Subgroups — #22582924

Order 8992.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17985138	C₂	1	2	yes	maximal, complemented
2	#17985137	C₂	1	4	yes	maximal, complemented
3	C₂×C₂²⋊D₂₈₁	C₂	1	4	yes	maximal, complemented
4	D₄×D₂₈₁	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₈₁	C₂²	1	4	yes	complemented
7	Dic₅₆₂	C₂²	1	2	yes	complemented
8	D₁₁₂₄	C₂²	1	2	yes	complemented
9	C₂²⋊D₂₈₁	C₂²	1	8	yes	complemented
10	C₂²⋊D₂₈₁	C₂²	1	4	yes	complemented
11	C₂×Dic₂₈₁	C₂²	1	4	yes	complemented
12	C₄×D₂₈₁	C₂²	1	4	yes	complemented
13	C₂³×C₂₈₁	C₂²	1	2	yes	complemented
14	D₄×C₂₈₁	C₂²	1	4	yes	complemented
15	C₂×C₄×C₂₈₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₂₈₁	—	4	2	no	—
17	C₂²×C₂₈₁	C₂³	1	1	yes	characteristic
18	C₂²×C₂₈₁	C₂³	1	4	yes	—
19	C₂×D₂₈₁	—	4	4	no	—
20	C₂×D₂₈₁	C₂³	1	4	yes	—
21	C₄×C₂₈₁	C₂³	1	2	yes	—
22	Dic₂₈₁	C₂³	1	4	yes	—
23	C₂×C₂₈₁	—	2	4	no	—
24	C₂×C₂₈₁	—	2	1	no	—
25	C₂×C₂₈₁	C₂⁴	1	1	yes	characteristic
26	D₂₈₁	—	2	4	no	—
27	C₂₈₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
28	H(5,𝔽₂)	—	281	1	no	maximal, Sylow
29	#41	—	281	4	no	—
30	#41	—	281	2	no	—
31	C₂×D₄	—	281	4	no	—
32	C₂×D₄	—	281	4	no	—
33	C₂×D₄	D₂₈₁	1	1	yes	characteristic, complemented
34	C₂³	—	281	4	no	—
35	C₂³	C₂×D₂₈₁	1	2	yes	complemented
36	Q₈	—	281	2	no	—
37	D₄	—	281	8	no	—
38	D₄	—	281	4	no	—
39	D₄	—	281	2	no	—
40	D₄	C₂×D₂₈₁	1	4	yes	complemented
41	C₂×C₄	—	281	4	no	—
42	C₂×C₄	—	281	4	no	—
43	C₂×C₄	C₂×D₂₈₁	1	1	yes	characteristic, complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	281	4	no	—
46	C₂²	—	1124	4	no	—
47	C₂²	C₂²×D₂₈₁	1	4	yes	—
48	C₂²	C₂²×D₂₈₁	1	1	yes	characteristic
49	C₄	—	281	4	no	—
50	C₄	C₂²×D₂₈₁	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	562	4	no	—
54	C₂	C₂³×D₂₈₁	1	1	yes	characteristic

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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #17985138	C₂	semidirect
2. #17985137	C₂	semidirect
3. C₂×C₂²⋊D₂₈₁	C₂	semidirect
4. D₄×D₂₈₁	C₂	semidirect
5. C₂×D₄×C₂₈₁	C₂	semidirect
6. C₂²×D₂₈₁	C₂²	semidirect
7. Dic₅₆₂	C₂²	semidirect
8. D₁₁₂₄	C₂²	semidirect
9. C₂²⋊D₂₈₁	C₂²	semidirect
10. C₂²⋊D₂₈₁	C₂²	semidirect
11. C₂×Dic₂₈₁	C₂²	semidirect
12. C₄×D₂₈₁	C₂²	semidirect
13. C₂³×C₂₈₁	C₂²	semidirect
14. D₄×C₂₈₁	C₂²	semidirect
15. C₂×C₄×C₂₈₁	C₂²	semidirect
27. C₂₈₁	H(5,𝔽₂)	semidirect
33. C₂×D₄	D₂₈₁	semidirect
35. C₂³	C₂×D₂₈₁	semidirect
40. D₄	C₂×D₂₈₁	semidirect
43. C₂×C₄	C₂×D₂₈₁	semidirect