Subgroups — Dic₄₁₇₀ #72215539

Order 16680.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₁₃₉₀	—	3	1	no	maximal, Hall
2	Dic₈₃₄	—	5	1	no	maximal, Hall
3	Dic₂₀₈₅	C₂	1	2	yes	maximal
4	C₃×C₄×C₅×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×C₅×C₁₃₉	C₂²	1	1	yes	characteristic
6	Dic₆₉₅	—	3	2	no	—
7	C₄×C₅×C₁₃₉	D₃	1	1	yes	characteristic
8	C₃×C₅×C₁₃₉	Q₈	1	1	yes	characteristic, complemented, Hall
9	C₃×C₄×C₁₃₉	D₅	1	1	yes	characteristic
10	Dic₄₁₇	—	5	2	no	—
11	C₂×C₅×C₁₃₉	C₂×D₃	1	1	yes	characteristic
12	Dic₂₇₈	—	15	1	no	Hall
13	C₂×C₃×C₁₃₉	C₂×D₅	1	1	yes	characteristic
14	C₅×C₁₃₉	Dic₆	1	1	yes	characteristic, complemented, Hall
15	C₄×C₁₃₉	D₁₅	1	1	yes	characteristic
16	Dic₁₃₉	—	15	2	no	—
17	C₃×C₁₃₉	Dic₁₀	1	1	yes	characteristic, complemented, Hall
18	C₂×C₁₃₉	C₂×D₁₅	1	1	yes	characteristic
19	C₁₃₉	Dic₃₀	1	1	yes	characteristic, complemented, Sylow
20	Dic₃₀	—	139	1	no	maximal, Hall
21	C₃×C₄×C₅	D₁₃₉	1	1	yes	characteristic
22	Dic₁₅	—	139	2	no	—
23	Dic₁₀	—	417	1	no	Hall
24	C₂×C₃×C₅	C₂×D₁₃₉	1	1	yes	characteristic
25	Dic₆	—	695	1	no	Hall
26	C₄×C₅	D₄₁₇	1	1	yes	characteristic
27	Dic₅	—	417	2	no	—
28	C₃×C₅	Dic₂₇₈	1	1	yes	characteristic, complemented, Hall
29	C₃×C₄	D₆₉₅	1	1	yes	characteristic
30	Dic₃	—	695	2	no	—
31	C₂×C₅	C₂×D₄₁₇	1	1	yes	characteristic
32	Q₈	—	2085	1	no	Sylow
33	C₂×C₃	C₂×D₆₉₅	1	1	yes	characteristic
34	C₅	Dic₈₃₄	1	1	yes	characteristic, complemented, Sylow
35	C₄	—	2085	2	no	—
36	C₄	D₂₀₈₅	1	1	yes	characteristic
37	C₃	Dic₁₃₉₀	1	1	yes	characteristic, complemented, Sylow
38	C₂	C₂×D₂₀₈₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₈₅
commutator, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₃₉	C₂²
Fitting	C₃×C₄×C₅×C₁₃₉	C₂
hypercenter	C₄	D₂₀₈₅
hypocenter	C₃×C₅×C₁₃₉	Q₈
LCS	G ▷ C₂×C₃×C₅×C₁₃₉ ▷ C₃×C₅×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
8. C₃×C₅×C₁₃₉	Q₈	semidirect
14. C₅×C₁₃₉	Dic₆	semidirect
17. C₃×C₁₃₉	Dic₁₀	semidirect
19. C₁₃₉	Dic₃₀	semidirect
28. C₃×C₅	Dic₂₇₈	semidirect
34. C₅	Dic₈₃₄	semidirect
37. C₃	Dic₁₃₉₀	semidirect

Maximal subgroups

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

subgroup	index
Dic₁₃₉₀	3
Dic₈₃₄	5
Dic₂₀₈₅	2
C₃×C₄×C₅×C₁₃₉	2
Dic₃₀	139

Subgroups — Dic4170 #72215539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₄₁₇₀ #72215539