Group #31824274 — subgroups

Subgroups — #31824274

Order 17984.

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	#22582926	C₂	1	1	yes	maximal, characteristic, complemented
2	#22582909	C₂	1	3	yes	maximal, complemented
3	#22582905	C₂	1	3	yes	maximal, complemented
4	#22582899	C₂	1	3	yes	maximal, complemented
5	C₂×#17985126	C₂	1	3	yes	maximal, complemented
6	#22582886	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#17985138	C₂²	1	3	yes	complemented
9	#17985137	—	2	3	no	—
10	C₂×Dic₅₆₂	C₂²	1	3	yes	complemented
11	Q₈×D₂₈₁	—	2	1	no	—
12	#17985128	C₂²	1	3	yes	complemented
13	#17985126	C₂²	1	3	yes	complemented
14	#17985126	C₂²	1	6	yes	complemented
15	#17985125	C₂²	1	1	yes	characteristic, complemented
16	#17985123	C₂²	1	3	yes	complemented
17	#17985122	C₂²	1	3	yes	complemented
18	C₂×#14555886	C₂²	1	3	yes	complemented
19	#41×C₂₈₁	—	2	1	no	—
20	#41×C₂₈₁	C₂²	1	3	yes	—
21	#41×C₂₈₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₂₈₁	—	2	3	no	—
23	C₂×D₄×C₂₈₁	C₂²	1	3	yes	—
24	Dic₅₆₂	—	2	3	no	—
25	Dic₅₆₂	C₂³	1	3	yes	—
26	D₁₁₂₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₂₈₁	—	2	3	no	—
28	C₂×Dic₂₈₁	—	2	3	no	—
29	C₄×D₂₈₁	—	2	3	no	—
30	#14555886	C₂³	1	1	yes	characteristic
31	#14555886	C₂³	1	3	yes	—
32	C₂³×C₂₈₁	—	2	3	no	—
33	Q₈×C₂₈₁	D₄	1	1	yes	characteristic
34	Q₈×C₂₈₁	C₂³	1	1	yes	characteristic
35	D₄×C₂₈₁	—	2	3	no	—
36	D₄×C₂₈₁	—	2	3	no	—
37	D₄×C₂₈₁	D₄	1	3	yes	—
38	D₄×C₂₈₁	C₂³	1	3	yes	—
39	C₂×C₄×C₂₈₁	—	2	3	no	—
40	C₂×C₄×C₂₈₁	C₂³	1	3	yes	—
41	C₂²×C₂₈₁	—	2	3	no	—
42	C₂²×C₂₈₁	—	8	3	no	—
43	C₂²×C₂₈₁	C₂×D₄	1	3	yes	—
44	C₂×D₂₈₁	—	2	1	no	—
45	C₄×C₂₈₁	—	2	1	no	—
46	C₄×C₂₈₁	C₂×D₄	1	3	yes	—
47	C₄×C₂₈₁	C₂⁴	1	1	yes	characteristic
48	Dic₂₈₁	—	2	3	no	—
49	C₂×C₂₈₁	—	2	3	no	—
50	C₂×C₂₈₁	—	4	3	no	—
51	C₂×C₂₈₁	C₂²×D₄	1	1	yes	characteristic
52	D₂₈₁	—	4	1	no	—
53	C₂₈₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	281	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	281	1	no	—
56	H(5,𝔽₂)	D₂₈₁	1	1	yes	characteristic, complemented
57	#137	—	281	3	no	—
58	AGL(1,ℤ/8)	—	281	3	no	—
59	#135	—	281	3	no	—
60	C₂×SD₁₆	—	281	3	no	—
61	#131	—	281	1	no	—
62	#41	—	2	1	no	—
63	#41	—	281	3	no	—
64	#41	—	562	3	no	—
65	#41	C₂×D₂₈₁	1	3	yes	—
66	#41	C₂×D₂₈₁	1	1	yes	characteristic
67	C₂×Q₈	—	281	3	no	—
68	C₂×Q₈	—	562	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₂₈₁	1	3	yes	—
71	Q₁₆	—	281	3	no	—
72	SD₁₆	—	281	1	no	—
73	SD₁₆	—	281	3	no	—
74	SD₁₆	—	281	6	no	—
75	D₈	—	281	3	no	—
76	M₁₆	—	281	3	no	—
77	C₂×C₈	—	281	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	281	3	no	—
80	Q₈	—	562	3	no	—
81	Q₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
82	Q₈	C₂²×D₂₈₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	281	1	no	—
86	D₄	—	562	3	no	—
87	D₄	C₂²⋊D₂₈₁	1	3	yes	—
88	D₄	C₂²×D₂₈₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	562	3	no	—
91	C₂×C₄	—	562	3	no	—
92	C₂×C₄	C₂²×D₂₈₁	1	3	yes	—
93	C₈	—	281	1	no	—
94	C₈	—	281	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	562	1	no	—
98	C₂²	C₂×C₂²⋊D₂₈₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	562	3	no	—
101	C₄	C₂×C₂²⋊D₂₈₁	1	3	yes	—
102	C₄	C₂³×D₂₈₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1124	1	no	—
106	C₂	C₂²×C₂²⋊D₂₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582926

