Subgroups — C₅×C₃₇×#4380 #45686365

Order 33300.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₇×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
3	C₃₇×#4380	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×#157×C₃₇	—	5	1	no	maximal
5	C₅×D₁₅×C₃₇	—	6	2	no	—
6	C₅×AGL(1,𝔽₅)×C₃₇	—	9	1	no	maximal, Hall
7	C₃₇×C₃⋊D₁₅	C₂×C₅	1	1	yes	characteristic
8	C₅×C₃⋊D₃×C₃₇	—	5	1	no	—
9	C₃×C₅²×C₃₇	—	2	2	no	—
10	C₅×D₅×C₃₇	—	9	1	no	—
11	C₃²×C₅×C₃₇	—	4	1	no	—
12	C₃²×C₅×C₃₇	C₄×C₅	1	1	yes	characteristic, complemented
13	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	#157×C₃₇	—	5	1	no	Hall
15	D₁₅×C₃₇	—	6	2	no	—
16	C₅×D₃×C₃₇	—	30	2	no	—
17	C₅²×C₃₇	#157	1	1	yes	characteristic, complemented, Hall
18	C₅×#4380	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	AGL(1,𝔽₅)×C₃₇	—	9	1	no	—
20	C₄×C₅×C₃₇	—	45	1	no	—
21	C₃⋊D₃×C₃₇	—	5	1	no	—
22	C₃×C₅×C₃₇	—	2	2	no	—
23	C₃×C₅×C₃₇	—	2	2	no	—
24	C₃×C₅×C₃₇	—	4	4	no	—
25	C₅×C₃⋊D₁₅	C₂×C₃₇	1	1	yes	characteristic
26	C₂×C₅×C₃₇	—	45	1	no	—
27	D₅×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
29	C₃²×C₅²	C₄×C₃₇	1	1	yes	characteristic, complemented, Hall
30	D₃×C₃₇	—	30	2	no	—
31	C₅×C₃₇	—	4	1	no	—
32	C₅×C₃₇	C₅×#157	1	1	yes	characteristic, complemented
33	C₅×C₃₇	#4380	1	1	yes	characteristic, direct product factor
34	#4380	C₅×C₃₇	1	1	yes	characteristic, direct product factor
35	C₅×#157	—	5	1	no	—
36	C₅×D₁₅	—	6	2	no	—
37	C₄×C₃₇	—	45	1	no	Hall
38	C₃×C₃₇	—	2	2	no	—
39	C₅×AGL(1,𝔽₅)	—	9	1	no	Hall
40	C₃⋊D₁₅	C₂×C₅×C₃₇	1	1	yes	characteristic
41	C₅×C₃⋊D₃	—	5	1	no	—
42	C₃×C₅²	—	2	2	no	—
43	C₂×C₃₇	—	45	1	no	—
44	C₅×D₅	—	9	1	no	—
45	C₃²×C₅	—	4	1	no	—
46	C₃²×C₅	C₄×C₅×C₃₇	1	1	yes	characteristic, complemented
47	C₃²×C₅	AGL(1,𝔽₅)×C₃₇	1	1	yes	characteristic, complemented
48	C₃₇	C₅×#4380	1	1	yes	characteristic, direct product factor, Sylow
49	#157	—	5	1	no	Hall
50	D₁₅	—	6	2	no	—
51	C₅×D₃	—	30	2	no	—
52	C₅²	#157×C₃₇	1	1	yes	characteristic, complemented, Sylow
53	AGL(1,𝔽₅)	—	9	1	no	—
54	C₄×C₅	—	45	1	no	—
55	C₃⋊D₃	—	5	1	no	—
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	2	2	no	—
58	C₃×C₅	—	4	4	no	—
59	C₂×C₅	—	45	1	no	—
60	D₅	—	9	1	no	—
61	C₃²	C₅×AGL(1,𝔽₅)×C₃₇	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	30	2	no	—
63	C₅	—	4	1	no	—
64	C₅	C₅×#157×C₃₇	1	1	yes	characteristic, complemented
65	C₅	C₃₇×#4380	1	1	yes	characteristic, direct product factor
66	C₄	—	45	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	45	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₃₇	#4380
commutator, hypocenter	C₃²×C₅	C₄×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₅×C₃₇×C₃⋊D₁₅	C₂
cosocle	C₃⋊D₁₅	C₂×C₅×C₃₇
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₅×C₃₇

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₇	C₄	semidirect
3. C₃₇×#4380	C₅	direct (N × H)
12. C₃²×C₅×C₃₇	C₄×C₅	semidirect
13. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
17. C₅²×C₃₇	#157	semidirect
18. C₅×#4380	C₃₇	direct (N × H)
28. C₃²×C₃₇	C₅×AGL(1,𝔽₅)	semidirect
29. C₃²×C₅²	C₄×C₃₇	semidirect
32. C₅×C₃₇	C₅×#157	semidirect
33. C₅×C₃₇	#4380	direct (N × H)
34. #4380	C₅×C₃₇	direct (N × H)
46. C₃²×C₅	C₄×C₅×C₃₇	semidirect
47. C₃²×C₅	AGL(1,𝔽₅)×C₃₇	semidirect
48. C₃₇	C₅×#4380	direct (N × H)
52. C₅²	#157×C₃₇	semidirect
61. C₃²	C₅×AGL(1,𝔽₅)×C₃₇	semidirect
64. C₅	C₅×#157×C₃₇	semidirect
65. C₅	C₃₇×#4380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₃₇×C₃⋊D₁₅	2
C₃₇×#4380	5
C₅×#157×C₃₇	5
C₅×AGL(1,𝔽₅)×C₃₇	9
C₅×#4380	37

Subgroups — C5×C37×#4380 #45686365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₃₇×#4380 #45686365