Subgroups — C₂³×D₁₂₆₈ #35204045

Order 20288.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃₁₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₂₆₈	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₁₇	—	2	16	no	—
5	C₂³×D₃₁₇	C₂²	1	28	yes	complemented
6	C₂×D₁₂₆₈	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₃₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₃₁₇	C₂²	1	14	yes	complemented
9	C₂²×D₃₁₇	—	2	112	no	—
10	C₂²×D₃₁₇	C₂³	1	56	yes	complemented
11	D₁₂₆₈	C₂³	1	64	yes	direct product factor
12	C₂³×C₃₁₇	D₄	1	8	yes	complemented
13	C₂³×C₃₁₇	C₂³	1	7	yes	—
14	C₂×C₄×C₃₁₇	C₂³	1	28	yes	complemented
15	C₂²×C₃₁₇	C₂×D₄	1	28	yes	complemented
16	C₂²×C₃₁₇	C₂⁴	1	7	yes	—
17	C₂×D₃₁₇	—	2	112	no	—
18	C₂×D₃₁₇	C₂⁴	1	16	yes	complemented
19	C₄×C₃₁₇	C₂⁴	1	8	yes	complemented
20	C₂×C₃₁₇	C₂²×D₄	1	14	yes	complemented
21	C₂×C₃₁₇	C₂⁵	1	1	yes	characteristic
22	D₃₁₇	—	2	16	no	—
23	C₃₁₇	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×D₄	—	317	1	no	maximal, Sylow
25	C₂⁵	—	317	2	no	—
26	C₂²×D₄	—	317	28	no	—
27	C₂³×C₄	D₃₁₇	1	1	yes	characteristic, complemented
28	C₂⁴	—	317	28	no	—
29	C₂⁴	—	634	16	no	—
30	C₂⁴	C₂×D₃₁₇	1	1	yes	characteristic
31	C₂×D₄	—	317	112	no	—
32	C₂²×C₄	C₂×D₃₁₇	1	14	yes	complemented
33	C₂³	—	317	56	no	—
34	C₂³	—	634	112	no	—
35	C₂³	D₁₂₆₈	1	8	yes	direct product factor
36	C₂³	C₂²×D₃₁₇	1	7	yes	—
37	D₄	—	317	64	no	—
38	C₂×C₄	C₂²×D₃₁₇	1	28	yes	complemented
39	C₂²	—	317	16	no	—
40	C₂²	—	634	112	no	—
41	C₂²	C₂×D₁₂₆₈	1	28	yes	direct product factor
42	C₂²	C₂³×D₃₁₇	1	7	yes	—
43	C₄	C₂³×D₃₁₇	1	8	yes	complemented
44	C₂	—	634	16	no	—
45	C₂	C₂²×D₁₂₆₈	1	14	yes	direct product factor
46	C₂	C₂⁴×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₇
commutator, cosocle	C₂×C₃₁₇	C₂⁵
Frattini	C₂	C₂⁴×D₃₁₇
Fitting	C₂³×C₄×C₃₁₇	C₂
socle	C₂⁴×C₃₁₇	C₂²
hypercenter	C₂³×C₄	D₃₁₇
hypocenter	C₃₁₇	C₂³×D₄
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₃₁₇	C₂	semidirect
2. C₂²×D₁₂₆₈	C₂	direct (N × H)
3. C₂³×C₄×C₃₁₇	C₂	semidirect
5. C₂³×D₃₁₇	C₂²	semidirect
6. C₂×D₁₂₆₈	C₂²	direct (N × H)
8. C₂²×C₄×C₃₁₇	C₂²	semidirect
10. C₂²×D₃₁₇	C₂³	semidirect
11. D₁₂₆₈	C₂³	direct (N × H)
12. C₂³×C₃₁₇	D₄	semidirect
14. C₂×C₄×C₃₁₇	C₂³	semidirect
15. C₂²×C₃₁₇	C₂×D₄	semidirect
18. C₂×D₃₁₇	C₂⁴	semidirect
19. C₄×C₃₁₇	C₂⁴	semidirect
20. C₂×C₃₁₇	C₂²×D₄	semidirect
23. C₃₁₇	C₂³×D₄	semidirect
27. C₂³×C₄	D₃₁₇	semidirect
32. C₂²×C₄	C₂×D₃₁₇	semidirect
35. C₂³	D₁₂₆₈	direct (N × H)
38. C₂×C₄	C₂²×D₃₁₇	semidirect
41. C₂²	C₂×D₁₂₆₈	direct (N × H)
43. C₄	C₂³×D₃₁₇	semidirect
45. C₂	C₂²×D₁₂₆₈	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×D₃₁₇	2
C₂²×D₁₂₆₈	2
C₂³×C₄×C₃₁₇	2
C₂³×D₄	317

Subgroups — C23×D1268 #35204045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₂₆₈ #35204045