Subgroups — C₂×C₇²×D₁₅₉ #44528489

Order 31164.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×D₁₅₉	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₇²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇²×D₅₃	—	3	1	no	maximal, Hall
4	C₃×C₇²×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇²×D₅₃	—	3	2	no	—
6	C₂×C₇²×C₅₃	D₃	1	1	yes	characteristic, complemented
7	C₂×C₇×D₁₅₉	C₇	1	8	yes	maximal, direct product factor
8	C₇²×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
9	C₇×D₁₅₉	C₂×C₇	1	16	yes	direct product factor
10	C₂×C₃×C₇×C₅₃	C₂×C₇	1	8	yes	complemented
11	C₂×C₇×D₅₃	—	3	8	no	—
12	C₃×C₇×C₅₃	C₂²×C₇	1	8	yes	complemented
13	C₂×C₇×C₅₃	D₃×C₇	1	8	yes	complemented
14	C₇×D₅₃	—	3	16	no	—
15	C₂×D₁₅₉	C₇²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×D₃×C₇²	—	53	1	no	maximal, Hall
17	C₇×C₅₃	C₂×D₃×C₇	1	8	yes	complemented
18	C₂×C₃×C₅₃	C₂×C₇²	1	1	yes	characteristic, complemented
19	D₁₅₉	C₂×C₇²	1	2	yes	direct product factor
20	C₂×C₃×C₇²	D₅₃	1	1	yes	characteristic, complemented
21	D₃×C₇²	—	53	2	no	—
22	C₂×D₅₃	—	3	1	no	Hall
23	C₂²×C₇²	—	159	1	no	Hall
24	C₃×C₅₃	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
25	C₃×C₇²	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₅₃	D₃×C₇²	1	1	yes	characteristic, complemented
27	D₅₃	—	3	2	no	—
28	C₂×C₇²	—	159	2	no	—
29	C₂×C₇²	D₁₅₉	1	1	yes	characteristic, direct product factor
30	C₂×D₃×C₇	—	53	8	no	—
31	C₅₃	C₂×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
32	C₇²	C₂×D₁₅₉	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃×C₇	C₇×D₅₃	1	8	yes	complemented
34	D₃×C₇	—	53	16	no	—
35	C₂²×C₇	—	159	8	no	—
36	C₃×C₇	C₂×C₇×D₅₃	1	8	yes	complemented
37	C₂×C₇	—	159	16	no	—
38	C₂×C₇	C₇×D₁₅₉	1	8	yes	direct product factor
39	C₂×D₃	—	53	1	no	Hall
40	C₇	C₂×C₇×D₁₅₉	1	8	yes	direct product factor
41	C₂×C₃	C₇²×D₅₃	1	1	yes	characteristic, complemented
42	D₃	—	53	2	no	—
43	C₂²	—	159	1	no	Sylow
44	C₃	C₂×C₇²×D₅₃	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	159	2	no	—
46	C₂	C₇²×D₁₅₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇²	D₁₅₉
commutator, cosocle, hypocenter	C₃×C₅₃	C₂²×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₅₃	C₂
LCS	G ▷ C₃×C₅₃
UCS	C₁ ◁ C₂×C₇²

Complements

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

N	H ≅ G/N	type
1. C₇²×D₁₅₉	C₂	direct (N × H)
2. C₂×C₃×C₇²×C₅₃	C₂	semidirect
4. C₃×C₇²×C₅₃	C₂²	semidirect
6. C₂×C₇²×C₅₃	D₃	semidirect
7. C₂×C₇×D₁₅₉	C₇	direct (N × H)
8. C₇²×C₅₃	C₂×D₃	semidirect
9. C₇×D₁₅₉	C₂×C₇	direct (N × H)
10. C₂×C₃×C₇×C₅₃	C₂×C₇	semidirect
12. C₃×C₇×C₅₃	C₂²×C₇	semidirect
13. C₂×C₇×C₅₃	D₃×C₇	semidirect
15. C₂×D₁₅₉	C₇²	direct (N × H)
17. C₇×C₅₃	C₂×D₃×C₇	semidirect
18. C₂×C₃×C₅₃	C₂×C₇²	semidirect
19. D₁₅₉	C₂×C₇²	direct (N × H)
20. C₂×C₃×C₇²	D₅₃	semidirect
24. C₃×C₅₃	C₂²×C₇²	semidirect
25. C₃×C₇²	C₂×D₅₃	semidirect
26. C₂×C₅₃	D₃×C₇²	semidirect
29. C₂×C₇²	D₁₅₉	direct (N × H)
31. C₅₃	C₂×D₃×C₇²	semidirect
32. C₇²	C₂×D₁₅₉	direct (N × H)
33. C₂×C₃×C₇	C₇×D₅₃	semidirect
36. C₃×C₇	C₂×C₇×D₅₃	semidirect
38. C₂×C₇	C₇×D₁₅₉	direct (N × H)
40. C₇	C₂×C₇×D₁₅₉	direct (N × H)
41. C₂×C₃	C₇²×D₅₃	semidirect
44. C₃	C₂×C₇²×D₅₃	semidirect
46. C₂	C₇²×D₁₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×D₁₅₉	2
C₂×C₃×C₇²×C₅₃	2
C₂×C₇²×D₅₃	3
C₂×C₇×D₁₅₉	7
C₂×D₃×C₇²	53

Subgroups — C2×C72×D159 #44528489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×D₁₅₉ #44528489