C₂

yes

maximal, characteristic, complemented

#22582909

C₂

yes

maximal, complemented

#22582905

C₂

yes

maximal, complemented

#22582899

C₂

yes

maximal, complemented

C₂×#17985126

C₂

yes

maximal, complemented

#22582886

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₂₈₁

C₂

yes

maximal, characteristic, complemented

#17985138

C₂²

yes

complemented

#17985137

—

C₂×Dic₅₆₂

C₂²

yes

complemented

Q₈×D₂₈₁

—

#17985128

C₂²

yes

complemented

#17985126

C₂²

yes

complemented

#17985126

C₂²

yes

complemented

#17985125

C₂²

yes

characteristic, complemented

#17985123

C₂²

yes

complemented

#17985122

C₂²

yes

complemented

C₂×#14555886

C₂²

yes

complemented

#41×C₂₈₁

—

#41×C₂₈₁

C₂²

yes

—

#41×C₂₈₁

C₂²

yes

characteristic

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

C₂²

yes

—

Dic₅₆₂

—

Dic₅₆₂

C₂³

yes

—

D₁₁₂₄

C₂³

yes

characteristic

C₂²⋊D₂₈₁

—

C₂×Dic₂₈₁

—

C₄×D₂₈₁

—

#14555886

C₂³

yes

characteristic

#14555886

C₂³

yes

—

C₂³×C₂₈₁

—

Q₈×C₂₈₁

D₄

yes

characteristic

Q₈×C₂₈₁

C₂³

yes

characteristic

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

D₄

yes

—

D₄×C₂₈₁

C₂³

yes

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂³

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂×D₄

yes

—

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×D₄

yes

—

C₄×C₂₈₁

C₂⁴

yes

characteristic

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₂²×D₄

yes

characteristic

D₂₈₁

—

C₂₈₁

#577

yes

characteristic, complemented, Sylow

#577

—

281

maximal, Sylow

2₋¹⁺⁴

—

281

—

H(5,𝔽₂)

D₂₈₁

yes

characteristic, complemented

#137

—

281

—

AGL(1,ℤ/8)

—

281

—

#135

—

281

—

C₂×SD₁₆

—

281

—

#131

—

281

—

#41

—

#41

—

281

—

#41

—

562

—

#41

C₂×D₂₈₁

yes

—

#41

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

281

—

C₂×Q₈

—

562

—

C₂×D₄

—

C₂×D₄

C₂×D₂₈₁

yes

—

Q₁₆

—

281

—

SD₁₆

—

281

—

SD₁₆

—

281

—

SD₁₆

—

281

—

D₈

—

281

—

M₁₆

—

281

—

C₂×C₈

—

281

—

C₂³

—

Q₈

—

281

—

Q₈

—

562

—

Q₈

C₂²⋊D₂₈₁

yes

characteristic

Q₈

C₂²×D₂₈₁

yes

characteristic

D₄

—

D₄

—

D₄

—

281

—

D₄

—

562

—

D₄

C₂²⋊D₂₈₁

yes

—

D₄

C₂²×D₂₈₁

yes

—

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

—

562

—

C₂×C₄

C₂²×D₂₈₁

yes

—

C₈

—

281

—

C₈

—

281

—

C₂²

—

C₂²

—

C₂²

—

562

—

C₂²

C₂×C₂²⋊D₂₈₁

yes

—

C₄

—

100

C₄

—

562

—

101

C₄

C₂×C₂²⋊D₂₈₁

yes

—

102

C₄

C₂³×D₂₈₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1124

—

106

C₂

C₂²×C₂²⋊D₂₈₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₂₈₁
commutator, cosocle	C₄×C₂₈₁	C₂⁴
Frattini	C₄	C₂³×D₂₈₁
Fitting	H(5,𝔽₂)×C₂₈₁	C₂
socle	C₂×C₂₈₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₂₈₁
hypocenter	C₂₈₁	#577
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₂₈₁

commutator, cosocle

C₄×C₂₈₁

C₂⁴

Frattini

C₄

C₂³×D₂₈₁

Fitting

H(5,𝔽₂)×C₂₈₁

C₂

socle

C₂×C₂₈₁

C₂²×D₄

D₂₈₁

C₂₈₁

G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #22582926	C₂	semidirect
2. #22582909	C₂	semidirect
3. #22582905	C₂	semidirect
4. #22582899	C₂	semidirect
5. C₂×#17985126	C₂	semidirect
6. #22582886	C₂	semidirect
7. H(5,𝔽₂)×C₂₈₁	C₂	semidirect
8. #17985138	C₂²	semidirect
10. C₂×Dic₅₆₂	C₂²	semidirect
12. #17985128	C₂²	semidirect
13. #17985126	C₂²	semidirect
14. #17985126	C₂²	semidirect
15. #17985125	C₂²	semidirect
16. #17985123	C₂²	semidirect
17. #17985122	C₂²	semidirect
18. C₂×#14555886	C₂²	semidirect
53. C₂₈₁	#577	semidirect
56. H(5,𝔽₂)	D₂₈₁	semidirect

H ≅ G/N

type

1. #22582926

C₂

semidirect

2. #22582909

C₂

semidirect

3. #22582905

C₂

semidirect

4. #22582899

C₂

semidirect

5. C₂×#17985126

C₂

semidirect

6. #22582886

C₂

semidirect

7. H(5,𝔽₂)×C₂₈₁

C₂

semidirect

8. #17985138

C₂²

semidirect

10. C₂×Dic₅₆₂

C₂²

semidirect

12. #17985128

C₂²

semidirect

13. #17985126

C₂²

semidirect

14. #17985126

C₂²

semidirect

15. #17985125

C₂²

semidirect

16. #17985123

C₂²

semidirect

17. #17985122

C₂²

semidirect

18. C₂×#14555886

C₂²

semidirect

53. C₂₈₁

#577

semidirect

56. H(5,𝔽₂)

D₂₈₁

semidirect

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

subgroup	index
#22582926	2
#22582909	2
#22582905	2
#22582899	2
C₂×#17985126	2
#22582886	2
H(5,𝔽₂)×C₂₈₁	2
#577	281

subgroup

index

H(5,𝔽₂)×C₂₈₁

281

Subgroups — #31824274

Conjugacy classes of subgroups

Complements

Maximal subgroups