Subgroups — C₃²×C₃₇×#1039 #45686135

Order 33300.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃₇×#1039	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×Dic₅×C₃₇	—	5	6	no	maximal
5	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	4	yes	—
6	C₃₇×#1039	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃²×C₅×C₃₇	D₅	1	6	yes	—
8	C₃×C₅²×C₃₇	C₃×C₄	1	4	yes	complemented
9	C₃×Dic₅×C₃₇	—	5	24	no	—
10	C₂×C₅²×C₃₇	C₂×C₃²	1	1	yes	characteristic
11	C₃²×C₅×C₃₇	Dic₅	1	6	yes	complemented
12	C₃²×C₄×C₃₇	—	25	1	no	Hall
13	C₂×C₃×C₅×C₃₇	C₃×D₅	1	24	yes	—
14	C₅²×C₃₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
15	C₃²×#1039	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	Dic₅×C₃₇	—	5	6	no	—
17	C₂×C₃²×C₃₇	C₅⋊D₅	1	1	yes	characteristic
18	C₃×C₅×C₃₇	C₃×Dic₅	1	24	yes	complemented
19	C₂×C₃²×C₅²	C₂×C₃₇	1	1	yes	characteristic
20	C₃×C₄×C₃₇	—	25	4	no	—
21	C₂×C₅×C₃₇	C₃²×D₅	1	6	yes	—
22	C₃²×C₃₇	#1039	1	1	yes	characteristic, direct product factor, Hall
23	C₃×#1039	C₃×C₃₇	1	4	yes	direct product factor
24	C₃²×C₅²	C₄×C₃₇	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₃₇	C₃×C₅⋊D₅	1	4	yes	—
26	C₅×C₃₇	C₃²×Dic₅	1	6	yes	complemented
27	C₃²×Dic₅	—	5	6	no	—
28	C₂×C₃×C₅²	C₂×C₃×C₃₇	1	4	yes	—
29	C₄×C₃₇	—	25	1	no	Hall
30	C₃×C₃₇	C₃×#1039	1	4	yes	direct product factor
31	#1039	C₃²×C₃₇	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₃²×C₅	D₅×C₃₇	1	6	yes	—
33	C₃×C₅²	C₃×C₄×C₃₇	1	4	yes	complemented
34	C₂×C₃₇	C₃²×C₅⋊D₅	1	1	yes	characteristic
35	C₃×Dic₅	—	5	24	no	—
36	C₂×C₅²	C₂×C₃²×C₃₇	1	1	yes	characteristic
37	C₃²×C₅	Dic₅×C₃₇	1	6	yes	complemented
38	C₃₇	C₃²×#1039	1	1	yes	characteristic, direct product factor, Sylow
39	C₃²×C₄	—	25	1	no	Hall
40	C₂×C₃×C₅	C₃×D₅×C₃₇	1	24	yes	—
41	C₅²	C₃²×C₄×C₃₇	1	1	yes	characteristic, complemented, Sylow
42	Dic₅	—	5	6	no	—
43	C₂×C₃²	C₃₇×C₅⋊D₅	1	1	yes	characteristic
44	C₃×C₅	C₃×Dic₅×C₃₇	1	24	yes	complemented
45	C₃×C₄	—	25	4	no	—
46	C₂×C₅	C₃²×D₅×C₃₇	1	6	yes	—
47	C₃²	C₃₇×#1039	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	C₃×C₃₇×C₅⋊D₅	1	4	yes	—
49	C₅	C₃²×Dic₅×C₃₇	1	6	yes	complemented
50	C₄	—	25	1	no	Sylow
51	C₃	C₃×C₃₇×#1039	1	4	yes	direct product factor
52	C₂	C₃²×C₃₇×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₃₇	C₅⋊D₅
commutator, hypocenter	C₅²	C₃²×C₄×C₃₇
Frattini	C₂	C₃²×C₃₇×C₅⋊D₅
Fitting, socle, prime radical	C₂×C₃²×C₅²×C₃₇	C₂
cosocle	C₂×C₅²	C₂×C₃²×C₃₇
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₃²×C₃₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₇×#1039	C₃	direct (N × H)
3. C₃²×C₅²×C₃₇	C₄	semidirect
6. C₃₇×#1039	C₃²	direct (N × H)
8. C₃×C₅²×C₃₇	C₃×C₄	semidirect
11. C₃²×C₅×C₃₇	Dic₅	semidirect
14. C₅²×C₃₇	C₃²×C₄	semidirect
15. C₃²×#1039	C₃₇	direct (N × H)
18. C₃×C₅×C₃₇	C₃×Dic₅	semidirect
22. C₃²×C₃₇	#1039	direct (N × H)
23. C₃×#1039	C₃×C₃₇	direct (N × H)
24. C₃²×C₅²	C₄×C₃₇	semidirect
26. C₅×C₃₇	C₃²×Dic₅	semidirect
30. C₃×C₃₇	C₃×#1039	direct (N × H)
31. #1039	C₃²×C₃₇	direct (N × H)
33. C₃×C₅²	C₃×C₄×C₃₇	semidirect
37. C₃²×C₅	Dic₅×C₃₇	semidirect
38. C₃₇	C₃²×#1039	direct (N × H)
41. C₅²	C₃²×C₄×C₃₇	semidirect
44. C₃×C₅	C₃×Dic₅×C₃₇	semidirect
47. C₃²	C₃₇×#1039	direct (N × H)
49. C₅	C₃²×Dic₅×C₃₇	semidirect
51. C₃	C₃×C₃₇×#1039	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅²×C₃₇	2
C₃×C₃₇×#1039	3
C₃²×Dic₅×C₃₇	5
C₃²×#1039	37

Subgroups — C32×C37×#1039 #45686135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×#1039 #45686135