Subgroups — C₂×C₃₁×D₂₅₅ #44539410

Order 31620.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×D₂₅₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₁₇×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×D₈₅	—	3	1	no	maximal, Hall
4	C₃×C₅×C₁₇×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃₁×D₅₁	—	5	1	no	maximal, Hall
6	C₃₁×D₈₅	—	3	2	no	—
7	C₂×C₅×C₁₇×C₃₁	D₃	1	1	yes	characteristic, complemented
8	C₃₁×D₅₁	—	5	2	no	—
9	C₂×C₃×C₁₇×C₃₁	D₅	1	1	yes	characteristic, complemented
10	C₅×C₁₇×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃₁×D₁₇	—	15	1	no	Hall
12	C₂×D₁₅×C₃₁	—	17	1	no	maximal, Hall
13	C₃×C₁₇×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₁₇×C₃₁	D₁₅	1	1	yes	characteristic, complemented
15	C₃₁×D₁₇	—	15	2	no	—
16	C₂×D₂₅₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₁	D₁₇	1	1	yes	characteristic, complemented
18	D₁₅×C₃₁	—	17	2	no	—
19	C₂×D₅×C₃₁	—	51	1	no	Hall
20	C₁₇×C₃₁	C₂×D₁₅	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₅×C₁₇	C₂×C₃₁	1	1	yes	characteristic, complemented
22	D₂₅₅	C₂×C₃₁	1	2	yes	direct product factor
23	C₃×C₅×C₃₁	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
24	C₂×D₃×C₃₁	—	85	1	no	Hall
25	C₂×D₈₅	—	3	1	no	Hall
26	C₂×C₅×C₃₁	D₅₁	1	1	yes	characteristic, complemented
27	D₅×C₃₁	—	51	2	no	—
28	C₃×C₅×C₁₇	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
29	C₂×D₅₁	—	5	1	no	Hall
30	C₂×C₃×C₃₁	D₈₅	1	1	yes	characteristic, complemented
31	D₃×C₃₁	—	85	2	no	—
32	C₂×C₅×C₁₇	D₃×C₃₁	1	1	yes	characteristic, complemented
33	D₈₅	—	3	2	no	—
34	C₅×C₃₁	C₂×D₅₁	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₃₁	—	255	1	no	Hall
36	C₂×C₃×C₁₇	D₅×C₃₁	1	1	yes	characteristic, complemented
37	D₅₁	—	5	2	no	—
38	C₃×C₃₁	C₂×D₈₅	1	1	yes	characteristic, complemented, Hall
39	C₅×C₁₇	C₂×D₃×C₃₁	1	1	yes	characteristic, complemented, Hall
40	C₂×D₁₇	—	15	1	no	Hall
41	C₂×C₃₁	—	255	2	no	—
42	C₂×C₃₁	D₂₅₅	1	1	yes	characteristic, direct product factor
43	C₂×D₁₅	—	17	1	no	Hall
44	C₃×C₁₇	C₂×D₅×C₃₁	1	1	yes	characteristic, complemented, Hall
45	C₂×C₁₇	D₁₅×C₃₁	1	1	yes	characteristic, complemented
46	D₁₇	—	15	2	no	—
47	C₃₁	C₂×D₂₅₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃×C₅	C₃₁×D₁₇	1	1	yes	characteristic, complemented
49	D₁₅	—	17	2	no	—
50	C₂×D₅	—	51	1	no	Hall
51	C₁₇	C₂×D₁₅×C₃₁	1	1	yes	characteristic, complemented, Sylow
52	C₃×C₅	C₂×C₃₁×D₁₇	1	1	yes	characteristic, complemented, Hall
53	C₂×D₃	—	85	1	no	Hall
54	C₂×C₅	C₃₁×D₅₁	1	1	yes	characteristic, complemented
55	D₅	—	51	2	no	—
56	C₂×C₃	C₃₁×D₈₅	1	1	yes	characteristic, complemented
57	D₃	—	85	2	no	—
58	C₅	C₂×C₃₁×D₅₁	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	255	1	no	Sylow
60	C₃	C₂×C₃₁×D₈₅	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	255	2	no	—
62	C₂	C₃₁×D₂₅₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃₁	D₂₅₅
commutator, cosocle, hypocenter	C₃×C₅×C₁₇	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₇×C₃₁	C₂
LCS	G ▷ C₃×C₅×C₁₇
UCS	C₁ ◁ C₂×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃₁×D₂₅₅	C₂	direct (N × H)
2. C₂×C₃×C₅×C₁₇×C₃₁	C₂	semidirect
4. C₃×C₅×C₁₇×C₃₁	C₂²	semidirect
7. C₂×C₅×C₁₇×C₃₁	D₃	semidirect
9. C₂×C₃×C₁₇×C₃₁	D₅	semidirect
10. C₅×C₁₇×C₃₁	C₂×D₃	semidirect
13. C₃×C₁₇×C₃₁	C₂×D₅	semidirect
14. C₂×C₁₇×C₃₁	D₁₅	semidirect
16. C₂×D₂₅₅	C₃₁	direct (N × H)
17. C₂×C₃×C₅×C₃₁	D₁₇	semidirect
20. C₁₇×C₃₁	C₂×D₁₅	semidirect
21. C₂×C₃×C₅×C₁₇	C₂×C₃₁	semidirect
22. D₂₅₅	C₂×C₃₁	direct (N × H)
23. C₃×C₅×C₃₁	C₂×D₁₇	semidirect
26. C₂×C₅×C₃₁	D₅₁	semidirect
28. C₃×C₅×C₁₇	C₂²×C₃₁	semidirect
30. C₂×C₃×C₃₁	D₈₅	semidirect
32. C₂×C₅×C₁₇	D₃×C₃₁	semidirect
34. C₅×C₃₁	C₂×D₅₁	semidirect
36. C₂×C₃×C₁₇	D₅×C₃₁	semidirect
38. C₃×C₃₁	C₂×D₈₅	semidirect
39. C₅×C₁₇	C₂×D₃×C₃₁	semidirect
42. C₂×C₃₁	D₂₅₅	direct (N × H)
44. C₃×C₁₇	C₂×D₅×C₃₁	semidirect
45. C₂×C₁₇	D₁₅×C₃₁	semidirect
47. C₃₁	C₂×D₂₅₅	direct (N × H)
48. C₂×C₃×C₅	C₃₁×D₁₇	semidirect
51. C₁₇	C₂×D₁₅×C₃₁	semidirect
52. C₃×C₅	C₂×C₃₁×D₁₇	semidirect
54. C₂×C₅	C₃₁×D₅₁	semidirect
56. C₂×C₃	C₃₁×D₈₅	semidirect
58. C₅	C₂×C₃₁×D₅₁	semidirect
60. C₃	C₂×C₃₁×D₈₅	semidirect
62. C₂	C₃₁×D₂₅₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×D₂₅₅	2
C₂×C₃×C₅×C₁₇×C₃₁	2
C₂×C₃₁×D₈₅	3
C₂×C₃₁×D₅₁	5
C₂×D₁₅×C₃₁	17
C₂×D₂₅₅	31

Subgroups — C2×C31×D255 #44539410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×D₂₅₅ #44539410