Subgroups — C₃×C₇₁×Dic₂₁ #31822048

Order 17892.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×Dic₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×Dic₇×C₇₁	—	3	1	no	maximal
4	C₃²×C₇×C₇₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇×C₇₁	—	2	1	no	—
6	C₂×C₃×C₇×C₇₁	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic
8	C₃×Dic₃×C₇₁	—	7	1	no	maximal, Hall
9	Dic₇×C₇₁	—	3	1	no	Hall
10	C₃×C₇×C₇₁	—	2	1	no	—
11	C₃×C₇×C₇₁	Dic₃	1	1	yes	characteristic, complemented
12	C₃×C₇×C₇₁	C₃×C₄	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₇₁	D₇	1	1	yes	characteristic
14	C₂×C₇×C₇₁	C₃×D₃	1	1	yes	characteristic
15	C₃×C₄×C₇₁	—	21	1	no	—
16	Dic₃×C₇₁	—	7	1	no	—
17	C₃²×C₇₁	Dic₇	1	1	yes	characteristic, complemented, Hall
18	C₇×C₇₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₇₁	—	2	1	no	—
20	C₂×C₃×C₇₁	C₃×D₇	1	1	yes	characteristic
21	C₂×C₃×C₇₁	D₂₁	1	1	yes	characteristic
22	C₄×C₇₁	—	21	1	no	Hall
23	C₃×Dic₂₁	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₃×C₇₁	—	2	1	no	—
25	C₃×C₇₁	C₃×Dic₇	1	1	yes	characteristic, complemented
26	C₃×C₇₁	Dic₂₁	1	1	yes	characteristic, direct product factor
27	C₂×C₇₁	C₃×D₂₁	1	1	yes	characteristic
28	C₂×C₃²×C₇	C₂×C₇₁	1	1	yes	characteristic
29	Dic₂₁	C₃×C₇₁	1	1	yes	characteristic, direct product factor
30	C₃×Dic₇	—	3	1	no	—
31	C₇₁	C₃×Dic₂₁	1	1	yes	characteristic, direct product factor, Sylow
32	C₃²×C₇	C₄×C₇₁	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇	—	2	1	no	—
34	C₂×C₃×C₇	D₃×C₇₁	1	1	yes	characteristic
35	C₂×C₃×C₇	C₂×C₃×C₇₁	1	1	yes	characteristic
36	C₃×Dic₃	—	7	1	no	Hall
37	Dic₇	—	3	1	no	Hall
38	C₃×C₇	—	2	1	no	—
39	C₃×C₇	Dic₃×C₇₁	1	1	yes	characteristic, complemented
40	C₃×C₇	C₃×C₄×C₇₁	1	1	yes	characteristic, complemented
41	C₂×C₃²	D₇×C₇₁	1	1	yes	characteristic
42	C₂×C₇	C₃×D₃×C₇₁	1	1	yes	characteristic
43	C₃×C₄	—	21	1	no	—
44	Dic₃	—	7	1	no	—
45	C₃²	Dic₇×C₇₁	1	1	yes	characteristic, complemented, Sylow
46	C₇	C₃×Dic₃×C₇₁	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₃×D₇×C₇₁	1	1	yes	characteristic
49	C₂×C₃	D₂₁×C₇₁	1	1	yes	characteristic
50	C₄	—	21	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×Dic₇×C₇₁	1	1	yes	characteristic, complemented
53	C₃	C₇₁×Dic₂₁	1	1	yes	characteristic, direct product factor
54	C₂	C₃×D₂₁×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇₁	D₂₁
commutator, hypocenter	C₃×C₇	C₃×C₄×C₇₁
Frattini	C₂	C₃×D₂₁×C₇₁
Fitting, socle, prime radical	C₂×C₃²×C₇×C₇₁	C₂
cosocle	C₂×C₃×C₇	C₂×C₃×C₇₁
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃×C₇₁

Complements

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

N	H ≅ G/N	type
2. C₇₁×Dic₂₁	C₃	direct (N × H)
4. C₃²×C₇×C₇₁	C₄	semidirect
11. C₃×C₇×C₇₁	Dic₃	semidirect
12. C₃×C₇×C₇₁	C₃×C₄	semidirect
17. C₃²×C₇₁	Dic₇	semidirect
18. C₇×C₇₁	C₃×Dic₃	semidirect
23. C₃×Dic₂₁	C₇₁	direct (N × H)
25. C₃×C₇₁	C₃×Dic₇	semidirect
26. C₃×C₇₁	Dic₂₁	direct (N × H)
29. Dic₂₁	C₃×C₇₁	direct (N × H)
31. C₇₁	C₃×Dic₂₁	direct (N × H)
32. C₃²×C₇	C₄×C₇₁	semidirect
39. C₃×C₇	Dic₃×C₇₁	semidirect
40. C₃×C₇	C₃×C₄×C₇₁	semidirect
45. C₃²	Dic₇×C₇₁	semidirect
46. C₇	C₃×Dic₃×C₇₁	semidirect
52. C₃	C₃×Dic₇×C₇₁	semidirect
53. C₃	C₇₁×Dic₂₁	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×C₃²×C₇×C₇₁	2
C₇₁×Dic₂₁	3
C₃×Dic₇×C₇₁	3
C₃×Dic₃×C₇₁	7
C₃×Dic₂₁	71

Subgroups — C3×C71×Dic21 #31822048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇₁×Dic₂₁ #31822048