Subgroups — C₃²×D₇×D₇₁ #31822084

Order 17892.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₄₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×D₇×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₇×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×D₇₁	C₃	1	4	yes	maximal, direct product factor
5	C₃²×C₇×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×D₄₉₇	C₂×C₃	1	4	yes	complemented
7	C₃×D₇×C₇₁	C₂×C₃	1	4	yes	complemented
8	C₃×C₇×D₇₁	C₂×C₃	1	4	yes	complemented
9	C₂×C₃²×D₇₁	—	7	1	no	maximal, Hall
10	D₇×D₇₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₇×C₇₁	C₂²×C₃	1	4	yes	complemented
12	C₂×C₃²×C₇₁	—	7	1	no	—
13	C₃²×D₇₁	—	7	1	no	—
14	C₃²×D₇₁	D₇	1	1	yes	characteristic, direct product factor
15	D₄₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
16	C₇×D₇₁	C₂×C₃²	1	1	yes	characteristic, complemented
17	D₇×C₇₁	C₂×C₃²	1	1	yes	characteristic, complemented
18	C₂×C₃×D₇₁	—	7	4	no	—
19	C₃²×C₇₁	C₂×D₇	1	1	yes	characteristic, complemented, Hall
20	C₇×C₇₁	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₇₁	—	7	4	no	—
22	C₃×D₇₁	—	7	4	no	—
23	C₃×D₇₁	C₃×D₇	1	4	yes	direct product factor
24	C₂×D₇₁	—	7	1	no	Hall
25	C₂×C₃²×D₇	—	71	1	no	maximal, Hall
26	C₃×C₇₁	C₂×C₃×D₇	1	4	yes	complemented
27	C₂×C₇₁	—	7	1	no	—
28	D₇₁	—	7	1	no	—
29	D₇₁	C₃²×D₇	1	1	yes	characteristic, direct product factor
30	C₂×C₃²×C₇	—	71	1	no	—
31	C₃²×D₇	—	71	1	no	—
32	C₃²×D₇	D₇₁	1	1	yes	characteristic, direct product factor
33	C₂×C₃×D₇	—	71	4	no	—
34	C₇₁	C₂×C₃²×D₇	1	1	yes	characteristic, complemented, Sylow
35	C₃²×C₇	C₂×D₇₁	1	1	yes	characteristic, complemented, Hall
36	C₂×C₃×C₇	—	71	4	no	—
37	C₃×D₇	—	71	4	no	—
38	C₃×D₇	C₃×D₇₁	1	4	yes	direct product factor
39	C₂²×C₃²	—	497	1	no	Hall
40	C₂×D₇	—	71	1	no	Hall
41	C₃×C₇	C₂×C₃×D₇₁	1	4	yes	complemented
42	C₂×C₃²	—	7	1	no	—
43	C₂×C₃²	—	71	1	no	—
44	C₂×C₃²	—	497	1	no	—
45	C₂×C₇	—	71	1	no	—
46	D₇	—	71	1	no	—
47	D₇	C₃²×D₇₁	1	1	yes	characteristic, direct product factor
48	C₂²×C₃	—	497	4	no	—
49	C₃²	D₇×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
50	C₇	C₂×C₃²×D₇₁	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	7	4	no	—
52	C₂×C₃	—	71	4	no	—
53	C₂×C₃	—	497	4	no	—
54	C₂²	—	497	1	no	Sylow
55	C₃	C₃×D₇×D₇₁	1	4	yes	direct product factor
56	C₂	—	7	1	no	—
57	C₂	—	71	1	no	—
58	C₂	—	497	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	D₇×D₇₁
commutator, cosocle, hypocenter	C₇×C₇₁	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₇₁	C₂²
LCS	G ▷ C₇×C₇₁
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×D₄₉₇	C₂	semidirect
2. C₃²×D₇×C₇₁	C₂	semidirect
3. C₃²×C₇×D₇₁	C₂	semidirect
4. C₃×D₇×D₇₁	C₃	direct (N × H)
5. C₃²×C₇×C₇₁	C₂²	semidirect
6. C₃×D₄₉₇	C₂×C₃	semidirect
7. C₃×D₇×C₇₁	C₂×C₃	semidirect
8. C₃×C₇×D₇₁	C₂×C₃	semidirect
10. D₇×D₇₁	C₃²	direct (N × H)
11. C₃×C₇×C₇₁	C₂²×C₃	semidirect
14. C₃²×D₇₁	D₇	direct (N × H)
15. D₄₉₇	C₂×C₃²	semidirect
16. C₇×D₇₁	C₂×C₃²	semidirect
17. D₇×C₇₁	C₂×C₃²	semidirect
19. C₃²×C₇₁	C₂×D₇	semidirect
20. C₇×C₇₁	C₂²×C₃²	semidirect
23. C₃×D₇₁	C₃×D₇	direct (N × H)
26. C₃×C₇₁	C₂×C₃×D₇	semidirect
29. D₇₁	C₃²×D₇	direct (N × H)
32. C₃²×D₇	D₇₁	direct (N × H)
34. C₇₁	C₂×C₃²×D₇	semidirect
35. C₃²×C₇	C₂×D₇₁	semidirect
38. C₃×D₇	C₃×D₇₁	direct (N × H)
41. C₃×C₇	C₂×C₃×D₇₁	semidirect
47. D₇	C₃²×D₇₁	direct (N × H)
49. C₃²	D₇×D₇₁	direct (N × H)
50. C₇	C₂×C₃²×D₇₁	semidirect
55. C₃	C₃×D₇×D₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₄₉₇	2
C₃²×D₇×C₇₁	2
C₃²×C₇×D₇₁	2
C₃×D₇×D₇₁	3
C₂×C₃²×D₇₁	7
C₂×C₃²×D₇	71

Subgroups — C32×D7×D71 #31822084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₇×D₇₁ #31822084