Subgroups — C₂×C₃×C₇²×D₅₃ #44528483

Order 31164.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇²	D₅₃
commutator, cosocle, hypocenter	C₅₃	C₂²×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₅₃	C₂
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₃×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×D₅₃	C₂	direct (N × H)
2. C₂×C₃×C₇²×C₅₃	C₂	semidirect
3. C₂×C₇²×D₅₃	C₃	direct (N × H)
4. C₃×C₇²×C₅₃	C₂²	semidirect
5. C₇²×D₅₃	C₂×C₃	direct (N × H)
6. C₂×C₇²×C₅₃	C₂×C₃	semidirect
7. C₂×C₃×C₇×D₅₃	C₇	direct (N × H)
8. C₇²×C₅₃	C₂²×C₃	semidirect
9. C₃×C₇×D₅₃	C₂×C₇	direct (N × H)
10. C₂×C₃×C₇×C₅₃	C₂×C₇	semidirect
11. C₂×C₇×D₅₃	C₃×C₇	direct (N × H)
12. C₃×C₇×C₅₃	C₂²×C₇	semidirect
13. C₂×C₇×C₅₃	C₂×C₃×C₇	semidirect
14. C₇×D₅₃	C₂×C₃×C₇	direct (N × H)
15. C₂×C₃×D₅₃	C₇²	direct (N × H)
17. C₇×C₅₃	C₂²×C₃×C₇	semidirect
18. C₂×C₃×C₅₃	C₂×C₇²	semidirect
19. C₃×D₅₃	C₂×C₇²	direct (N × H)
21. C₂×C₃×C₇²	D₅₃	direct (N × H)
22. C₂×D₅₃	C₃×C₇²	direct (N × H)
24. C₃×C₅₃	C₂²×C₇²	semidirect
25. C₃×C₇²	C₂×D₅₃	direct (N × H)
26. C₂×C₅₃	C₂×C₃×C₇²	semidirect
27. D₅₃	C₂×C₃×C₇²	direct (N × H)
29. C₂×C₇²	C₃×D₅₃	direct (N × H)
31. C₅₃	C₂²×C₃×C₇²	semidirect
32. C₇²	C₂×C₃×D₅₃	direct (N × H)
34. C₂×C₃×C₇	C₇×D₅₃	direct (N × H)
36. C₃×C₇	C₂×C₇×D₅₃	direct (N × H)
38. C₂×C₇	C₃×C₇×D₅₃	direct (N × H)
40. C₇	C₂×C₃×C₇×D₅₃	direct (N × H)
42. C₂×C₃	C₇²×D₅₃	direct (N × H)
44. C₃	C₂×C₇²×D₅₃	direct (N × H)
46. C₂	C₃×C₇²×D₅₃	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×C₇²×D₅₃	2
C₂×C₃×C₇²×C₅₃	2
C₂×C₇²×D₅₃	3
C₂×C₃×C₇×D₅₃	7
C₂²×C₃×C₇²	53

Subgroups — C2×C3×C72×D53 #44528483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇²×D₅₃ #44528483