Subgroups — #29506059

Order 15300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₅⋊D₈₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
3	#15690391	—	5	1	no	maximal
4	#15690390	—	5	1	no	maximal
5	C₃⋊(C₅⋊D₈₅)	—	6	2	no	—
6	#13133347	—	9	1	no	maximal, Hall
7	C₃⋊D₂₅₅	—	5	1	no	—
8	C₃⋊D₂₅₅	—	5	1	no	—
9	C₃⋊D₂₅₅	—	10	2	no	—
10	C₃×C₅²×C₁₇	—	2	2	no	—
11	#11744708	—	17	1	no	maximal, Hall
12	C₅⋊D₈₅	—	9	1	no	—
13	C₃²×C₅×C₁₇	—	2	2	no	—
14	C₃²×C₅×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	C₃²×C₅×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
16	#10597479	—	25	1	no	Hall
17	D₂₅₅	—	30	2	no	—
18	D₂₅₅	—	30	2	no	—
19	D₂₅₅	—	30	8	no	—
20	C₃⋊(C₃⋊(C₅⋊D₅))	—	17	1	no	—
21	C₅²×C₁₇	#157	1	1	yes	characteristic, complemented, Hall
22	#66776	—	45	1	no	—
23	#66775	—	45	1	no	—
24	C₃⋊D₅₁	—	25	1	no	—
25	C₃×C₅×C₁₇	—	2	8	no	—
26	C₃×C₅×C₁₇	—	2	2	no	—
27	C₃×C₅×C₁₇	—	2	2	no	—
28	C₃²×C₅²	#595	1	1	yes	characteristic, complemented, Hall
29	#4380	—	85	1	no	—
30	#4380	—	85	1	no	—
31	D₈₅	—	45	1	no	—
32	D₈₅	—	45	1	no	—
33	D₈₅	—	90	2	no	—
34	C₃²×C₁₇	#1044	1	1	yes	characteristic, complemented, Hall
35	C₃⋊(C₅⋊D₅)	—	102	2	no	—
36	D₅₁	—	150	2	no	—
37	#1044	—	153	1	no	Hall
38	C₃⋊D₁₅	—	85	1	no	—
39	C₃⋊D₁₅	—	85	1	no	—
40	C₃⋊D₁₅	—	170	2	no	—
41	C₅×C₁₇	—	2	2	no	—
42	C₅×C₁₇	#4380	1	1	yes	characteristic, complemented
43	C₅×C₁₇	#4380	1	1	yes	characteristic, complemented
44	C₃×C₅²	—	2	2	no	—
45	#595	—	225	1	no	Hall
46	C₃×C₁₇	—	2	2	no	—
47	C₅⋊D₅	—	153	1	no	—
48	C₃²×C₅	—	2	2	no	—
49	C₃²×C₅	#66775	1	1	yes	characteristic, complemented
50	C₃²×C₅	#66776	1	1	yes	characteristic, complemented
51	#157	—	425	1	no	Hall
52	D₁₇	—	225	1	no	—
53	D₁₅	—	510	8	no	—
54	D₁₅	—	510	2	no	—
55	D₁₅	—	510	2	no	—
56	C₅²	#10597479	1	1	yes	characteristic, complemented, Sylow
57	AGL(1,𝔽₅)	—	765	1	no	—
58	AGL(1,𝔽₅)	—	765	1	no	—
59	C₃⋊D₃	—	425	1	no	—
60	C₁₇	#11744708	1	1	yes	characteristic, complemented, Sylow
61	C₃×C₅	—	2	8	no	—
62	C₃×C₅	—	2	2	no	—
63	C₃×C₅	—	2	2	no	—
64	D₅	—	1530	2	no	—
65	D₅	—	765	1	no	—
66	D₅	—	765	1	no	—
67	C₃²	#13133347	1	1	yes	characteristic, complemented, Sylow
68	D₃	—	2550	2	no	—
69	C₅	—	2	2	no	—
70	C₅	#15690390	1	1	yes	characteristic, complemented
71	C₅	#15690391	1	1	yes	characteristic, complemented
72	C₄	—	3825	1	no	Sylow
73	C₃	—	2	2	no	—
74	C₂	—	3825	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₁₇	C₄
prime radical, cosocle	C₃⋊(C₃⋊(C₅⋊D₈₅))	C₂
LCS	G ▷ C₃²×C₅²×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₁₇	C₄	semidirect
14. C₃²×C₅×C₁₇	AGL(1,𝔽₅)	semidirect
15. C₃²×C₅×C₁₇	AGL(1,𝔽₅)	semidirect
21. C₅²×C₁₇	#157	semidirect
28. C₃²×C₅²	#595	semidirect
34. C₃²×C₁₇	#1044	semidirect
42. C₅×C₁₇	#4380	semidirect
43. C₅×C₁₇	#4380	semidirect
49. C₃²×C₅	#66775	semidirect
50. C₃²×C₅	#66776	semidirect
56. C₅²	#10597479	semidirect
60. C₁₇	#11744708	semidirect
67. C₃²	#13133347	semidirect
70. C₅	#15690390	semidirect
71. C₅	#15690391	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊(C₅⋊D₈₅))	2
#15690391	5
#15690390	5
#13133347	9
#11744708	17