Subgroups — C₂×C₄×C₅×D₁₇ #13062819

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	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₁₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₁₇	C₄	1	4	yes	direct product factor
7	C₂×C₅×D₁₇	C₂²	1	2	yes	—
8	C₄×C₅×C₁₇	C₂²	1	2	yes	complemented
9	C₅×Dic₁₇	C₂²	1	2	yes	complemented
10	C₂×C₄×D₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₅×C₁₇	C₂×C₄	1	2	yes	complemented
12	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
13	C₅×D₁₇	C₂×C₄	1	4	yes	direct product factor
14	C₂²×D₁₇	C₂×C₅	1	1	yes	characteristic
15	C₂×C₄×C₁₇	C₂×C₅	1	1	yes	characteristic, complemented
16	C₂×Dic₁₇	C₂×C₅	1	1	yes	characteristic, complemented
17	C₄×D₁₇	C₂×C₅	1	4	yes	direct product factor
18	C₅×C₁₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₄×C₅	—	17	1	no	maximal, Hall
20	C₂²×C₁₇	C₂²×C₅	1	1	yes	characteristic
21	C₂×D₁₇	C₄×C₅	1	4	yes	direct product factor
22	C₂×D₁₇	C₂²×C₅	1	2	yes	—
23	C₄×C₁₇	C₂²×C₅	1	2	yes	complemented
24	Dic₁₇	C₂²×C₅	1	2	yes	complemented
25	C₂³×C₅	—	17	1	no	—
26	C₂×C₄×C₅	—	17	1	no	—
27	C₂×C₄×C₅	—	17	4	no	—
28	C₂×C₄×C₅	D₁₇	1	1	yes	characteristic, direct product factor
29	C₂×C₁₇	C₂×C₄×C₅	1	2	yes	complemented
30	C₂×C₁₇	C₂³×C₅	1	1	yes	characteristic
31	D₁₇	C₂×C₄×C₅	1	4	yes	direct product factor
32	C₂²×C₅	—	17	4	no	—
33	C₂²×C₅	—	17	2	no	—
34	C₂²×C₅	C₂×D₁₇	1	1	yes	characteristic
35	C₄×C₅	—	17	2	no	—
36	C₄×C₅	C₂×D₁₇	1	2	yes	direct product factor
37	C₁₇	C₂²×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₄	—	17	1	no	Sylow
39	C₂×C₅	—	17	4	no	—
40	C₂×C₅	C₄×D₁₇	1	2	yes	direct product factor
41	C₂×C₅	C₂²×D₁₇	1	1	yes	characteristic
42	C₂³	—	17	1	no	—
43	C₂×C₄	—	17	1	no	—
44	C₂×C₄	—	17	4	no	—
45	C₂×C₄	C₅×D₁₇	1	1	yes	characteristic, direct product factor
46	C₅	C₂×C₄×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²	—	17	4	no	—
48	C₂²	—	17	2	no	—
49	C₂²	C₂×C₅×D₁₇	1	1	yes	characteristic
50	C₄	—	17	2	no	—
51	C₄	C₂×C₅×D₁₇	1	2	yes	direct product factor
52	C₂	—	17	4	no	—
53	C₂	C₄×C₅×D₁₇	1	2	yes	direct product factor
54	C₂	C₂²×C₅×D₁₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₁₇	C₂	semidirect
3. C₂×C₅×Dic₁₇	C₂	semidirect
4. C₄×C₅×D₁₇	C₂	direct (N × H)
6. C₂×C₅×D₁₇	C₄	direct (N × H)
8. C₄×C₅×C₁₇	C₂²	semidirect
9. C₅×Dic₁₇	C₂²	semidirect
10. C₂×C₄×D₁₇	C₅	direct (N × H)
11. C₂×C₅×C₁₇	C₂×C₄	semidirect
13. C₅×D₁₇	C₂×C₄	direct (N × H)
15. C₂×C₄×C₁₇	C₂×C₅	semidirect
16. C₂×Dic₁₇	C₂×C₅	semidirect
17. C₄×D₁₇	C₂×C₅	direct (N × H)
18. C₅×C₁₇	C₂²×C₄	semidirect
21. C₂×D₁₇	C₄×C₅	direct (N × H)
23. C₄×C₁₇	C₂²×C₅	semidirect
24. Dic₁₇	C₂²×C₅	semidirect
28. C₂×C₄×C₅	D₁₇	direct (N × H)
29. C₂×C₁₇	C₂×C₄×C₅	semidirect
31. D₁₇	C₂×C₄×C₅	direct (N × H)
36. C₄×C₅	C₂×D₁₇	direct (N × H)
37. C₁₇	C₂²×C₄×C₅	semidirect
40. C₂×C₅	C₄×D₁₇	direct (N × H)
45. C₂×C₄	C₅×D₁₇	direct (N × H)
46. C₅	C₂×C₄×D₁₇	direct (N × H)
51. C₄	C₂×C₅×D₁₇	direct (N × H)
53. C₂	C₄×C₅×D₁₇	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
C₂×C₅×Dic₁₇	2
C₄×C₅×D₁₇	2
C₂×C₄×D₁₇	5
C₂²×C₄×C₅	17

Subgroups — C2×C4×C5×D17 #13062819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₁₇ #13062819