Group #37589162 — subgroups

Subgroups — #37589162

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	#27133871	C₂	1	3	yes	maximal, complemented
3	#27133867	C₂	1	3	yes	maximal, complemented
4	#27133861	C₂	1	3	yes	maximal, complemented
5	C₂×#20259445	C₂	1	3	yes	maximal, complemented
6	#27133848	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20259457	C₂²	1	3	yes	complemented
9	#20259456	—	2	3	no	—
10	C₂×Dic₇₆₆	C₂²	1	3	yes	complemented
11	Q₈×D₃₈₃	—	2	1	no	—
12	#20259447	C₂²	1	3	yes	complemented
13	#20259445	C₂²	1	3	yes	complemented
14	#20259445	C₂²	1	6	yes	complemented
15	#20259444	C₂²	1	1	yes	characteristic, complemented
16	#20259442	C₂²	1	3	yes	complemented
17	#20259441	C₂²	1	3	yes	complemented
18	C₂×C₃₈₃⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₃₈₃	—	2	1	no	—
20	#41×C₃₈₃	C₂²	1	3	yes	—
21	#41×C₃₈₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₃₈₃	—	2	3	no	—
23	C₂×D₄×C₃₈₃	C₂²	1	3	yes	—
24	Dic₇₆₆	—	2	3	no	—
25	Dic₇₆₆	C₂³	1	3	yes	—
26	D₁₅₃₂	C₂³	1	1	yes	characteristic
27	C₂²⋊D₃₈₃	—	2	3	no	—
28	C₂×Dic₃₈₃	—	2	3	no	—
29	C₄×D₃₈₃	—	2	3	no	—
30	C₃₈₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₃₈₃⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₃₈₃	—	2	3	no	—
33	Q₈×C₃₈₃	D₄	1	1	yes	characteristic
34	Q₈×C₃₈₃	C₂³	1	1	yes	characteristic
35	D₄×C₃₈₃	—	2	3	no	—
36	D₄×C₃₈₃	—	2	3	no	—
37	D₄×C₃₈₃	D₄	1	3	yes	—
38	D₄×C₃₈₃	C₂³	1	3	yes	—
39	C₂×C₄×C₃₈₃	—	2	3	no	—
40	C₂×C₄×C₃₈₃	C₂³	1	3	yes	—
41	C₂²×C₃₈₃	—	2	3	no	—
42	C₂²×C₃₈₃	—	8	3	no	—
43	C₂²×C₃₈₃	C₂×D₄	1	3	yes	—
44	C₂×D₃₈₃	—	2	1	no	—
45	C₄×C₃₈₃	—	2	1	no	—
46	C₄×C₃₈₃	C₂×D₄	1	3	yes	—
47	C₄×C₃₈₃	C₂⁴	1	1	yes	characteristic
48	Dic₃₈₃	—	2	3	no	—
49	C₂×C₃₈₃	—	2	3	no	—
50	C₂×C₃₈₃	—	4	3	no	—
51	C₂×C₃₈₃	C₂²×D₄	1	1	yes	characteristic
52	D₃₈₃	—	4	1	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)	D₃₈₃	1	1	yes	characteristic, complemented
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	—	383	1	no	—
62	#41	—	2	1	no	—
63	#41	—	383	3	no	—
64	#41	—	766	3	no	—
65	#41	C₂×D₃₈₃	1	3	yes	—
66	#41	C₂×D₃₈₃	1	1	yes	characteristic
67	C₂×Q₈	—	383	3	no	—
68	C₂×Q₈	—	766	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₃₈₃	1	3	yes	—
71	Q₁₆	—	383	3	no	—
72	SD₁₆	—	383	1	no	—
73	SD₁₆	—	383	3	no	—
74	SD₁₆	—	383	6	no	—
75	D₈	—	383	3	no	—
76	M₁₆	—	383	3	no	—
77	C₂×C₈	—	383	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	383	3	no	—
80	Q₈	—	766	3	no	—
81	Q₈	C₂²⋊D₃₈₃	1	1	yes	characteristic
82	Q₈	C₂²×D₃₈₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	383	1	no	—
86	D₄	—	766	3	no	—
87	D₄	C₂²⋊D₃₈₃	1	3	yes	—
88	D₄	C₂²×D₃₈₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	766	3	no	—
91	C₂×C₄	—	766	3	no	—
92	C₂×C₄	C₂²×D₃₈₃	1	3	yes	—
93	C₈	—	383	1	no	—
94	C₈	—	383	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	766	1	no	—
98	C₂²	C₂×C₂²⋊D₃₈₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	766	3	no	—
101	C₄	C₂×C₂²⋊D₃₈₃	1	3	yes	—
102	C₄	C₂³×D₃₈₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1532	1	no	—
106	C₂	C₂²×C₂²⋊D₃₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133888

