Subgroups — Dic₁₃₅₁₅ #72289596

Order 54060.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₇×C₅₃	C₂	1	1	yes	maximal, characteristic
2	Dic₄₅₀₅	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₇×C₅₃	C₄	1	1	yes	characteristic, complemented, Hall
4	Dic₂₇₀₃	—	5	1	no	maximal, Hall
5	C₂×C₅×C₁₇×C₅₃	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₇×C₅₃	D₅	1	1	yes	characteristic
7	C₅×C₁₇×C₅₃	Dic₃	1	1	yes	characteristic, complemented, Hall
8	Dic₉₀₁	—	15	1	no	Hall
9	Dic₇₉₅	—	17	1	no	maximal, Hall
10	C₃×C₁₇×C₅₃	Dic₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₇×C₅₃	D₁₅	1	1	yes	characteristic
12	C₂×C₃×C₅×C₅₃	D₁₇	1	1	yes	characteristic
13	Dic₂₆₅	—	51	1	no	Hall
14	Dic₂₅₅	—	53	1	no	maximal, Hall
15	C₁₇×C₅₃	Dic₁₅	1	1	yes	characteristic, complemented, Hall
16	C₃×C₅×C₅₃	Dic₁₇	1	1	yes	characteristic, complemented, Hall
17	Dic₁₅₉	—	85	1	no	Hall
18	C₂×C₅×C₅₃	D₅₁	1	1	yes	characteristic
19	C₂×C₃×C₅×C₁₇	D₅₃	1	1	yes	characteristic
20	Dic₈₅	—	159	1	no	Hall
21	C₂×C₃×C₅₃	D₈₅	1	1	yes	characteristic
22	C₅×C₅₃	Dic₅₁	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅×C₁₇	Dic₅₃	1	1	yes	characteristic, complemented, Hall
24	Dic₅₃	—	255	1	no	Hall
25	Dic₅₁	—	265	1	no	Hall
26	C₂×C₅×C₁₇	D₁₅₉	1	1	yes	characteristic
27	C₃×C₅₃	Dic₈₅	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅₃	D₂₅₅	1	1	yes	characteristic
29	C₂×C₃×C₁₇	D₂₆₅	1	1	yes	characteristic
30	C₅×C₁₇	Dic₁₅₉	1	1	yes	characteristic, complemented, Hall
31	Dic₁₇	—	795	1	no	Hall
32	Dic₁₅	—	901	1	no	Hall
33	C₅₃	Dic₂₅₅	1	1	yes	characteristic, complemented, Sylow
34	C₃×C₁₇	Dic₂₆₅	1	1	yes	characteristic, complemented, Hall
35	C₂×C₁₇	D₇₉₅	1	1	yes	characteristic
36	C₂×C₃×C₅	D₉₀₁	1	1	yes	characteristic
37	Dic₅	—	2703	1	no	Hall
38	C₁₇	Dic₇₉₅	1	1	yes	characteristic, complemented, Sylow
39	C₃×C₅	Dic₉₀₁	1	1	yes	characteristic, complemented, Hall
40	Dic₃	—	4505	1	no	Hall
41	C₂×C₅	D₂₇₀₃	1	1	yes	characteristic
42	C₂×C₃	D₄₅₀₅	1	1	yes	characteristic
43	C₅	Dic₂₇₀₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	13515	1	no	Sylow
45	C₃	Dic₄₅₀₅	1	1	yes	characteristic, complemented, Sylow
46	C₂	D₁₃₅₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₅₁₅
commutator, hypocenter	C₃×C₅×C₁₇×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₇×C₅₃	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₅₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₇×C₅₃	C₄	semidirect
7. C₅×C₁₇×C₅₃	Dic₃	semidirect
10. C₃×C₁₇×C₅₃	Dic₅	semidirect
15. C₁₇×C₅₃	Dic₁₅	semidirect
16. C₃×C₅×C₅₃	Dic₁₇	semidirect
22. C₅×C₅₃	Dic₅₁	semidirect
23. C₃×C₅×C₁₇	Dic₅₃	semidirect
27. C₃×C₅₃	Dic₈₅	semidirect
30. C₅×C₁₇	Dic₁₅₉	semidirect
33. C₅₃	Dic₂₅₅	semidirect
34. C₃×C₁₇	Dic₂₆₅	semidirect
38. C₁₇	Dic₇₉₅	semidirect
39. C₃×C₅	Dic₉₀₁	semidirect
43. C₅	Dic₂₇₀₃	semidirect
45. C₃	Dic₄₅₀₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₇×C₅₃	2
Dic₄₅₀₅	3
Dic₂₇₀₃	5
Dic₇₉₅	17
Dic₂₅₅	53

Subgroups — Dic13515 #72289596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃₅₁₅ #72289596