Subgroups — C₃×C₅×#13095532 #37552071

Order 23310.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#11715912	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×#13095532	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₅×C₇×D₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#13095532	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×#11715912	C₂×C₃	1	3	yes	complemented
6	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₅×D₃₇⋊C₃	—	7	1	no	maximal, Hall
8	C₅×C₇×D₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×#11715912	C₂×C₅	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅×C₃₇⋊C₃	—	7	1	no	Hall
11	C₃×C₇×D₃₇	C₃×C₅	1	1	yes	characteristic, complemented
12	#13095532	C₃×C₅	1	3	yes	direct product factor
13	C₅×C₇×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₃×C₅×D₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₅×D₃₇⋊C₃	—	7	3	no	—
16	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
17	#11715912	C₂×C₃×C₅	1	3	yes	complemented
18	C₃×D₃₇⋊C₃	—	7	1	no	Hall
19	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	37	1	no	maximal, Hall
20	C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₅×C₃₇⋊C₃	—	7	3	no	—
22	C₇×D₃₇	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
23	C₅×D₃₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
24	C₃×C₃₇⋊C₃	—	7	1	no	Hall
25	C₃×C₅×ΓL(1,𝔽₈)	—	37	1	no	Hall
26	C₇×C₃₇	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
27	C₃×D₃₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	D₃₇⋊C₃	—	7	3	no	—
29	C₂×C₃×C₅×C₇	—	111	1	no	—
30	C₂×C₅×ΓL(1,𝔽₈)	—	37	3	no	—
31	C₅×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×ΓL(1,𝔽₈)	—	37	1	no	Hall
33	C₃×C₃₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₃₇⋊C₃	—	7	3	no	—
35	C₃×C₅×C₇	D₃₇⋊C₃	1	1	yes	characteristic, complemented
36	C₅×ΓL(1,𝔽₈)	—	37	3	no	—
37	C₂×C₃²×C₅	—	259	1	no	Hall
38	D₃₇	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
39	C₂×C₅×C₇	—	111	1	no	Hall
40	C₃×ΓL(1,𝔽₈)	—	37	1	no	Hall
41	C₃²×C₅	—	259	1	no	Hall
42	C₂×C₃×C₇	—	111	1	no	—
43	C₂×ΓL(1,𝔽₈)	—	37	3	no	—
44	C₃₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
45	C₅×C₇	C₃×D₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₅	—	259	3	no	—
47	C₂×C₃×C₅	—	37	1	no	—
48	C₃×C₇	C₅×D₃₇⋊C₃	1	1	yes	characteristic, complemented
49	ΓL(1,𝔽₈)	—	37	3	no	—
50	C₂×C₃²	—	259	1	no	Hall
51	C₃×C₅	—	259	3	no	—
52	C₃×C₅	#13095532	1	1	yes	characteristic, direct product factor
53	C₂×C₇	—	111	1	no	Hall
54	C₂×C₅	—	37	1	no	Hall
55	C₃²	—	259	1	no	Sylow
56	C₇	C₃×C₅×D₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	259	3	no	—
58	C₂×C₃	—	111	1	no	—
59	C₅	C₃×#13095532	1	1	yes	characteristic, direct product factor, Sylow
60	C₃	—	259	3	no	—
61	C₃	C₅×#13095532	1	1	yes	characteristic, direct product factor
62	C₂	—	111	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅	#13095532
commutator, cosocle, hypocenter	C₇×C₃₇	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₇×C₃₇
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#11715912	C₂	semidirect
2. C₅×#13095532	C₃	direct (N × H)
3. C₃×C₅×C₇×D₃₇	C₃	semidirect
4. C₃×#13095532	C₅	direct (N × H)
5. C₅×#11715912	C₂×C₃	semidirect
6. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
8. C₅×C₇×D₃₇	C₃²	semidirect
9. C₃×#11715912	C₂×C₅	semidirect
11. C₃×C₇×D₃₇	C₃×C₅	semidirect
12. #13095532	C₃×C₅	direct (N × H)
13. C₅×C₇×C₃₇	C₂×C₃²	semidirect
14. C₃×C₅×D₃₇	ΓL(1,𝔽₈)	semidirect
16. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
17. #11715912	C₂×C₃×C₅	semidirect
20. C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
22. C₇×D₃₇	C₃²×C₅	semidirect
23. C₅×D₃₇	C₃×ΓL(1,𝔽₈)	semidirect
26. C₇×C₃₇	C₂×C₃²×C₅	semidirect
27. C₃×D₃₇	C₅×ΓL(1,𝔽₈)	semidirect
31. C₅×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
33. C₃×C₃₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
35. C₃×C₅×C₇	D₃₇⋊C₃	semidirect
38. D₃₇	C₃×C₅×ΓL(1,𝔽₈)	semidirect
44. C₃₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
45. C₅×C₇	C₃×D₃₇⋊C₃	semidirect
48. C₃×C₇	C₅×D₃₇⋊C₃	semidirect
52. C₃×C₅	#13095532	direct (N × H)
56. C₇	C₃×C₅×D₃₇⋊C₃	semidirect
59. C₅	C₃×#13095532	direct (N × H)
61. C₃	C₅×#13095532	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×C₅×#11715912	2
C₅×#13095532	3
C₃×C₅×C₇×D₃₇	3
C₃×#13095532	5
C₃×C₅×D₃₇⋊C₃	7
C₂×C₃×C₅×ΓL(1,𝔽₈)	37

Subgroups — C3×C5×#13095532 #37552071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#13095532 #37552071