Subgroups — C₃²×C₅×#10625192 #45686388

Order 33300.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₁₈₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#10625192	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×#10625192	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₅×#3866	—	5	1	no	maximal
6	C₃×C₅×D₁₈₅	C₂×C₃	1	4	yes	—
7	C₅×#10625192	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×D₁₈₅	C₂×C₅	1	1	yes	characteristic
9	C₃²×C₅×D₃₇	—	5	1	no	—
10	C₃×C₅²×C₃₇	C₃×C₄	1	4	yes	complemented
11	C₃×#10625192	C₃×C₅	1	4	yes	direct product factor
12	C₃×C₅×#3866	—	5	4	no	—
13	C₅×D₁₈₅	C₂×C₃²	1	1	yes	characteristic
14	C₃²×C₅×C₃₇	—	4	1	no	—
15	C₃²×C₅×C₃₇	C₄×C₅	1	1	yes	characteristic, complemented
16	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	C₃²×#3866	—	5	1	no	Hall
18	C₃×D₁₈₅	C₂×C₃×C₅	1	4	yes	—
19	C₃×C₅×D₃₇	—	5	4	no	—
20	C₅²×C₃₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₃²×C₅×AGL(1,𝔽₅)	—	37	1	no	maximal, Hall
22	#10625192	C₃²×C₅	1	1	yes	characteristic, direct product factor
23	C₅×#3866	—	5	1	no	—
24	C₃²×D₃₇	—	5	1	no	—
25	C₃×C₅×C₃₇	—	4	4	no	—
26	C₃×C₅×C₃₇	C₃×C₄×C₅	1	4	yes	complemented
27	C₃×C₅×C₃₇	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
28	C₃²×C₅×D₅	—	37	1	no	—
29	C₃×#3866	—	5	4	no	—
30	D₁₈₅	C₂×C₃²×C₅	1	1	yes	characteristic
31	C₅×D₃₇	—	5	1	no	—
32	C₃²×C₃₇	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅×AGL(1,𝔽₅)	—	37	4	no	—
34	C₃²×C₅²	#3866	1	1	yes	characteristic, complemented, Hall
35	C₃×D₃₇	—	5	4	no	—
36	C₅×C₃₇	—	4	1	no	—
37	C₅×C₃₇	C₃²×C₄×C₅	1	1	yes	characteristic, complemented
38	C₅×C₃₇	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₃²×AGL(1,𝔽₅)	—	37	1	no	—
40	C₃²×C₄×C₅	—	185	1	no	—
41	C₃×C₅×D₅	—	37	4	no	—
42	#3866	—	5	1	no	Hall
43	C₃×C₃₇	C₃×C₅×AGL(1,𝔽₅)	1	4	yes	complemented
44	C₅×AGL(1,𝔽₅)	—	37	1	no	Hall
45	C₂×C₃²×C₅	—	185	1	no	—
46	C₃²×D₅	—	37	1	no	—
47	C₃×C₅²	C₃×#3866	1	4	yes	complemented
48	D₃₇	—	5	1	no	—
49	C₃×AGL(1,𝔽₅)	—	37	4	no	—
50	C₃×C₄×C₅	—	185	4	no	—
51	C₅×D₅	—	37	1	no	—
52	C₃²×C₅	—	4	1	no	—
53	C₃²×C₅	C₅×#3866	1	1	yes	characteristic, complemented
54	C₃²×C₅	#10625192	1	1	yes	characteristic, direct product factor
55	C₃₇	C₃²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
56	C₃²×C₄	—	185	1	no	Hall
57	C₂×C₃×C₅	—	185	4	no	—
58	C₃×D₅	—	37	4	no	—
59	C₅²	C₃²×#3866	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	37	1	no	—
61	C₄×C₅	—	185	1	no	—
62	C₂×C₃²	—	185	1	no	—
63	C₃×C₅	—	4	4	no	—
64	C₃×C₅	C₃×C₅×#3866	1	4	yes	complemented
65	C₃×C₅	C₃×#10625192	1	4	yes	direct product factor
66	C₃×C₄	—	185	4	no	—
67	C₂×C₅	—	185	1	no	—
68	D₅	—	37	1	no	—
69	C₃²	C₅×#10625192	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	—	185	4	no	—
71	C₅	—	4	1	no	—
72	C₅	C₃²×C₅×#3866	1	1	yes	characteristic, complemented
73	C₅	C₃²×#10625192	1	1	yes	characteristic, direct product factor
74	C₄	—	185	1	no	Sylow
75	C₃	C₃×C₅×#10625192	1	4	yes	direct product factor
76	C₂	—	185	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₅	#10625192
commutator, hypocenter	C₅×C₃₇	C₃²×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅×D₁₈₅	C₂
cosocle	D₁₈₅	C₂×C₃²×C₅
LCS	G ▷ C₅×C₃₇
UCS	C₁ ◁ C₃²×C₅

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#10625192	C₃	direct (N × H)
3. C₃²×C₅²×C₃₇	C₄	semidirect
4. C₃²×#10625192	C₅	direct (N × H)
7. C₅×#10625192	C₃²	direct (N × H)
10. C₃×C₅²×C₃₇	C₃×C₄	semidirect
11. C₃×#10625192	C₃×C₅	direct (N × H)
15. C₃²×C₅×C₃₇	C₄×C₅	semidirect
16. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
20. C₅²×C₃₇	C₃²×C₄	semidirect
22. #10625192	C₃²×C₅	direct (N × H)
26. C₃×C₅×C₃₇	C₃×C₄×C₅	semidirect
27. C₃×C₅×C₃₇	C₃×AGL(1,𝔽₅)	semidirect
32. C₃²×C₃₇	C₅×AGL(1,𝔽₅)	semidirect
34. C₃²×C₅²	#3866	semidirect
37. C₅×C₃₇	C₃²×C₄×C₅	semidirect
38. C₅×C₃₇	C₃²×AGL(1,𝔽₅)	semidirect
43. C₃×C₃₇	C₃×C₅×AGL(1,𝔽₅)	semidirect
47. C₃×C₅²	C₃×#3866	semidirect
53. C₃²×C₅	C₅×#3866	semidirect
54. C₃²×C₅	#10625192	direct (N × H)
55. C₃₇	C₃²×C₅×AGL(1,𝔽₅)	semidirect
59. C₅²	C₃²×#3866	semidirect
64. C₃×C₅	C₃×C₅×#3866	semidirect
65. C₃×C₅	C₃×#10625192	direct (N × H)
69. C₃²	C₅×#10625192	direct (N × H)
72. C₅	C₃²×C₅×#3866	semidirect
73. C₅	C₃²×#10625192	direct (N × H)
75. C₃	C₃×C₅×#10625192	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×C₅×D₁₈₅	2
C₃×C₅×#10625192	3
C₃²×#10625192	5
C₃²×C₅×#3866	5
C₃²×C₅×AGL(1,𝔽₅)	37

Subgroups — C32×C5×#10625192 #45686388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×#10625192 #45686388