Subgroups — C₂×C₃×C₅²×D₇₁ #36319150

Order 21300.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×D₇₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₇₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅²×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₅×D₇₁	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₇₁	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₅²×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₇₁	C₂×C₅	1	12	yes	direct product factor
9	C₂×C₃×C₅×C₇₁	C₂×C₅	1	6	yes	complemented
10	C₅²×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₅×D₇₁	C₃×C₅	1	6	yes	direct product factor
12	C₃×C₅×C₇₁	C₂²×C₅	1	6	yes	complemented
13	C₂×C₃×D₇₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₅×C₇₁	C₂×C₃×C₅	1	6	yes	complemented
15	C₅×D₇₁	C₂×C₃×C₅	1	12	yes	direct product factor
16	C₂×C₃×C₇₁	C₂×C₅²	1	1	yes	characteristic, complemented
17	C₃×D₇₁	C₂×C₅²	1	2	yes	direct product factor
18	C₅×C₇₁	C₂²×C₃×C₅	1	6	yes	complemented
19	C₂²×C₃×C₅²	—	71	1	no	maximal, Hall
20	C₂×D₇₁	C₃×C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₇₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅²	—	71	2	no	—
23	C₂×C₃×C₅²	D₇₁	1	1	yes	characteristic, direct product factor
24	C₂×C₇₁	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
25	D₇₁	C₂×C₃×C₅²	1	2	yes	direct product factor
26	C₂²×C₅²	—	71	1	no	Hall
27	C₃×C₅²	C₂×D₇₁	1	1	yes	characteristic, direct product factor, Hall
28	C₇₁	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₃×C₅	—	71	6	no	—
30	C₂×C₅²	—	71	2	no	—
31	C₂×C₅²	C₃×D₇₁	1	1	yes	characteristic, direct product factor
32	C₂×C₃×C₅	—	71	12	no	—
33	C₂×C₃×C₅	C₅×D₇₁	1	6	yes	direct product factor
34	C₅²	C₂×C₃×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₅	—	71	6	no	—
36	C₃×C₅	C₂×C₅×D₇₁	1	6	yes	direct product factor
37	C₂²×C₃	—	71	1	no	Hall
38	C₂×C₅	—	71	12	no	—
39	C₂×C₅	C₃×C₅×D₇₁	1	6	yes	direct product factor
40	C₂×C₃	—	71	2	no	—
41	C₂×C₃	C₅²×D₇₁	1	1	yes	characteristic, direct product factor
42	C₅	C₂×C₃×C₅×D₇₁	1	6	yes	direct product factor
43	C₂²	—	71	1	no	Sylow
44	C₃	C₂×C₅²×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	—	71	2	no	—
46	C₂	C₃×C₅²×D₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²	D₇₁
commutator, cosocle, hypocenter	C₇₁	C₂²×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇₁	C₂
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×D₇₁	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₇₁	C₂	semidirect
3. C₂×C₅²×D₇₁	C₃	direct (N × H)
4. C₃×C₅²×C₇₁	C₂²	semidirect
5. C₂×C₃×C₅×D₇₁	C₅	direct (N × H)
6. C₅²×D₇₁	C₂×C₃	direct (N × H)
7. C₂×C₅²×C₇₁	C₂×C₃	semidirect
8. C₃×C₅×D₇₁	C₂×C₅	direct (N × H)
9. C₂×C₃×C₅×C₇₁	C₂×C₅	semidirect
10. C₅²×C₇₁	C₂²×C₃	semidirect
11. C₂×C₅×D₇₁	C₃×C₅	direct (N × H)
12. C₃×C₅×C₇₁	C₂²×C₅	semidirect
13. C₂×C₃×D₇₁	C₅²	direct (N × H)
14. C₂×C₅×C₇₁	C₂×C₃×C₅	semidirect
15. C₅×D₇₁	C₂×C₃×C₅	direct (N × H)
16. C₂×C₃×C₇₁	C₂×C₅²	semidirect
17. C₃×D₇₁	C₂×C₅²	direct (N × H)
18. C₅×C₇₁	C₂²×C₃×C₅	semidirect
20. C₂×D₇₁	C₃×C₅²	direct (N × H)
21. C₃×C₇₁	C₂²×C₅²	semidirect
23. C₂×C₃×C₅²	D₇₁	direct (N × H)
24. C₂×C₇₁	C₂×C₃×C₅²	semidirect
25. D₇₁	C₂×C₃×C₅²	direct (N × H)
27. C₃×C₅²	C₂×D₇₁	direct (N × H)
28. C₇₁	C₂²×C₃×C₅²	semidirect
31. C₂×C₅²	C₃×D₇₁	direct (N × H)
33. C₂×C₃×C₅	C₅×D₇₁	direct (N × H)
34. C₅²	C₂×C₃×D₇₁	direct (N × H)
36. C₃×C₅	C₂×C₅×D₇₁	direct (N × H)
39. C₂×C₅	C₃×C₅×D₇₁	direct (N × H)
41. C₂×C₃	C₅²×D₇₁	direct (N × H)
42. C₅	C₂×C₃×C₅×D₇₁	direct (N × H)
44. C₃	C₂×C₅²×D₇₁	direct (N × H)
46. C₂	C₃×C₅²×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
C₃×C₅²×D₇₁	2
C₂×C₃×C₅²×C₇₁	2
C₂×C₅²×D₇₁	3
C₂×C₃×C₅×D₇₁	5
C₂²×C₃×C₅²	71

Subgroups — C2×C3×C52×D71 #36319150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅²×D₇₁ #36319150