Subgroups — C₃₁×Dic₂₅₅ #44539261

Order 31620.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₇×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×Dic₈₅	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₇×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃₁×Dic₅₁	—	5	1	no	maximal, Hall
5	C₂×C₅×C₁₇×C₃₁	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₇×C₃₁	D₅	1	1	yes	characteristic
7	C₅×C₁₇×C₃₁	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₃₁×Dic₁₇	—	15	1	no	Hall
9	C₃₁×Dic₁₅	—	17	1	no	maximal, Hall
10	C₃×C₁₇×C₃₁	Dic₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₇×C₃₁	D₁₅	1	1	yes	characteristic
12	Dic₂₅₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₃₁	D₁₇	1	1	yes	characteristic
14	Dic₅×C₃₁	—	51	1	no	Hall
15	C₁₇×C₃₁	Dic₁₅	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₅×C₁₇	C₂×C₃₁	1	1	yes	characteristic
17	C₃×C₅×C₃₁	Dic₁₇	1	1	yes	characteristic, complemented, Hall
18	Dic₃×C₃₁	—	85	1	no	Hall
19	Dic₈₅	—	3	1	no	Hall
20	C₂×C₅×C₃₁	D₅₁	1	1	yes	characteristic
21	C₃×C₅×C₁₇	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
22	Dic₅₁	—	5	1	no	Hall
23	C₂×C₃×C₃₁	D₈₅	1	1	yes	characteristic
24	C₂×C₅×C₁₇	D₃×C₃₁	1	1	yes	characteristic
25	C₅×C₃₁	Dic₅₁	1	1	yes	characteristic, complemented, Hall
26	C₄×C₃₁	—	255	1	no	Hall
27	C₂×C₃×C₁₇	D₅×C₃₁	1	1	yes	characteristic
28	C₃×C₃₁	Dic₈₅	1	1	yes	characteristic, complemented, Hall
29	C₅×C₁₇	Dic₃×C₃₁	1	1	yes	characteristic, complemented, Hall
30	Dic₁₇	—	15	1	no	Hall
31	C₂×C₃₁	D₂₅₅	1	1	yes	characteristic
32	Dic₁₅	—	17	1	no	Hall
33	C₃×C₁₇	Dic₅×C₃₁	1	1	yes	characteristic, complemented, Hall
34	C₂×C₁₇	D₁₅×C₃₁	1	1	yes	characteristic
35	C₃₁	Dic₂₅₅	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃×C₅	C₃₁×D₁₇	1	1	yes	characteristic
37	Dic₅	—	51	1	no	Hall
38	C₁₇	C₃₁×Dic₁₅	1	1	yes	characteristic, complemented, Sylow
39	C₃×C₅	C₃₁×Dic₁₇	1	1	yes	characteristic, complemented, Hall
40	Dic₃	—	85	1	no	Hall
41	C₂×C₅	C₃₁×D₅₁	1	1	yes	characteristic
42	C₂×C₃	C₃₁×D₈₅	1	1	yes	characteristic
43	C₅	C₃₁×Dic₅₁	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	255	1	no	Sylow
45	C₃	C₃₁×Dic₈₅	1	1	yes	characteristic, complemented, Sylow
46	C₂	C₃₁×D₂₅₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃₁	D₂₅₅
commutator, hypocenter	C₃×C₅×C₁₇	C₄×C₃₁
Frattini	C₂	C₃₁×D₂₅₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₁₇×C₃₁	C₂
cosocle	C₂×C₃×C₅×C₁₇	C₂×C₃₁
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
3. C₃×C₅×C₁₇×C₃₁	C₄	semidirect
7. C₅×C₁₇×C₃₁	Dic₃	semidirect
10. C₃×C₁₇×C₃₁	Dic₅	semidirect
12. Dic₂₅₅	C₃₁	direct (N × H)
15. C₁₇×C₃₁	Dic₁₅	semidirect
17. C₃×C₅×C₃₁	Dic₁₇	semidirect
21. C₃×C₅×C₁₇	C₄×C₃₁	semidirect
25. C₅×C₃₁	Dic₅₁	semidirect
28. C₃×C₃₁	Dic₈₅	semidirect
29. C₅×C₁₇	Dic₃×C₃₁	semidirect
33. C₃×C₁₇	Dic₅×C₃₁	semidirect
35. C₃₁	Dic₂₅₅	direct (N × H)
38. C₁₇	C₃₁×Dic₁₅	semidirect
39. C₃×C₅	C₃₁×Dic₁₇	semidirect
43. C₅	C₃₁×Dic₅₁	semidirect
45. C₃	C₃₁×Dic₈₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₇×C₃₁	2
C₃₁×Dic₈₅	3
C₃₁×Dic₅₁	5
C₃₁×Dic₁₅	17
Dic₂₅₅	31

Subgroups — C31×Dic255 #44539261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×Dic₂₅₅ #44539261