Subgroups — D₃²×C₇×C₇₁ #31822097

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊D₃×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₂×D₃×C₇×C₇₁	—	3	2	no	maximal
4	C₃²×C₇×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₃×C₇×C₇₁	—	3	2	no	—
6	D₃×C₇×C₇₁	—	6	1	no	—
7	D₃×C₇×C₇₁	D₃	1	2	yes	direct product factor
8	C₂×C₃×C₇×C₇₁	—	3	2	no	—
9	D₃²×C₇₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₇×C₇₁	—	9	1	no	Hall
11	C₃×C₇×C₇₁	—	2	1	no	—
12	C₃×C₇×C₇₁	C₂×D₃	1	2	yes	complemented
13	C₃⋊D₃×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
14	C₃×D₃×C₇₁	C₂×C₇	1	2	yes	complemented
15	C₂×C₇×C₇₁	—	3	2	no	—
16	C₂×C₇×C₇₁	—	9	1	no	—
17	C₂×D₃×C₇₁	—	3	2	no	—
18	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
19	C₇×C₇₁	D₃²	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₃×C₇₁	—	3	2	no	—
21	D₃×C₇₁	—	3	2	no	—
22	D₃×C₇₁	—	6	1	no	—
23	D₃×C₇₁	D₃×C₇	1	2	yes	direct product factor
24	C₂²×C₇₁	—	9	1	no	Hall
25	D₃²×C₇	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₃×C₇₁	—	2	1	no	—
27	C₃×C₇₁	C₂×D₃×C₇	1	2	yes	complemented
28	C₂×C₇₁	—	3	2	no	—
29	C₂×C₇₁	—	9	1	no	—
30	C₇×C₃⋊D₃	C₂×C₇₁	1	1	yes	characteristic, complemented
31	C₃×D₃×C₇	C₂×C₇₁	1	2	yes	complemented
32	C₂×D₃×C₇	—	3	2	no	—
33	C₇₁	D₃²×C₇	1	1	yes	characteristic, direct product factor, Sylow
34	C₃²×C₇	C₂²×C₇₁	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₇	—	3	2	no	—
36	D₃×C₇	—	3	2	no	—
37	D₃×C₇	—	6	1	no	—
38	D₃×C₇	D₃×C₇₁	1	2	yes	direct product factor
39	D₃²	C₇×C₇₁	1	1	yes	characteristic, direct product factor, Hall
40	C₂²×C₇	—	9	1	no	Hall
41	C₃×C₇	—	2	1	no	—
42	C₃×C₇	C₂×D₃×C₇₁	1	2	yes	complemented
43	C₃⋊D₃	C₂×C₇×C₇₁	1	1	yes	characteristic, complemented
44	C₃×D₃	C₂×C₇×C₇₁	1	2	yes	complemented
45	C₂×C₇	—	3	2	no	—
46	C₂×C₇	—	9	1	no	—
47	C₂×D₃	—	3	2	no	—
48	C₃²	C₂²×C₇×C₇₁	1	1	yes	characteristic, complemented, Sylow
49	C₇	D₃²×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₃	—	3	2	no	—
51	D₃	—	3	2	no	—
52	D₃	—	6	1	no	—
53	D₃	D₃×C₇×C₇₁	1	2	yes	direct product factor
54	C₂²	—	9	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	C₂×D₃×C₇×C₇₁	1	2	yes	complemented
57	C₂	—	3	2	no	—
58	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₇×C₇₁	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₇×C₇₁
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₇₁	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇×C₇₁

Complements

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

N	H ≅ G/N	type
1. C₇×C₃⋊D₃×C₇₁	C₂	semidirect
2. C₃×D₃×C₇×C₇₁	C₂	semidirect
4. C₃²×C₇×C₇₁	C₂²	semidirect
7. D₃×C₇×C₇₁	D₃	direct (N × H)
9. D₃²×C₇₁	C₇	direct (N × H)
12. C₃×C₇×C₇₁	C₂×D₃	semidirect
13. C₃⋊D₃×C₇₁	C₂×C₇	semidirect
14. C₃×D₃×C₇₁	C₂×C₇	semidirect
18. C₃²×C₇₁	C₂²×C₇	semidirect
19. C₇×C₇₁	D₃²	direct (N × H)
23. D₃×C₇₁	D₃×C₇	direct (N × H)
25. D₃²×C₇	C₇₁	direct (N × H)
27. C₃×C₇₁	C₂×D₃×C₇	semidirect
30. C₇×C₃⋊D₃	C₂×C₇₁	semidirect
31. C₃×D₃×C₇	C₂×C₇₁	semidirect
33. C₇₁	D₃²×C₇	direct (N × H)
34. C₃²×C₇	C₂²×C₇₁	semidirect
38. D₃×C₇	D₃×C₇₁	direct (N × H)
39. D₃²	C₇×C₇₁	direct (N × H)
42. C₃×C₇	C₂×D₃×C₇₁	semidirect
43. C₃⋊D₃	C₂×C₇×C₇₁	semidirect
44. C₃×D₃	C₂×C₇×C₇₁	semidirect
48. C₃²	C₂²×C₇×C₇₁	semidirect
49. C₇	D₃²×C₇₁	direct (N × H)
53. D₃	D₃×C₇×C₇₁	direct (N × H)
56. C₃	C₂×D₃×C₇×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃⋊D₃×C₇₁	2
C₃×D₃×C₇×C₇₁	2
C₂×D₃×C₇×C₇₁	3
D₃²×C₇₁	7
D₃²×C₇	71

Subgroups — D32×C7×C71 #31822097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃²×C₇×C₇₁ #31822097