Subgroups — C₅×#15690391 #29506049

Order 15300.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊D₂₅₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
3	#15690391	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×#10597479	—	5	1	no	maximal
5	C₅×D₂₅₅	—	6	2	no	—
6	C₅×#66776	—	9	1	no	maximal, Hall
7	C₃⋊D₂₅₅	C₂×C₅	1	1	yes	characteristic
8	C₅×C₃⋊D₅₁	—	5	1	no	—
9	C₃×C₅²×C₁₇	—	2	2	no	—
10	C₅×#4380	—	17	1	no	maximal, Hall
11	C₅×D₈₅	—	9	1	no	—
12	C₃²×C₅×C₁₇	—	4	1	no	—
13	C₃²×C₅×C₁₇	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃²×C₅×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	#10597479	—	5	1	no	Hall
16	D₂₅₅	—	6	2	no	—
17	C₅×D₅₁	—	30	2	no	—
18	C₅×C₃⋊D₁₅	—	17	1	no	—
19	C₅²×C₁₇	#157	1	1	yes	characteristic, complemented, Hall
20	#66776	—	9	1	no	—
21	C₅×#595	—	45	1	no	—
22	C₃⋊D₅₁	—	5	1	no	—
23	C₃×C₅×C₁₇	—	2	2	no	—
24	C₃×C₅×C₁₇	—	2	2	no	—
25	C₃×C₅×C₁₇	—	4	4	no	—
26	C₃²×C₅²	#595	1	1	yes	characteristic, complemented, Hall
27	#4380	—	17	1	no	—
28	C₅×#157	—	85	1	no	—
29	D₈₅	—	9	1	no	—
30	C₅×D₁₇	—	45	1	no	—
31	C₃²×C₁₇	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
32	C₅×D₁₅	—	102	2	no	—
33	D₅₁	—	30	2	no	—
34	C₅×AGL(1,𝔽₅)	—	153	1	no	Hall
35	C₃⋊D₁₅	—	17	1	no	—
36	C₅×C₃⋊D₃	—	85	1	no	—
37	C₅×C₁₇	—	4	1	no	—
38	C₅×C₁₇	C₅×#157	1	1	yes	characteristic, complemented
39	C₅×C₁₇	#4380	1	1	yes	characteristic, complemented
40	C₃×C₅²	—	2	2	no	—
41	#595	—	45	1	no	Hall
42	C₃×C₁₇	—	2	2	no	—
43	C₅×D₅	—	153	1	no	—
44	C₃²×C₅	—	4	1	no	—
45	C₃²×C₅	C₅×#595	1	1	yes	characteristic, complemented
46	C₃²×C₅	#66776	1	1	yes	characteristic, complemented
47	#157	—	85	1	no	Hall
48	D₁₇	—	45	1	no	—
49	D₁₅	—	102	2	no	—
50	C₅×D₃	—	510	2	no	—
51	C₅²	#10597479	1	1	yes	characteristic, complemented, Sylow
52	AGL(1,𝔽₅)	—	153	1	no	—
53	C₄×C₅	—	765	1	no	—
54	C₃⋊D₃	—	85	1	no	—
55	C₁₇	C₅×#4380	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	2	2	no	—
58	C₃×C₅	—	4	4	no	—
59	C₂×C₅	—	765	1	no	—
60	D₅	—	153	1	no	—
61	C₃²	C₅×#66776	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	510	2	no	—
63	C₅	—	4	1	no	—
64	C₅	C₅×#10597479	1	1	yes	characteristic, complemented
65	C₅	#15690391	1	1	yes	characteristic, direct product factor
66	C₄	—	765	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	765	1	no	—

	subgroup	G/H
center, hypercenter	C₅	#15690391
commutator, hypocenter	C₃²×C₅×C₁₇	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₇	C₄
prime radical	C₅×C₃⋊D₂₅₅	C₂
cosocle	C₃⋊D₂₅₅	C₂×C₅
LCS	G ▷ C₃²×C₅×C₁₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₁₇	C₄	semidirect
3. #15690391	C₅	direct (N × H)
13. C₃²×C₅×C₁₇	C₄×C₅	semidirect
14. C₃²×C₅×C₁₇	AGL(1,𝔽₅)	semidirect
19. C₅²×C₁₇	#157	semidirect
26. C₃²×C₅²	#595	semidirect
31. C₃²×C₁₇	C₅×AGL(1,𝔽₅)	semidirect
38. C₅×C₁₇	C₅×#157	semidirect
39. C₅×C₁₇	#4380	semidirect
45. C₃²×C₅	C₅×#595	semidirect
46. C₃²×C₅	#66776	semidirect
51. C₅²	#10597479	semidirect
55. C₁₇	C₅×#4380	semidirect
61. C₃²	C₅×#66776	semidirect
64. C₅	C₅×#10597479	semidirect
65. C₅	#15690391	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×C₃⋊D₂₅₅	2
#15690391	5
C₅×#10597479	5
C₅×#66776	9
C₅×#4380	17

Subgroups — C5×#15690391 #29506049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#15690391 #29506049