Subgroups — C₂×C₇×C₉×D₇₁ #31822021

Order 17892.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₉×D₇₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₇×C₉×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×D₇₁	C₃	1	1	yes	maximal, characteristic
4	C₇×C₉×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×D₇₁	C₂×C₃	1	2	yes	—
6	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic
7	C₂×C₉×D₇₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₇×D₇₁	C₉	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic
10	C₂×C₉×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
11	C₉×D₇₁	C₂×C₇	1	2	yes	direct product factor
12	C₂×C₇×C₇₁	C₂×C₉	1	1	yes	characteristic, complemented
13	C₇×D₇₁	C₂×C₉	1	2	yes	direct product factor
14	C₂×C₃×D₇₁	C₃×C₇	1	1	yes	characteristic
15	C₉×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
16	C₇×C₇₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇₁	C₂×C₃×C₇	1	1	yes	characteristic
18	C₃×D₇₁	C₂×C₃×C₇	1	2	yes	—
19	C₂×D₇₁	C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₇×C₉	—	71	1	no	maximal, Hall
21	C₃×C₇₁	C₂²×C₃×C₇	1	1	yes	characteristic
22	C₂×C₇₁	C₂×C₇×C₉	1	1	yes	characteristic, complemented
23	D₇₁	C₂×C₇×C₉	1	2	yes	direct product factor
24	C₂×C₇×C₉	—	71	2	no	—
25	C₂×C₇×C₉	D₇₁	1	1	yes	characteristic, direct product factor
26	C₂²×C₃×C₇	—	71	1	no	—
27	C₇₁	C₂²×C₇×C₉	1	1	yes	characteristic, complemented, Sylow
28	C₇×C₉	C₂×D₇₁	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₇	—	71	2	no	—
30	C₂×C₃×C₇	C₃×D₇₁	1	1	yes	characteristic
31	C₂²×C₉	—	71	1	no	Hall
32	C₂²×C₇	—	71	1	no	Hall
33	C₃×C₇	C₂×C₃×D₇₁	1	1	yes	characteristic
34	C₂×C₉	—	71	2	no	—
35	C₂×C₉	C₇×D₇₁	1	1	yes	characteristic, direct product factor
36	C₂×C₇	—	71	2	no	—
37	C₂×C₇	C₉×D₇₁	1	1	yes	characteristic, direct product factor
38	C₂²×C₃	—	71	1	no	—
39	C₉	C₂×C₇×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
40	C₇	C₂×C₉×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃	—	71	2	no	—
42	C₂×C₃	C₃×C₇×D₇₁	1	1	yes	characteristic
43	C₂²	—	71	1	no	Sylow
44	C₃	C₂×C₃×C₇×D₇₁	1	1	yes	characteristic
45	C₂	—	71	2	no	—
46	C₂	C₇×C₉×D₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇×C₉	D₇₁
commutator, hypocenter	C₇₁	C₂²×C₇×C₉
Frattini	C₃	C₂×C₃×C₇×D₇₁
Fitting	C₂×C₇×C₉×C₇₁	C₂
socle	C₂×C₃×C₇×C₇₁	C₂×C₃
prime radical	C₂×C₃×C₇×D₇₁	C₃
cosocle	C₃×C₇₁	C₂²×C₃×C₇
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂×C₇×C₉

Complements

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

N	H ≅ G/N	type
1. C₇×C₉×D₇₁	C₂	direct (N × H)
2. C₂×C₇×C₉×C₇₁	C₂	semidirect
4. C₇×C₉×C₇₁	C₂²	semidirect
7. C₂×C₉×D₇₁	C₇	direct (N × H)
8. C₂×C₇×D₇₁	C₉	direct (N × H)
10. C₂×C₉×C₇₁	C₂×C₇	semidirect
11. C₉×D₇₁	C₂×C₇	direct (N × H)
12. C₂×C₇×C₇₁	C₂×C₉	semidirect
13. C₇×D₇₁	C₂×C₉	direct (N × H)
15. C₉×C₇₁	C₂²×C₇	semidirect
16. C₇×C₇₁	C₂²×C₉	semidirect
19. C₂×D₇₁	C₇×C₉	direct (N × H)
22. C₂×C₇₁	C₂×C₇×C₉	semidirect
23. D₇₁	C₂×C₇×C₉	direct (N × H)
25. C₂×C₇×C₉	D₇₁	direct (N × H)
27. C₇₁	C₂²×C₇×C₉	semidirect
28. C₇×C₉	C₂×D₇₁	direct (N × H)
35. C₂×C₉	C₇×D₇₁	direct (N × H)
37. C₂×C₇	C₉×D₇₁	direct (N × H)
39. C₉	C₂×C₇×D₇₁	direct (N × H)
40. C₇	C₂×C₉×D₇₁	direct (N × H)
46. C₂	C₇×C₉×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
C₇×C₉×D₇₁	2
C₂×C₇×C₉×C₇₁	2
C₂×C₃×C₇×D₇₁	3
C₂×C₉×D₇₁	7
C₂²×C₇×C₉	71

Subgroups — C2×C7×C9×D71 #31822021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₉×D₇₁ #31822021