Subgroups — D₅×D₆₈ #13062803

Order 1360.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

Maximal subgroups

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

subgroup	index
C₂×D₅×D₁₇	2
D₃₄₀	2
C₄×D₅×C₁₇	2
C₅×D₆₈	2
#10621755	2
C₂×D₆₈	5
D₄×D₅	17

Subgroups — D5×D68 #13062803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×D₆₈ #13062803