Subgroups — C₃²×C₅×AGL(1,𝔽₅)×C₃₇ #45686386

Order 33300.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₃²×C₅×C₃₇	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₄×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅×D₅×C₃₇	C₂
cosocle	D₅	C₂×C₃²×C₅×C₃₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₅×C₃₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×AGL(1,𝔽₅)×C₃₇	C₃	direct (N × H)
3. C₃²×C₅²×C₃₇	C₄	semidirect
4. C₃²×AGL(1,𝔽₅)×C₃₇	C₅	direct (N × H)
7. C₅×AGL(1,𝔽₅)×C₃₇	C₃²	direct (N × H)
10. C₃×C₅²×C₃₇	C₃×C₄	semidirect
11. C₃×AGL(1,𝔽₅)×C₃₇	C₃×C₅	direct (N × H)
15. C₃²×C₅×C₃₇	C₄×C₅	semidirect
16. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	direct (N × H)
20. C₅²×C₃₇	C₃²×C₄	semidirect
21. C₃²×C₅×AGL(1,𝔽₅)	C₃₇	direct (N × H)
22. AGL(1,𝔽₅)×C₃₇	C₃²×C₅	direct (N × H)
26. C₃×C₅×C₃₇	C₃×C₄×C₅	semidirect
27. C₃×C₅×C₃₇	C₃×AGL(1,𝔽₅)	direct (N × H)
32. C₃²×C₃₇	C₅×AGL(1,𝔽₅)	direct (N × H)
33. C₃×C₅×AGL(1,𝔽₅)	C₃×C₃₇	direct (N × H)
34. C₃²×C₅²	C₄×C₃₇	semidirect
37. C₅×C₃₇	C₃²×C₄×C₅	semidirect
38. C₅×C₃₇	C₃²×AGL(1,𝔽₅)	direct (N × H)
39. C₃²×AGL(1,𝔽₅)	C₅×C₃₇	direct (N × H)
43. C₃×C₃₇	C₃×C₅×AGL(1,𝔽₅)	direct (N × H)
44. C₅×AGL(1,𝔽₅)	C₃²×C₃₇	direct (N × H)
47. C₃×C₅²	C₃×C₄×C₃₇	semidirect
49. C₃×AGL(1,𝔽₅)	C₃×C₅×C₃₇	direct (N × H)
53. C₃²×C₅	C₄×C₅×C₃₇	semidirect
54. C₃²×C₅	AGL(1,𝔽₅)×C₃₇	direct (N × H)
55. C₃₇	C₃²×C₅×AGL(1,𝔽₅)	direct (N × H)
59. C₅²	C₃²×C₄×C₃₇	semidirect
60. AGL(1,𝔽₅)	C₃²×C₅×C₃₇	direct (N × H)
64. C₃×C₅	C₃×C₄×C₅×C₃₇	semidirect
65. C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₇	direct (N × H)
69. C₃²	C₅×AGL(1,𝔽₅)×C₃₇	direct (N × H)
72. C₅	C₃²×C₄×C₅×C₃₇	semidirect
73. C₅	C₃²×AGL(1,𝔽₅)×C₃₇	direct (N × H)
75. C₃	C₃×C₅×AGL(1,𝔽₅)×C₃₇	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×D₅×C₃₇	2
C₃×C₅×AGL(1,𝔽₅)×C₃₇	3
C₃²×AGL(1,𝔽₅)×C₃₇	5
C₃²×C₄×C₅×C₃₇	5
C₃²×C₅×AGL(1,𝔽₅)	37

Subgroups — C32×C5×AGL(1,𝔽5)×C37 #45686386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×AGL(1,𝔽₅)×C₃₇ #45686386