Subgroups — C₃²×C₅²×#3866 #45686383

Order 33300.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅²×#3866	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₅×#3866	C₅	1	6	yes	maximal, direct product factor
5	C₃×C₅²×D₃₇	C₂×C₃	1	4	yes	—
6	C₅²×#3866	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₅×D₃₇	C₂×C₅	1	6	yes	—
8	C₃×C₅²×C₃₇	C₃×C₄	1	4	yes	complemented
9	C₃×C₅×#3866	C₃×C₅	1	24	yes	direct product factor
10	C₅²×D₃₇	C₂×C₃²	1	1	yes	characteristic
11	C₃²×C₅×C₃₇	C₄×C₅	1	6	yes	complemented
12	C₃²×#3866	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×D₃₇	C₂×C₃×C₅	1	24	yes	—
14	C₅²×C₃₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
15	C₃²×C₄×C₅²	—	37	1	no	maximal, Hall
16	C₅×#3866	C₃²×C₅	1	6	yes	direct product factor
17	C₃²×D₃₇	C₂×C₅²	1	1	yes	characteristic
18	C₃×C₅×C₃₇	C₃×C₄×C₅	1	24	yes	complemented
19	C₂×C₃²×C₅²	—	37	1	no	—
20	C₃×#3866	C₃×C₅²	1	4	yes	direct product factor
21	C₅×D₃₇	C₂×C₃²×C₅	1	6	yes	—
22	C₃²×C₃₇	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
23	C₃×C₄×C₅²	—	37	4	no	—
24	C₃²×C₅²	#3866	1	1	yes	characteristic, direct product factor, Hall
25	C₃×D₃₇	C₂×C₃×C₅²	1	4	yes	—
26	C₅×C₃₇	C₃²×C₄×C₅	1	6	yes	complemented
27	C₃²×C₄×C₅	—	37	6	no	—
28	C₂×C₃×C₅²	—	37	4	no	—
29	#3866	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Hall
30	C₃×C₃₇	C₃×C₄×C₅²	1	4	yes	complemented
31	C₄×C₅²	—	37	1	no	Hall
32	C₂×C₃²×C₅	—	37	6	no	—
33	C₃×C₅²	C₃×#3866	1	4	yes	direct product factor
34	D₃₇	C₂×C₃²×C₅²	1	1	yes	characteristic
35	C₃×C₄×C₅	—	37	24	no	—
36	C₂×C₅²	—	37	1	no	—
37	C₃²×C₅	C₅×#3866	1	6	yes	direct product factor
38	C₃₇	C₃²×C₄×C₅²	1	1	yes	characteristic, complemented, Sylow
39	C₃²×C₄	—	37	1	no	Hall
40	C₂×C₃×C₅	—	37	24	no	—
41	C₅²	C₃²×#3866	1	1	yes	characteristic, direct product factor, Sylow
42	C₄×C₅	—	37	6	no	—
43	C₂×C₃²	—	37	1	no	—
44	C₃×C₅	C₃×C₅×#3866	1	24	yes	direct product factor
45	C₃×C₄	—	37	4	no	—
46	C₂×C₅	—	37	6	no	—
47	C₃²	C₅²×#3866	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	—	37	4	no	—
49	C₅	C₃²×C₅×#3866	1	6	yes	direct product factor
50	C₄	—	37	1	no	Sylow
51	C₃	C₃×C₅²×#3866	1	4	yes	direct product factor
52	C₂	—	37	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₅²	#3866
commutator, hypocenter	C₃₇	C₃²×C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅²×D₃₇	C₂
cosocle	D₃₇	C₂×C₃²×C₅²
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃²×C₅²

Complements

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

N	H ≅ G/N	type
2. C₃×C₅²×#3866	C₃	direct (N × H)
3. C₃²×C₅²×C₃₇	C₄	semidirect
4. C₃²×C₅×#3866	C₅	direct (N × H)
6. C₅²×#3866	C₃²	direct (N × H)
8. C₃×C₅²×C₃₇	C₃×C₄	semidirect
9. C₃×C₅×#3866	C₃×C₅	direct (N × H)
11. C₃²×C₅×C₃₇	C₄×C₅	semidirect
12. C₃²×#3866	C₅²	direct (N × H)
14. C₅²×C₃₇	C₃²×C₄	semidirect
16. C₅×#3866	C₃²×C₅	direct (N × H)
18. C₃×C₅×C₃₇	C₃×C₄×C₅	semidirect
20. C₃×#3866	C₃×C₅²	direct (N × H)
22. C₃²×C₃₇	C₄×C₅²	semidirect
24. C₃²×C₅²	#3866	direct (N × H)
26. C₅×C₃₇	C₃²×C₄×C₅	semidirect
29. #3866	C₃²×C₅²	direct (N × H)
30. C₃×C₃₇	C₃×C₄×C₅²	semidirect
33. C₃×C₅²	C₃×#3866	direct (N × H)
37. C₃²×C₅	C₅×#3866	direct (N × H)
38. C₃₇	C₃²×C₄×C₅²	semidirect
41. C₅²	C₃²×#3866	direct (N × H)
44. C₃×C₅	C₃×C₅×#3866	direct (N × H)
47. C₃²	C₅²×#3866	direct (N × H)
49. C₅	C₃²×C₅×#3866	direct (N × H)
51. C₃	C₃×C₅²×#3866	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅²×D₃₇	2
C₃×C₅²×#3866	3
C₃²×C₅×#3866	5
C₃²×C₄×C₅²	37

Subgroups — C32×C52×#3866 #45686383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×#3866 #45686383