C₂

yes

maximal, characteristic, complemented

#27133871

C₂

yes

maximal, complemented

#27133867

C₂

yes

maximal, complemented

#27133861

C₂

yes

maximal, complemented

C₂×#20259445

C₂

yes

maximal, complemented

#27133848

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₈₃

C₂

yes

maximal, characteristic, complemented

#20259457

C₂²

yes

complemented

#20259456

—

C₂×Dic₇₆₆

C₂²

yes

complemented

Q₈×D₃₈₃

—

#20259447

C₂²

yes

complemented

#20259445

C₂²

yes

complemented

#20259445

C₂²

yes

complemented

#20259444

C₂²

yes

characteristic, complemented

#20259442

C₂²

yes

complemented

#20259441

C₂²

yes

complemented

C₂×C₃₈₃⋊C₈

C₂²

yes

complemented

#41×C₃₈₃

—

#41×C₃₈₃

C₂²

yes

—

#41×C₃₈₃

C₂²

yes

characteristic

C₂×D₄×C₃₈₃

—

C₂×D₄×C₃₈₃

C₂²

yes

—

Dic₇₆₆

—

Dic₇₆₆

C₂³

yes

—

D₁₅₃₂

C₂³

yes

characteristic

C₂²⋊D₃₈₃

—

C₂×Dic₃₈₃

—

C₄×D₃₈₃

—

C₃₈₃⋊C₈

C₂³

yes

characteristic

C₃₈₃⋊C₈

C₂³

yes

—

C₂³×C₃₈₃

—

Q₈×C₃₈₃

D₄

yes

characteristic

Q₈×C₃₈₃

C₂³

yes

characteristic

D₄×C₃₈₃

—

D₄×C₃₈₃

—

D₄×C₃₈₃

D₄

yes

—

D₄×C₃₈₃

C₂³

yes

—

C₂×C₄×C₃₈₃

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂²×C₃₈₃

—

C₂²×C₃₈₃

—

C₂²×C₃₈₃

C₂×D₄

yes

—

C₂×D₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

C₂×D₄

yes

—

C₄×C₃₈₃

C₂⁴

yes

characteristic

Dic₃₈₃

—

C₂×C₃₈₃

—

C₂×C₃₈₃

—

C₂×C₃₈₃

C₂²×D₄

yes

characteristic

D₃₈₃

—

C₃₈₃

#577

yes

characteristic, complemented, Sylow

#577

—

383

maximal, Sylow

2₋¹⁺⁴

—

383

—

H(5,𝔽₂)

D₃₈₃

yes

characteristic, complemented

#137

—

383

—

AGL(1,ℤ/8)

