Subgroups — Dic₁₄₁₇₀ #72290657

Order 56680.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₂₈₃₄	—	5	1	no	maximal, Hall
2	Dic₁₀₉₀	—	13	1	no	maximal, Hall
3	Dic₇₀₈₅	C₂	1	2	yes	maximal
4	C₄×C₅×C₁₃×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₁₃×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₅×C₁₃×C₁₀₉	Q₈	1	1	yes	characteristic, complemented, Hall
7	Dic₁₄₁₇	—	5	2	no	—
8	C₄×C₁₃×C₁₀₉	D₅	1	1	yes	characteristic
9	C₂×C₁₃×C₁₀₉	C₂×D₅	1	1	yes	characteristic
10	Dic₅₄₅	—	13	2	no	—
11	C₄×C₅×C₁₀₉	D₁₃	1	1	yes	characteristic
12	C₁₃×C₁₀₉	Dic₁₀	1	1	yes	characteristic, complemented, Hall
13	C₂×C₅×C₁₀₉	C₂×D₁₃	1	1	yes	characteristic
14	Dic₂₁₈	—	65	1	no	Hall
15	C₅×C₁₀₉	Dic₂₆	1	1	yes	characteristic, complemented, Hall
16	Dic₁₃₀	—	109	1	no	maximal, Hall
17	C₄×C₁₀₉	D₆₅	1	1	yes	characteristic
18	Dic₁₀₉	—	65	2	no	—
19	C₄×C₅×C₁₃	D₁₀₉	1	1	yes	characteristic
20	Dic₆₅	—	109	2	no	—
21	C₂×C₁₀₉	C₂×D₆₅	1	1	yes	characteristic
22	C₂×C₅×C₁₃	C₂×D₁₀₉	1	1	yes	characteristic
23	C₁₀₉	Dic₁₃₀	1	1	yes	characteristic, complemented, Sylow
24	Dic₂₆	—	545	1	no	Hall
25	C₅×C₁₃	Dic₂₁₈	1	1	yes	characteristic, complemented, Hall
26	C₄×C₁₃	D₅₄₅	1	1	yes	characteristic
27	Dic₁₃	—	545	2	no	—
28	Dic₁₀	—	1417	1	no	Hall
29	C₂×C₁₃	C₂×D₅₄₅	1	1	yes	characteristic
30	C₄×C₅	D₁₄₁₇	1	1	yes	characteristic
31	Dic₅	—	1417	2	no	—
32	C₁₃	Dic₁₀₉₀	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₅	C₂×D₁₄₁₇	1	1	yes	characteristic
34	Q₈	—	7085	1	no	Sylow
35	C₅	Dic₂₈₃₄	1	1	yes	characteristic, complemented, Sylow
36	C₄	—	7085	2	no	—
37	C₄	D₇₀₈₅	1	1	yes	characteristic
38	C₂	C₂×D₇₀₈₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₀₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₁₀₉	C₂²
Fitting	C₄×C₅×C₁₃×C₁₀₉	C₂
hypercenter	C₄	D₇₀₈₅
hypocenter	C₅×C₁₃×C₁₀₉	Q₈
LCS	G ▷ C₂×C₅×C₁₃×C₁₀₉ ▷ C₅×C₁₃×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
6. C₅×C₁₃×C₁₀₉	Q₈	semidirect
12. C₁₃×C₁₀₉	Dic₁₀	semidirect
15. C₅×C₁₀₉	Dic₂₆	semidirect
23. C₁₀₉	Dic₁₃₀	semidirect
25. C₅×C₁₃	Dic₂₁₈	semidirect
32. C₁₃	Dic₁₀₉₀	semidirect
35. C₅	Dic₂₈₃₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₂₈₃₄	5
Dic₁₀₉₀	13
Dic₇₀₈₅	2
C₄×C₅×C₁₃×C₁₀₉	2
Dic₁₃₀	109

Subgroups — Dic14170 #72290657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₄₁₇₀ #72290657