383

—

AGL(1,ℤ/8)

—

383

—

#135

—

383

—

C₂×SD₁₆

—

383

—

#131

D₃₈₃

yes

characteristic, complemented

#41

—

383

—

#41

—

383

—

#41

—

766

—

#41

—

766

—

#41

C₂×D₃₈₃

yes

characteristic

C₂×Q₈

—

383

—

C₂×Q₈

—

766

—

C₂×D₄

—

383

—

C₂×D₄

—

766

—

Q₁₆

—

383

—

SD₁₆

—

383

—

SD₁₆

—

383

—

SD₁₆

—

383

—

D₈

—

383

—

M₁₆

C₂×D₃₈₃

yes

complemented

C₂×C₈

C₂×D₃₈₃

yes

complemented

C₂³

—

766

—

Q₈

—

383

—

Q₈

—

383

—

Q₈

—

766

—

Q₈

D₁₅₃₂

yes

characteristic

D₄

—

383

—

D₄

—

383

—

D₄

—

766

—

D₄

—

766

—

D₄

—

766

—

D₄

D₁₅₃₂

yes

—

C₂×C₄

—

766

—

C₂×C₄

—

766

—

C₂×C₄

—

766

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₈

C₂²×D₃₈₃

yes

characteristic

C₈

C₂²×D₃₈₃

yes

—

C₂²

—

3064

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

C₂×D₁₅₃₂

yes

—

C₄

—

766

—

100

C₄

—

766

—

101

C₄

C₂×D₁₅₃₂

yes

—

102

C₄

C₂³×D₃₈₃

yes

characteristic

103

C₂

—

104

C₂

—

1532

—

105

C₂

—

1532

—

106

C₂

C₂²×D₁₅₃₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₅₃₂
commutator, cosocle	C₄×C₃₈₃	C₂⁴
Frattini	C₄	C₂³×D₃₈₃
Fitting	#131×C₃₈₃	C₂
socle	C₂×C₃₈₃	C₂²×D₄
hypercenter	#131	D₃₈₃
hypocenter	C₃₈₃	#577
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₁₅₃₂

commutator, cosocle

C₄×C₃₈₃

C₂⁴

Frattini

C₄

C₂³×D₃₈₃

Fitting

#131×C₃₈₃

C₂

socle

C₂×C₃₈₃

C₂²×D₄

D₃₈₃

C₃₈₃

G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #27133888	C₂	semidirect
2. #27133887	C₂	semidirect
3. #27133874	C₂	semidirect
4. #27133869	C₂	semidirect
5. #27133863	C₂	semidirect
6. C₂×#20259446	C₂	semidirect
7. #131×C₃₈₃	C₂	semidirect
10. #20259457	C₂²	semidirect
12. C₂×Dic₇₆₆	C₂²	semidirect
16. Dic₁₅₃₂	C₂²	semidirect
17. #20259446	C₂²	semidirect
18. #20259446	C₂²	semidirect
19. #20259446	C₂²	semidirect
20. D₃₀₆₄	C₂²	semidirect
22. M₁₆×C₃₈₃	C₂²	semidirect
23. C₂×C₈×C₃₈₃	C₂²	semidirect
53. C₃₈₃	#577	semidirect
61. #131	D₃₈₃	semidirect
76. M₁₆	C₂×D₃₈₃	semidirect
77. C₂×C₈	C₂×D₃₈₃	semidirect

H ≅ G/N

type

1. #27133888

C₂

semidirect

2. #27133887

C₂

semidirect

3. #27133874

C₂

semidirect

4. #27133869

C₂

semidirect

5. #27133863

C₂

semidirect

6. C₂×#20259446

C₂

semidirect

7. #131×C₃₈₃

C₂

semidirect

10. #20259457

C₂²

semidirect

12. C₂×Dic₇₆₆

C₂²

semidirect

16. Dic₁₅₃₂

C₂²

semidirect

17. #20259446

C₂²

semidirect

18. #20259446

C₂²

semidirect

19. #20259446

C₂²

semidirect

20. D₃₀₆₄

C₂²

semidirect

22. M₁₆×C₃₈₃

C₂²

semidirect

23. C₂×C₈×C₃₈₃

C₂²

semidirect

53. C₃₈₃

#577

semidirect

61. #131

D₃₈₃

semidirect

76. M₁₆

C₂×D₃₈₃

semidirect

77. C₂×C₈

C₂×D₃₈₃

semidirect

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

subgroup	index
#27133888	2
#27133887	2
#27133874	2
#27133869	2
#27133863	2
C₂×#20259446	2
#131×C₃₈₃	2
#577	383

subgroup

index

#131×C₃₈₃

383

Subgroups — #37589163

Conjugacy classes of subgroups

Complements

Maximal subgroups