Subgroups — #13062800

Order 1360.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₄₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621755	C₂	1	2	yes	maximal, complemented
5	#10621753	C₂	1	2	yes	maximal, complemented
6	C₂×D₈₅	C₂²	1	2	yes	—
7	C₂×C₅×D₁₇	C₂²	1	1	yes	characteristic
8	C₄×C₅×C₁₇	C₂²	1	1	yes	characteristic
9	C₅×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×Dic₅	C₂²	1	2	yes	—
11	#63256	—	5	1	no	maximal, Hall
12	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
13	D₈₅	—	2	2	no	—
14	C₅×D₁₇	—	2	1	no	—
15	Q₈×C₁₇	—	5	1	no	—
16	D₆₈	—	5	2	no	—
17	D₆₈	—	5	1	no	—
18	C₄×D₁₇	—	5	2	no	—
19	C₄×D₁₇	D₅	1	1	yes	characteristic, complemented
20	C₅×C₁₇	#41	1	1	yes	characteristic, complemented, Hall
21	#709	—	17	1	no	maximal, Hall
22	C₂×D₁₇	—	5	2	no	—
23	C₂×D₁₇	C₂×D₅	1	1	yes	characteristic
24	C₄×C₁₇	—	5	2	no	—
25	C₄×C₁₇	C₂×D₅	1	1	yes	characteristic
26	Dic₁₇	C₂×D₅	1	1	yes	characteristic
27	C₂×C₄×C₅	—	17	1	no	—
28	C₂²⋊D₅	—	17	2	no	—
29	D₂₀	—	17	1	no	—
30	C₄×D₅	—	17	2	no	—
31	Dic₁₀	D₁₇	1	1	yes	characteristic, complemented
32	C₂×C₁₇	C₂²×D₅	1	1	yes	characteristic
33	D₁₇	—	2	1	no	—
34	D₁₇	—	10	2	no	—
35	C₂²×C₅	—	17	1	no	—
36	C₂×D₅	—	17	2	no	—
37	C₄×C₅	—	17	1	no	—
38	C₄×C₅	C₂×D₁₇	1	1	yes	characteristic
39	Dic₅	C₂×D₁₇	1	2	yes	—
40	C₁₇	#709	1	1	yes	characteristic, complemented, Sylow
41	#41	—	85	1	no	Sylow
42	C₂×C₅	—	34	1	no	—
43	C₂×C₅	C₂²×D₁₇	1	1	yes	characteristic
44	D₅	—	34	2	no	—
45	Q₈	—	5	1	no	—
46	D₄	—	85	2	no	—
47	D₄	—	85	1	no	—
48	C₂×C₄	—	17	1	no	—
49	C₂×C₄	—	85	2	no	—
50	C₅	#63256	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	17	1	no	—
52	C₂²	—	85	2	no	—
53	C₄	—	5	2	no	—
54	C₄	—	17	1	no	—
55	C₄	D₅×D₁₇	1	1	yes	characteristic
56	C₂	—	34	1	no	—
57	C₂	—	170	2	no	—
58	C₂	C₂×D₅×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×D₁₇
commutator, socle, cosocle	C₂×C₅×C₁₇	C₂³
Fitting	C₄×C₅×C₁₇	C₂²
hypercenter	C₄	D₅×D₁₇
hypocenter	C₅×C₁₇	#41
LCS	G ▷ C₂×C₅×C₁₇ ▷ C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. D₃₄₀	C₂	semidirect
2. C₁₇×Dic₁₀	C₂	semidirect
3. C₄×C₅×D₁₇	C₂	semidirect
4. #10621755	C₂	semidirect
5. #10621753	C₂	semidirect
19. C₄×D₁₇	D₅	semidirect
20. C₅×C₁₇	#41	semidirect
31. Dic₁₀	D₁₇	semidirect
40. C₁₇	#709	semidirect
50. C₅	#63256	semidirect

Maximal subgroups

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

subgroup	index
D₃₄₀	2
C₁₇×Dic₁₀	2
C₄×C₅×D₁₇	2
#10621755	2
#10621753	2
#63256	5
#709	17