Subgroups — C₃²×#16814882 #45686405

Order 33300.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₇×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#16814882	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×#10625193	—	5	2	no	maximal
5	C₃×C₃₇×C₅⋊D₅	C₂×C₃	1	4	yes	—
6	#16814882	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×D₅×C₃₇	—	5	2	no	—
8	C₃²×D₅×C₃₇	—	10	2	no	—
9	C₃×C₅²×C₃₇	C₃×C₄	1	4	yes	complemented
10	C₃×#10625193	—	5	8	no	—
11	C₃₇×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
12	C₃²×C₅×C₃₇	—	2	2	no	—
13	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	2	yes	complemented
14	C₃²×Dic₃₇	—	25	1	no	Hall
15	C₃×D₅×C₃₇	—	5	8	no	—
16	C₃×D₅×C₃₇	—	10	8	no	—
17	C₅²×C₃₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
18	C₃²×#1044	—	37	1	no	maximal, Hall
19	#10625193	—	5	2	no	—
20	C₂×C₃²×C₃₇	—	25	1	no	—
21	C₃×C₅×C₃₇	—	2	8	no	—
22	C₃×C₅×C₃₇	C₃×AGL(1,𝔽₅)	1	8	yes	complemented
23	C₃²×C₅⋊D₅	D₃₇	1	1	yes	characteristic
24	C₃×Dic₃₇	—	25	4	no	—
25	D₅×C₃₇	—	5	2	no	—
26	D₅×C₃₇	—	10	2	no	—
27	C₃²×C₃₇	#1044	1	1	yes	characteristic, complemented, Hall
28	C₃×#1044	—	37	4	no	—
29	C₃²×C₅²	Dic₃₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₃₇	—	25	4	no	—
31	C₅×C₃₇	—	2	2	no	—
32	C₅×C₃₇	C₃²×AGL(1,𝔽₅)	1	2	yes	complemented
33	C₃²×AGL(1,𝔽₅)	—	185	2	no	—
34	C₃×C₅⋊D₅	C₃×D₃₇	1	4	yes	—
35	Dic₃₇	—	25	1	no	Hall
36	C₃×C₃₇	C₃×#1044	1	4	yes	complemented
37	#1044	—	37	1	no	Hall
38	C₃²×D₅	—	5	2	no	—
39	C₃²×D₅	—	10	2	no	—
40	C₃×C₅²	C₃×Dic₃₇	1	4	yes	complemented
41	C₂×C₃₇	—	25	1	no	—
42	C₃×AGL(1,𝔽₅)	—	185	8	no	—
43	C₅⋊D₅	C₃²×D₃₇	1	1	yes	characteristic
44	C₃²×C₅	—	2	2	no	—
45	C₃²×C₅	#10625193	1	2	yes	complemented
46	C₃₇	C₃²×#1044	1	1	yes	characteristic, complemented, Sylow
47	C₃²×C₄	—	925	1	no	Hall
48	C₃×D₅	—	5	8	no	—
49	C₃×D₅	—	10	8	no	—
50	C₅²	C₃²×Dic₃₇	1	1	yes	characteristic, complemented, Sylow
51	AGL(1,𝔽₅)	—	185	2	no	—
52	C₂×C₃²	—	25	1	no	—
53	C₃×C₅	—	2	8	no	—
54	C₃×C₅	C₃×#10625193	1	8	yes	complemented
55	C₃×C₄	—	925	4	no	—
56	D₅	—	5	2	no	—
57	D₅	—	10	2	no	—
58	C₃²	#16814882	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃	—	25	4	no	—
60	C₅	—	2	2	no	—
61	C₅	C₃²×#10625193	1	2	yes	complemented
62	C₄	—	925	1	no	Sylow
63	C₃	C₃×#16814882	1	4	yes	direct product factor
64	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	#16814882
commutator, hypocenter	C₅²×C₃₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₃₇×C₅⋊D₅	C₂
cosocle	C₃₇×C₅⋊D₅	C₂×C₃²
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×#16814882	C₃	direct (N × H)
3. C₃²×C₅²×C₃₇	C₄	semidirect
6. #16814882	C₃²	direct (N × H)
9. C₃×C₅²×C₃₇	C₃×C₄	semidirect
13. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
17. C₅²×C₃₇	C₃²×C₄	semidirect
22. C₃×C₅×C₃₇	C₃×AGL(1,𝔽₅)	semidirect
27. C₃²×C₃₇	#1044	semidirect
29. C₃²×C₅²	Dic₃₇	semidirect
32. C₅×C₃₇	C₃²×AGL(1,𝔽₅)	semidirect
36. C₃×C₃₇	C₃×#1044	semidirect
40. C₃×C₅²	C₃×Dic₃₇	semidirect
45. C₃²×C₅	#10625193	semidirect
46. C₃₇	C₃²×#1044	semidirect
50. C₅²	C₃²×Dic₃₇	semidirect
54. C₃×C₅	C₃×#10625193	semidirect
58. C₃²	#16814882	direct (N × H)
61. C₅	C₃²×#10625193	semidirect
63. C₃	C₃×#16814882	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×C₃₇×C₅⋊D₅	2
C₃×#16814882	3
C₃²×#10625193	5
C₃²×#1044	37

Subgroups — C32×#16814882 #45686405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#16814882 #45686405