Subgroups — C₅²×C₃⋊(C₃⋊#3866) #45686466

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	C₅²×C₃⋊#3866	—	3	4	no	maximal
3	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₅×C₃⋊(C₃⋊#3866)	C₅	1	6	yes	maximal, direct product factor
5	C₃×C₅²×D₃₇	D₃	1	4	yes	—
6	C₅²×#3866	—	9	1	no	Hall
7	C₃²×C₅×D₃₇	C₂×C₅	1	6	yes	—
8	C₃×C₅²×C₃₇	Dic₃	1	4	yes	complemented
9	C₅×C₃⋊#3866	—	3	24	no	—
10	C₅²×D₃₇	C₃⋊D₃	1	1	yes	characteristic
11	C₃²×C₅×C₃₇	C₄×C₅	1	6	yes	complemented
12	C₃⋊(C₃⋊#3866)	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×D₃₇	C₅×D₃	1	24	yes	—
14	C₅²×C₃₇	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
15	C₅²×C₃⋊Dic₃	—	37	1	no	maximal, Hall
16	C₅×#3866	—	9	6	no	—
17	C₃²×D₃₇	C₂×C₅²	1	1	yes	characteristic
18	C₃×C₅×C₃₇	C₅×Dic₃	1	24	yes	complemented
19	C₂×C₃²×C₅²	—	37	1	no	—
20	C₃⋊#3866	—	3	4	no	—
21	C₅×D₃₇	C₅×C₃⋊D₃	1	6	yes	—
22	C₃²×C₃₇	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
23	C₅²×Dic₃	—	111	4	no	—
24	C₃²×C₅²	#3866	1	1	yes	characteristic, complemented, Hall
25	C₃×D₃₇	C₅²×D₃	1	4	yes	—
26	C₅×C₃₇	C₅×C₃⋊Dic₃	1	6	yes	complemented
27	C₅×C₃⋊Dic₃	—	37	6	no	—
28	C₂×C₃×C₅²	—	37	4	no	—
29	#3866	—	9	1	no	Hall
30	C₃×C₃₇	C₅²×Dic₃	1	4	yes	complemented
31	C₄×C₅²	—	333	1	no	Hall
32	C₂×C₃²×C₅	—	37	6	no	—
33	C₃×C₅²	C₃⋊#3866	1	4	yes	complemented
34	D₃₇	C₅²×C₃⋊D₃	1	1	yes	characteristic
35	C₅×Dic₃	—	111	24	no	—
36	C₂×C₅²	—	37	1	no	—
37	C₃²×C₅	C₅×#3866	1	6	yes	complemented
38	C₃₇	C₅²×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
39	C₃⋊Dic₃	—	37	1	no	Hall
40	C₂×C₃×C₅	—	37	24	no	—
41	C₅²	C₃⋊(C₃⋊#3866)	1	1	yes	characteristic, direct product factor, Sylow
42	C₄×C₅	—	333	6	no	—
43	C₂×C₃²	—	37	1	no	—
44	C₃×C₅	C₅×C₃⋊#3866	1	24	yes	complemented
45	Dic₃	—	111	4	no	—
46	C₂×C₅	—	37	6	no	—
47	C₃²	C₅²×#3866	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	37	4	no	—
49	C₅	C₅×C₃⋊(C₃⋊#3866)	1	6	yes	direct product factor
50	C₄	—	333	1	no	Sylow
51	C₃	C₅²×C₃⋊#3866	1	4	yes	complemented
52	C₂	—	37	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊(C₃⋊#3866)
commutator, hypocenter	C₃²×C₃₇	C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅²×D₃₇	C₂
cosocle	C₃²×D₃₇	C₂×C₅²
LCS	G ▷ C₃²×C₃₇
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₃₇	C₄	semidirect
4. C₅×C₃⋊(C₃⋊#3866)	C₅	direct (N × H)
8. C₃×C₅²×C₃₇	Dic₃	semidirect
11. C₃²×C₅×C₃₇	C₄×C₅	semidirect
12. C₃⋊(C₃⋊#3866)	C₅²	direct (N × H)
14. C₅²×C₃₇	C₃⋊Dic₃	semidirect
18. C₃×C₅×C₃₇	C₅×Dic₃	semidirect
22. C₃²×C₃₇	C₄×C₅²	semidirect
24. C₃²×C₅²	#3866	semidirect
26. C₅×C₃₇	C₅×C₃⋊Dic₃	semidirect
30. C₃×C₃₇	C₅²×Dic₃	semidirect
33. C₃×C₅²	C₃⋊#3866	semidirect
37. C₃²×C₅	C₅×#3866	semidirect
38. C₃₇	C₅²×C₃⋊Dic₃	semidirect
41. C₅²	C₃⋊(C₃⋊#3866)	direct (N × H)
44. C₃×C₅	C₅×C₃⋊#3866	semidirect
47. C₃²	C₅²×#3866	semidirect
49. C₅	C₅×C₃⋊(C₃⋊#3866)	direct (N × H)
51. C₃	C₅²×C₃⋊#3866	semidirect

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
C₅²×C₃⋊#3866	3
C₅×C₃⋊(C₃⋊#3866)	5
C₅²×C₃⋊Dic₃	37

Subgroups — C52×C3⋊(C3⋊#3866) #45686466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊(C₃⋊#3866) #45686466