Subgroups — C₁₇×#709 #13062831

Order 1360.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	C₁₇×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₅×C₁₇	C₂	1	2	yes	maximal, complemented
5	C₁₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×D₅×C₁₇	C₂²	1	2	yes	—
8	C₄×C₅×C₁₇	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×Dic₅	C₂²	1	2	yes	—
11	#41×C₁₇	—	5	1	no	maximal, Hall
12	C₂×C₅×C₁₇	—	2	1	no	—
13	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
14	D₅×C₁₇	—	2	2	no	—
15	Q₈×C₁₇	—	5	1	no	—
16	D₄×C₁₇	—	5	2	no	—
17	D₄×C₁₇	—	5	1	no	—
18	C₂×C₄×C₁₇	—	5	2	no	—
19	C₂×C₄×C₁₇	D₅	1	1	yes	characteristic, complemented
20	C₅×C₁₇	#41	1	1	yes	characteristic, complemented, Hall
21	#709	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₁₇	—	5	2	no	—
23	C₂²×C₁₇	C₂×D₅	1	1	yes	characteristic
24	C₄×C₁₇	—	5	2	no	—
25	C₄×C₁₇	C₂×D₅	1	1	yes	characteristic
26	C₄×C₁₇	C₂×D₅	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂×C₁₇	1	1	yes	characteristic, complemented
28	C₂²⋊D₅	C₂×C₁₇	1	2	yes	complemented
29	D₂₀	C₂×C₁₇	1	1	yes	characteristic, complemented
30	C₄×D₅	C₂×C₁₇	1	2	yes	complemented
31	Dic₁₀	C₂×C₁₇	1	1	yes	characteristic, complemented
32	C₂×C₁₇	—	2	1	no	—
33	C₂×C₁₇	—	10	2	no	—
34	C₂×C₁₇	C₂²×D₅	1	1	yes	characteristic
35	C₂²×C₅	C₂²×C₁₇	1	1	yes	characteristic
36	C₂×D₅	C₂²×C₁₇	1	2	yes	—
37	C₄×C₅	C₂²×C₁₇	1	1	yes	characteristic
38	C₄×C₅	C₂²×C₁₇	1	1	yes	characteristic
39	Dic₅	C₂²×C₁₇	1	2	yes	—
40	C₁₇	#709	1	1	yes	characteristic, direct product factor, Sylow
41	#41	—	5	1	no	Sylow
42	C₂×C₅	—	2	1	no	—
43	C₂×C₅	C₂³×C₁₇	1	1	yes	characteristic
44	D₅	—	2	2	no	—
45	Q₈	—	5	1	no	—
46	D₄	—	5	2	no	—
47	D₄	—	5	1	no	—
48	C₂×C₄	—	5	2	no	—
49	C₂×C₄	D₅×C₁₇	1	1	yes	characteristic, complemented
50	C₅	#41×C₁₇	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	5	2	no	—
52	C₂²	C₂×D₅×C₁₇	1	1	yes	characteristic
53	C₄	—	5	2	no	—
54	C₄	C₂×D₅×C₁₇	1	1	yes	characteristic
55	C₄	C₂×D₅×C₁₇	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	10	2	no	—
58	C₂	C₂²×D₅×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₇	C₂×D₅
commutator, cosocle	C₂×C₅	C₂³×C₁₇
Frattini	C₂	C₂²×D₅×C₁₇
Fitting	C₂×C₄×C₅×C₁₇	C₂
socle	C₂×C₅×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₇	D₅
hypocenter	C₅	#41×C₁₇
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅×C₁₇	C₂	semidirect
2. C₁₇×C₂²⋊D₅	C₂	semidirect
3. C₁₇×D₂₀	C₂	semidirect
4. C₄×D₅×C₁₇	C₂	semidirect
5. C₁₇×Dic₁₀	C₂	semidirect
19. C₂×C₄×C₁₇	D₅	semidirect
20. C₅×C₁₇	#41	semidirect
21. #709	C₁₇	direct (N × H)
27. C₂×C₄×C₅	C₂×C₁₇	semidirect
28. C₂²⋊D₅	C₂×C₁₇	semidirect
29. D₂₀	C₂×C₁₇	semidirect
30. C₄×D₅	C₂×C₁₇	semidirect
31. Dic₁₀	C₂×C₁₇	semidirect
40. C₁₇	#709	direct (N × H)
49. C₂×C₄	D₅×C₁₇	semidirect
50. C₅	#41×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₇	2
C₁₇×C₂²⋊D₅	2
C₁₇×D₂₀	2
C₄×D₅×C₁₇	2
C₁₇×Dic₁₀	2
#41×C₁₇	5
#709	17

Subgroups — C17×#709 #13062831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#709 #13062831