Subgroups — C₃×C₅×Dic₂₂₁ #27210455

Order 13260.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₃×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₂₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₁₃×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×Dic₂₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₁₃×C₁₇	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃×C₁₃×C₁₇	C₂×C₅	1	1	yes	characteristic
7	C₅×C₁₃×C₁₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅×Dic₁₇	—	13	1	no	maximal, Hall
9	Dic₂₂₁	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₅×Dic₁₃	—	17	1	no	maximal, Hall
11	C₃×C₁₃×C₁₇	C₄×C₅	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅×C₁₇	D₁₃	1	1	yes	characteristic
13	C₂×C₁₃×C₁₇	C₂×C₃×C₅	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₃	D₁₇	1	1	yes	characteristic
15	C₅×Dic₁₇	—	13	1	no	Hall
16	C₅×Dic₁₃	—	17	1	no	Hall
17	C₃×C₅×C₁₇	Dic₁₃	1	1	yes	characteristic, complemented, Hall
18	C₁₃×C₁₇	C₃×C₄×C₅	1	1	yes	characteristic, complemented, Hall
19	C₃×Dic₁₇	—	13	1	no	Hall
20	C₃×C₅×C₁₃	Dic₁₇	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅×C₁₇	C₃×D₁₃	1	1	yes	characteristic
22	C₃×Dic₁₃	—	17	1	no	Hall
23	C₂×C₅×C₁₃	C₃×D₁₇	1	1	yes	characteristic
24	C₂×C₃×C₁₇	C₅×D₁₃	1	1	yes	characteristic
25	C₅×C₁₇	C₃×Dic₁₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₁₃	C₅×D₁₇	1	1	yes	characteristic
27	Dic₁₇	—	13	1	no	Hall
28	C₅×C₁₃	C₃×Dic₁₇	1	1	yes	characteristic, complemented, Hall
29	C₃×C₄×C₅	—	221	1	no	Hall
30	Dic₁₃	—	17	1	no	Hall
31	C₃×C₁₇	C₅×Dic₁₃	1	1	yes	characteristic, complemented, Hall
32	C₃×C₁₃	C₅×Dic₁₇	1	1	yes	characteristic, complemented, Hall
33	C₂×C₁₇	C₃×C₅×D₁₃	1	1	yes	characteristic
34	C₂×C₃×C₅	D₂₂₁	1	1	yes	characteristic
35	C₂×C₁₃	C₃×C₅×D₁₇	1	1	yes	characteristic
36	C₄×C₅	—	221	1	no	Hall
37	C₁₇	C₃×C₅×Dic₁₃	1	1	yes	characteristic, complemented, Sylow
38	C₃×C₅	Dic₂₂₁	1	1	yes	characteristic, direct product factor, Hall
39	C₁₃	C₃×C₅×Dic₁₇	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₄	—	221	1	no	Hall
41	C₂×C₅	C₃×D₂₂₁	1	1	yes	characteristic
42	C₂×C₃	C₅×D₂₂₁	1	1	yes	characteristic
43	C₅	C₃×Dic₂₂₁	1	1	yes	characteristic, direct product factor, Sylow
44	C₄	—	221	1	no	Sylow
45	C₃	C₅×Dic₂₂₁	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	C₃×C₅×D₂₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₂₂₁
commutator, hypocenter	C₁₃×C₁₇	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×D₂₂₁
Fitting, socle, prime radical	C₂×C₃×C₅×C₁₃×C₁₇	C₂
cosocle	C₂×C₁₃×C₁₇	C₂×C₃×C₅
LCS	G ▷ C₁₃×C₁₇
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×Dic₂₂₁	C₃	direct (N × H)
3. C₃×C₅×C₁₃×C₁₇	C₄	semidirect
4. C₃×Dic₂₂₁	C₅	direct (N × H)
7. C₅×C₁₃×C₁₇	C₃×C₄	semidirect
9. Dic₂₂₁	C₃×C₅	direct (N × H)
11. C₃×C₁₃×C₁₇	C₄×C₅	semidirect
17. C₃×C₅×C₁₇	Dic₁₃	semidirect
18. C₁₃×C₁₇	C₃×C₄×C₅	semidirect
20. C₃×C₅×C₁₃	Dic₁₇	semidirect
25. C₅×C₁₇	C₃×Dic₁₃	semidirect
28. C₅×C₁₃	C₃×Dic₁₇	semidirect
31. C₃×C₁₇	C₅×Dic₁₃	semidirect
32. C₃×C₁₃	C₅×Dic₁₇	semidirect
37. C₁₇	C₃×C₅×Dic₁₃	semidirect
38. C₃×C₅	Dic₂₂₁	direct (N × H)
39. C₁₃	C₃×C₅×Dic₁₇	semidirect
43. C₅	C₃×Dic₂₂₁	direct (N × H)
45. C₃	C₅×Dic₂₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₃×C₁₇	2
C₅×Dic₂₂₁	3
C₃×Dic₂₂₁	5
C₃×C₅×Dic₁₇	13
C₃×C₅×Dic₁₃	17

Subgroups — C3×C5×Dic221 #27210455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Dic₂₂₁ #27210455