Subgroups — C₂×C₅×D₆₈ #13062820

Order 1360.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₁₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₅×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₆₈	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×C₁₇	C₂²	1	1	yes	characteristic
5	C₂×C₅×D₁₇	—	2	4	no	—
6	C₂×C₅×D₁₇	C₂²	1	4	yes	complemented
7	C₄×C₅×C₁₇	C₂²	1	2	yes	complemented
8	C₂×D₆₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₅×C₁₇	D₄	1	2	yes	complemented
10	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
11	C₅×D₁₇	—	2	4	no	—
12	C₂²×D₁₇	C₂×C₅	1	2	yes	complemented
13	C₂×C₄×C₁₇	C₂×C₅	1	1	yes	characteristic, complemented
14	D₆₈	C₂×C₅	1	4	yes	direct product factor
15	C₅×C₁₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
16	C₂×C₅×D₄	—	17	1	no	maximal, Hall
17	C₂²×C₁₇	C₂²×C₅	1	1	yes	characteristic
18	C₂×D₁₇	—	2	4	no	—
19	C₂×D₁₇	C₂²×C₅	1	4	yes	complemented
20	C₄×C₁₇	C₂²×C₅	1	2	yes	complemented
21	C₂³×C₅	—	17	2	no	—
22	C₅×D₄	—	17	4	no	—
23	C₂×C₄×C₅	D₁₇	1	1	yes	characteristic, complemented
24	C₂×C₁₇	C₅×D₄	1	2	yes	complemented
25	C₂×C₁₇	C₂³×C₅	1	1	yes	characteristic
26	D₁₇	—	2	4	no	—
27	C₂²×C₅	—	17	4	no	—
28	C₂²×C₅	—	34	4	no	—
29	C₂²×C₅	C₂×D₁₇	1	1	yes	characteristic
30	C₄×C₅	C₂×D₁₇	1	2	yes	complemented
31	C₁₇	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
32	C₂×D₄	—	17	1	no	Sylow
33	C₂×C₅	—	34	4	no	—
34	C₂×C₅	D₆₈	1	2	yes	direct product factor
35	C₂×C₅	C₂²×D₁₇	1	1	yes	characteristic
36	C₂³	—	17	2	no	—
37	D₄	—	17	4	no	—
38	C₂×C₄	C₅×D₁₇	1	1	yes	characteristic, complemented
39	C₅	C₂×D₆₈	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²	—	17	4	no	—
41	C₂²	—	34	4	no	—
42	C₂²	C₂×C₅×D₁₇	1	1	yes	characteristic
43	C₄	C₂×C₅×D₁₇	1	2	yes	complemented
44	C₂	—	34	4	no	—
45	C₂	C₅×D₆₈	1	2	yes	direct product factor
46	C₂	C₂²×C₅×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₁₇
commutator, cosocle	C₂×C₁₇	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₁₇
Fitting	C₂×C₄×C₅×C₁₇	C₂
socle	C₂²×C₅×C₁₇	C₂²
hypercenter	C₂×C₄×C₅	D₁₇
hypocenter	C₁₇	C₂×C₅×D₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₁₇	C₂	semidirect
2. C₂×C₄×C₅×C₁₇	C₂	semidirect
3. C₅×D₆₈	C₂	direct (N × H)
6. C₂×C₅×D₁₇	C₂²	semidirect
7. C₄×C₅×C₁₇	C₂²	semidirect
8. C₂×D₆₈	C₅	direct (N × H)
9. C₂×C₅×C₁₇	D₄	semidirect
12. C₂²×D₁₇	C₂×C₅	semidirect
13. C₂×C₄×C₁₇	C₂×C₅	semidirect
14. D₆₈	C₂×C₅	direct (N × H)
15. C₅×C₁₇	C₂×D₄	semidirect
19. C₂×D₁₇	C₂²×C₅	semidirect
20. C₄×C₁₇	C₂²×C₅	semidirect
23. C₂×C₄×C₅	D₁₇	semidirect
24. C₂×C₁₇	C₅×D₄	semidirect
30. C₄×C₅	C₂×D₁₇	semidirect
31. C₁₇	C₂×C₅×D₄	semidirect
34. C₂×C₅	D₆₈	direct (N × H)
38. C₂×C₄	C₅×D₁₇	semidirect
39. C₅	C₂×D₆₈	direct (N × H)
43. C₄	C₂×C₅×D₁₇	semidirect
45. C₂	C₅×D₆₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₁₇	2
C₂×C₄×C₅×C₁₇	2
C₅×D₆₈	2
C₂×D₆₈	5
C₂×C₅×D₄	17

Subgroups — C2×C5×D68 #13062820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₆₈ #13062820