Subgroups — #23782502

Order 10432.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19104783	C₂	1	5	yes	maximal, complemented
2	#19104781	C₂	1	10	yes	maximal, complemented
3	C₂×#14579700	C₂	1	5	yes	maximal, complemented
4	#41×D₁₆₃	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579700	C₂²	1	10	yes	complemented
7	#14579700	C₂²	1	20	yes	complemented
8	#14579699	C₂²	1	30	yes	complemented
9	#14579698	C₂²	1	10	yes	complemented
10	Q₈×D₁₆₃	C₂²	1	10	yes	complemented
11	C₂×D₆₅₂	C₂²	1	15	yes	complemented
12	D₄×D₁₆₃	C₂²	1	30	yes	complemented
13	C₂×C₄×D₁₆₃	C₂²	1	15	yes	complemented
14	#41×C₁₆₃	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₁₆₃	C₂²	1	5	yes	complemented
16	C₂²×D₁₆₃	C₂³	1	15	yes	—
17	Dic₃₂₆	C₂³	1	10	yes	—
18	D₆₅₂	C₂³	1	30	yes	—
19	C₂²⋊D₁₆₃	C₂³	1	20	yes	—
20	C₂×Dic₁₆₃	C₂³	1	5	yes	—
21	C₄×D₁₆₃	C₂³	1	10	yes	—
22	C₄×D₁₆₃	C₂³	1	30	yes	—
23	Q₈×C₁₆₃	C₂³	1	10	yes	—
24	D₄×C₁₆₃	C₂³	1	10	yes	—
25	C₂×C₄×C₁₆₃	C₂³	1	15	yes	—
26	C₂²×C₁₆₃	C₂⁴	1	5	yes	—
27	C₂×D₁₆₃	—	4	15	no	—
28	C₂×D₁₆₃	C₂⁴	1	10	yes	—
29	C₄×C₁₆₃	C₂⁴	1	10	yes	—
30	Dic₁₆₃	C₂⁴	1	1	yes	characteristic
31	Dic₁₆₃	C₂⁴	1	5	yes	—
32	C₂×C₁₆₃	—	2	5	no	—
33	C₂×C₁₆₃	C₂⁵	1	1	yes	characteristic
34	D₁₆₃	—	2	10	no	—
35	C₁₆₃	#585	1	1	yes	characteristic, complemented, Sylow
36	#585	—	163	1	no	maximal, Sylow
37	2₋¹⁺⁴	—	163	5	no	—
38	2₋¹⁺⁴	D₁₆₃	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	163	10	no	—
40	C₂×#41	—	163	10	no	—
41	C₂×#41	—	163	5	no	—
42	#41	—	163	10	no	—
43	#41	—	163	20	no	—
44	#41	—	163	10	no	—
45	#41	—	163	30	no	—
46	#41	C₂×D₁₆₃	1	10	yes	complemented
47	C₂×Q₈	—	163	10	no	—
48	C₂×Q₈	C₂×D₁₆₃	1	5	yes	complemented
49	C₂×D₄	—	163	30	no	—
50	C₂×D₄	—	163	15	no	—
51	C₂²×C₄	—	163	15	no	—
52	C₂³	—	163	15	no	—
53	Q₈	—	163	10	no	—
54	Q₈	C₂²×D₁₆₃	1	10	yes	—
55	D₄	—	163	20	no	—
56	D₄	—	163	30	no	—
57	D₄	C₂²×D₁₆₃	1	10	yes	—
58	C₂×C₄	—	163	5	no	—
59	C₂×C₄	—	163	10	no	—
60	C₂×C₄	—	163	30	no	—
61	C₂×C₄	C₂²×D₁₆₃	1	15	yes	—
62	C₂²	—	652	15	no	—
63	C₂²	—	163	10	no	—
64	C₂²	C₂³×D₁₆₃	1	5	yes	—
65	C₄	—	163	1	no	—
66	C₄	—	163	5	no	—
67	C₄	C₂³×D₁₆₃	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	326	10	no	—
70	C₂	C₂⁴×D₁₆₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₁₆₃
commutator, socle, cosocle	C₂×C₁₆₃	C₂⁵
Fitting	2₋¹⁺⁴×C₁₆₃	C₂
hypercenter	2₋¹⁺⁴	D₁₆₃
hypocenter	C₁₆₃	#585
LCS	G ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #19104783	C₂	semidirect
2. #19104781	C₂	semidirect
3. C₂×#14579700	C₂	semidirect
4. #41×D₁₆₃	C₂	semidirect
5. 2₋¹⁺⁴×C₁₆₃	C₂	semidirect
6. #14579700	C₂²	semidirect
7. #14579700	C₂²	semidirect
8. #14579699	C₂²	semidirect
9. #14579698	C₂²	semidirect
10. Q₈×D₁₆₃	C₂²	semidirect
11. C₂×D₆₅₂	C₂²	semidirect
12. D₄×D₁₆₃	C₂²	semidirect
13. C₂×C₄×D₁₆₃	C₂²	semidirect
14. #41×C₁₆₃	C₂²	semidirect
15. C₂×Q₈×C₁₆₃	C₂²	semidirect
35. C₁₆₃	#585	semidirect
38. 2₋¹⁺⁴	D₁₆₃	semidirect
46. #41	C₂×D₁₆₃	semidirect
48. C₂×Q₈	C₂×D₁₆₃	semidirect