Subgroups — #29506060

Order 15300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
3	#15690392	—	5	2	no	maximal
4	C₁₇×C₃⋊(C₅⋊D₅)	—	6	2	no	—
5	#13133348	—	9	1	no	maximal, Hall
6	C₁₇×C₃⋊D₁₅	—	5	2	no	—
7	C₁₇×C₃⋊D₁₅	—	10	2	no	—
8	C₃×C₅²×C₁₇	—	2	2	no	—
9	#11744708	—	17	1	no	maximal, Hall
10	C₁₇×C₅⋊D₅	—	9	1	no	—
11	C₃²×C₅×C₁₇	—	2	2	no	—
12	C₃²×C₅×C₁₇	AGL(1,𝔽₅)	1	2	yes	complemented
13	#10597480	—	25	1	no	Hall
14	C₁₇×D₁₅	—	30	4	no	—
15	C₁₇×D₁₅	—	30	8	no	—
16	C₃⋊(C₃⋊(C₅⋊D₅))	D₁₇	1	1	yes	characteristic
17	C₅²×C₁₇	#157	1	1	yes	characteristic, complemented, Hall
18	#66774	—	45	2	no	—
19	C₁₇×C₃⋊D₃	—	25	1	no	—
20	C₃×C₅×C₁₇	—	2	8	no	—
21	C₃×C₅×C₁₇	—	2	4	no	—
22	C₃²×C₅²	Dic₁₇	1	1	yes	characteristic, complemented, Hall
23	#4380	—	85	2	no	—
24	D₅×C₁₇	—	45	2	no	—
25	D₅×C₁₇	—	90	2	no	—
26	C₃²×C₁₇	#1044	1	1	yes	characteristic, complemented, Hall
27	C₃⋊(C₅⋊D₅)	—	6	2	no	—
28	D₃×C₁₇	—	150	2	no	—
29	#1044	—	153	1	no	Hall
30	C₃⋊D₁₅	—	5	2	no	—
31	C₃⋊D₁₅	—	10	2	no	—
32	C₅×C₁₇	—	2	2	no	—
33	C₅×C₁₇	#4380	1	2	yes	complemented
34	C₃×C₅²	—	2	2	no	—
35	Dic₁₇	—	225	1	no	Hall
36	C₃×C₁₇	—	2	2	no	—
37	C₅⋊D₅	—	9	1	no	—
38	C₃²×C₅	—	2	2	no	—
39	C₃²×C₅	#66774	1	2	yes	complemented
40	#157	—	425	1	no	Hall
41	C₂×C₁₇	—	225	1	no	—
42	D₁₅	—	30	4	no	—
43	D₁₅	—	30	8	no	—
44	C₅²	#10597480	1	1	yes	characteristic, complemented, Sylow
45	AGL(1,𝔽₅)	—	765	2	no	—
46	C₃⋊D₃	—	25	1	no	—
47	C₁₇	#11744708	1	1	yes	characteristic, complemented, Sylow
48	C₃×C₅	—	2	8	no	—
49	C₃×C₅	—	2	4	no	—
50	D₅	—	45	2	no	—
51	D₅	—	90	2	no	—
52	C₃²	#13133348	1	1	yes	characteristic, complemented, Sylow
53	D₃	—	150	2	no	—
54	C₅	—	2	2	no	—
55	C₅	#15690392	1	2	yes	complemented
56	C₄	—	3825	1	no	Sylow
57	C₃	—	2	2	no	—
58	C₂	—	225	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₁₇	C₄
prime radical, cosocle	C₁₇×C₃⋊(C₃⋊(C₅⋊D₅))	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
12. C₃²×C₅×C₁₇	AGL(1,𝔽₅)	semidirect
17. C₅²×C₁₇	#157	semidirect
22. C₃²×C₅²	Dic₁₇	semidirect
26. C₃²×C₁₇	#1044	semidirect
33. C₅×C₁₇	#4380	semidirect
39. C₃²×C₅	#66774	semidirect
44. C₅²	#10597480	semidirect
47. C₁₇	#11744708	semidirect
52. C₃²	#13133348	semidirect
55. C₅	#15690392	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₃⋊(C₃⋊(C₅⋊D₅))	2
#15690392	5
#13133348	9
#11744708	17