—

383

—

#135

—

383

—

C₂×SD₁₆

—

383

—

#131

—

383

—

#41

—

#41

—

383

—

#41

—

766

—

#41

C₂×D₃₈₃

yes

—

#41

C₂×D₃₈₃

yes

characteristic

C₂×Q₈

—

383

—

C₂×Q₈

—

766

—

C₂×D₄

—

C₂×D₄

C₂×D₃₈₃

yes

—

Q₁₆

—

383

—

SD₁₆

—

383

—

SD₁₆

—

383

—

SD₁₆

—

383

—

D₈

—

383

—

M₁₆

—

383

—

C₂×C₈

—

383

—

C₂³

—

Q₈

—

383

—

Q₈

—

766

—

Q₈

C₂²⋊D₃₈₃

yes

characteristic

Q₈

C₂²×D₃₈₃

yes

characteristic

D₄

—

D₄

—

D₄

—

383

—

D₄

—

766

—

D₄

C₂²⋊D₃₈₃

yes

—

D₄

C₂²×D₃₈₃

yes

—

C₂×C₄

—

C₂×C₄

—

766

—

C₂×C₄

—

766

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₈

—

383

—

C₈

—

383

—

C₂²

—

C₂²

—

C₂²

—

766

—

C₂²

C₂×C₂²⋊D₃₈₃

yes

—

C₄

—

100

C₄

—

766

—

101

C₄

C₂×C₂²⋊D₃₈₃

yes

—

102

C₄

C₂³×D₃₈₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1532

—

106

C₂

C₂²×C₂²⋊D₃₈₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₈₃
commutator, cosocle	C₄×C₃₈₃	C₂⁴
Frattini	C₄	C₂³×D₃₈₃
Fitting	H(5,𝔽₂)×C₃₈₃	C₂
socle	C₂×C₃₈₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₈₃
hypocenter	C₃₈₃	#577
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₈₃

commutator, cosocle

C₄×C₃₈₃

C₂⁴

Frattini

C₄

C₂³×D₃₈₃

Fitting

H(5,𝔽₂)×C₃₈₃

C₂

socle

C₂×C₃₈₃

C₂²×D₄

D₃₈₃

C₃₈₃

G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #27133888	C₂	semidirect
2. #27133871	C₂	semidirect
3. #27133867	C₂	semidirect
4. #27133861	C₂	semidirect
5. C₂×#20259445	C₂	semidirect
6. #27133848	C₂	semidirect
7. H(5,𝔽₂)×C₃₈₃	C₂	semidirect
8. #20259457	C₂²	semidirect
10. C₂×Dic₇₆₆	C₂²	semidirect
12. #20259447	C₂²	semidirect
13. #20259445	C₂²	semidirect
14. #20259445	C₂²	semidirect
15. #20259444	C₂²	semidirect
16. #20259442	C₂²	semidirect
17. #20259441	C₂²	semidirect
18. C₂×C₃₈₃⋊C₈	C₂²	semidirect
53. C₃₈₃	#577	semidirect
56. H(5,𝔽₂)	D₃₈₃	semidirect

H ≅ G/N

type

1. #27133888

C₂

semidirect

2. #27133871

C₂

semidirect

3. #27133867

C₂

semidirect

4. #27133861

C₂

semidirect

5. C₂×#20259445

C₂

semidirect

6. #27133848

C₂

semidirect

7. H(5,𝔽₂)×C₃₈₃

C₂

semidirect

8. #20259457

C₂²

semidirect

10. C₂×Dic₇₆₆

C₂²

semidirect

12. #20259447

C₂²

semidirect

13. #20259445

C₂²

semidirect

14. #20259445

C₂²

semidirect

15. #20259444

C₂²

semidirect

16. #20259442

C₂²

semidirect

17. #20259441

C₂²

semidirect

18. C₂×C₃₈₃⋊C₈

C₂²

semidirect

53. C₃₈₃

#577

semidirect

56. H(5,𝔽₂)

D₃₈₃

semidirect

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

subgroup	index
#27133888	2
#27133871	2
#27133867	2
#27133861	2
C₂×#20259445	2
#27133848	2
H(5,𝔽₂)×C₃₈₃	2
#577	383

subgroup

index

H(5,𝔽₂)×C₃₈₃

383

Subgroups — #37589162

Conjugacy classes of subgroups

Complements

Maximal subgroups