Subgroups — C₂³×D₂₁₆₄ #46830997

Order 34624.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅₄₁	C₂	1	2	yes	maximal, complemented
2	C₂²×D₂₁₆₄	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅₄₁	—	2	16	no	—
5	C₂³×D₅₄₁	C₂²	1	28	yes	complemented
6	C₂×D₂₁₆₄	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₅₄₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅₄₁	C₂²	1	14	yes	complemented
9	C₂²×D₅₄₁	—	2	112	no	—
10	C₂²×D₅₄₁	C₂³	1	56	yes	complemented
11	D₂₁₆₄	C₂³	1	64	yes	direct product factor
12	C₂³×C₅₄₁	D₄	1	8	yes	complemented
13	C₂³×C₅₄₁	C₂³	1	7	yes	—
14	C₂×C₄×C₅₄₁	C₂³	1	28	yes	complemented
15	C₂²×C₅₄₁	C₂×D₄	1	28	yes	complemented
16	C₂²×C₅₄₁	C₂⁴	1	7	yes	—
17	C₂×D₅₄₁	—	2	112	no	—
18	C₂×D₅₄₁	C₂⁴	1	16	yes	complemented
19	C₄×C₅₄₁	C₂⁴	1	8	yes	complemented
20	C₂×C₅₄₁	C₂²×D₄	1	14	yes	complemented
21	C₂×C₅₄₁	C₂⁵	1	1	yes	characteristic
22	D₅₄₁	—	2	16	no	—
23	C₅₄₁	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×D₄	—	541	1	no	maximal, Sylow
25	C₂⁵	—	541	2	no	—
26	C₂²×D₄	—	541	28	no	—
27	C₂³×C₄	D₅₄₁	1	1	yes	characteristic, complemented
28	C₂⁴	—	541	28	no	—
29	C₂⁴	—	1082	16	no	—
30	C₂⁴	C₂×D₅₄₁	1	1	yes	characteristic
31	C₂×D₄	—	541	112	no	—
32	C₂²×C₄	C₂×D₅₄₁	1	14	yes	complemented
33	C₂³	—	541	56	no	—
34	C₂³	—	1082	112	no	—
35	C₂³	D₂₁₆₄	1	8	yes	direct product factor
36	C₂³	C₂²×D₅₄₁	1	7	yes	—
37	D₄	—	541	64	no	—
38	C₂×C₄	C₂²×D₅₄₁	1	28	yes	complemented
39	C₂²	—	541	16	no	—
40	C₂²	—	1082	112	no	—
41	C₂²	C₂×D₂₁₆₄	1	28	yes	direct product factor
42	C₂²	C₂³×D₅₄₁	1	7	yes	—
43	C₄	C₂³×D₅₄₁	1	8	yes	complemented
44	C₂	—	1082	16	no	—
45	C₂	C₂²×D₂₁₆₄	1	14	yes	direct product factor
46	C₂	C₂⁴×D₅₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₅₄₁
commutator, cosocle	C₂×C₅₄₁	C₂⁵
Frattini	C₂	C₂⁴×D₅₄₁
Fitting	C₂³×C₄×C₅₄₁	C₂
socle	C₂⁴×C₅₄₁	C₂²
hypercenter	C₂³×C₄	D₅₄₁
hypocenter	C₅₄₁	C₂³×D₄
LCS	G ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₅₄₁	C₂	semidirect
2. C₂²×D₂₁₆₄	C₂	direct (N × H)
3. C₂³×C₄×C₅₄₁	C₂	semidirect
5. C₂³×D₅₄₁	C₂²	semidirect
6. C₂×D₂₁₆₄	C₂²	direct (N × H)
8. C₂²×C₄×C₅₄₁	C₂²	semidirect
10. C₂²×D₅₄₁	C₂³	semidirect
11. D₂₁₆₄	C₂³	direct (N × H)
12. C₂³×C₅₄₁	D₄	semidirect
14. C₂×C₄×C₅₄₁	C₂³	semidirect
15. C₂²×C₅₄₁	C₂×D₄	semidirect
18. C₂×D₅₄₁	C₂⁴	semidirect
19. C₄×C₅₄₁	C₂⁴	semidirect
20. C₂×C₅₄₁	C₂²×D₄	semidirect
23. C₅₄₁	C₂³×D₄	semidirect
27. C₂³×C₄	D₅₄₁	semidirect
32. C₂²×C₄	C₂×D₅₄₁	semidirect
35. C₂³	D₂₁₆₄	direct (N × H)
38. C₂×C₄	C₂²×D₅₄₁	semidirect
41. C₂²	C₂×D₂₁₆₄	direct (N × H)
43. C₄	C₂³×D₅₄₁	semidirect
45. C₂	C₂²×D₂₁₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₅₄₁	2
C₂²×D₂₁₆₄	2
C₂³×C₄×C₅₄₁	2
C₂³×D₄	541

Subgroups — C23×D2164 #46830997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₂₁₆₄ #46830997