Subgroups — C₅×C₃⋊(C₃⋊#66772) #29505990

Order 15300.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊#66772	—	3	4	no	maximal
3	C₃²×C₅²×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊(C₃⋊#66772)	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₅×C₃⋊(C₃⋊#595)	—	5	1	no	maximal
6	C₃×C₅²×D₁₇	D₃	1	4	yes	—
7	C₅×#66772	—	9	1	no	Hall
8	C₃²×C₅×D₁₇	—	2	2	no	—
9	C₃²×C₅×D₁₇	D₅	1	1	yes	characteristic
10	C₃²×C₅×D₁₇	C₂×C₅	1	1	yes	characteristic
11	C₃×C₅²×C₁₇	Dic₃	1	4	yes	complemented
12	C₃⋊#66772	—	3	4	no	—
13	C₅×C₃⋊#595	—	15	4	no	—
14	C₅×C₃⋊Dic₁₅	—	17	1	no	maximal, Hall
15	C₅²×D₁₇	C₃⋊D₃	1	1	yes	characteristic
16	C₃²×C₅×C₁₇	—	2	2	no	—
17	C₃²×C₅×C₁₇	Dic₅	1	1	yes	characteristic, complemented
18	C₃²×C₅×C₁₇	C₄×C₅	1	1	yes	characteristic, complemented
19	C₃⋊(C₃⋊#595)	—	5	1	no	Hall
20	C₃×C₅×D₁₇	—	2	8	no	—
21	C₃×C₅×D₁₇	C₅×D₃	1	4	yes	—
22	C₃×C₅×D₁₇	D₁₅	1	4	yes	—
23	C₂×C₃²×C₅²	—	17	1	no	—
24	C₅²×C₁₇	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
25	#66772	—	9	1	no	—
26	C₅×#595	—	45	1	no	—
27	C₃²×D₁₇	C₅×D₅	1	1	yes	characteristic
28	C₅×Dic₁₅	—	51	4	no	—
29	C₃×C₅×C₁₇	—	2	8	no	—
30	C₃×C₅×C₁₇	C₅×Dic₃	1	4	yes	complemented
31	C₃×C₅×C₁₇	Dic₁₅	1	4	yes	complemented
32	C₃²×C₅²	#595	1	1	yes	characteristic, complemented, Hall
33	C₃⋊#595	—	15	4	no	—
34	C₃⋊Dic₁₅	—	17	1	no	—
35	C₅×C₃⋊Dic₃	—	85	1	no	—
36	C₅×D₁₇	—	2	2	no	—
37	C₅×D₁₇	C₅×C₃⋊D₃	1	1	yes	characteristic
38	C₅×D₁₇	C₃⋊D₁₅	1	1	yes	characteristic
39	C₃²×C₁₇	C₅×Dic₅	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₅²	—	17	4	no	—
41	C₃×D₁₇	C₅×D₁₅	1	4	yes	—
42	C₅×Dic₅	—	153	1	no	Hall
43	C₂×C₃²×C₅	—	17	1	no	—
44	C₂×C₃²×C₅	—	17	1	no	—
45	C₂×C₃²×C₅	—	34	2	no	—
46	C₅×C₁₇	—	2	2	no	—
47	C₅×C₁₇	C₅×C₃⋊Dic₃	1	1	yes	characteristic, complemented
48	C₅×C₁₇	C₃⋊Dic₁₅	1	1	yes	characteristic, complemented
49	C₃×C₅²	C₃⋊#595	1	4	yes	complemented
50	#595	—	45	1	no	Hall
51	Dic₁₅	—	51	4	no	—
52	C₅×Dic₃	—	255	4	no	—
53	C₃×C₁₇	C₅×Dic₁₅	1	4	yes	complemented
54	C₂×C₅²	—	17	1	no	—
55	C₃²×C₅	—	2	2	no	—
56	C₃²×C₅	C₅×#595	1	1	yes	characteristic, complemented
57	C₃²×C₅	#66772	1	1	yes	characteristic, complemented
58	C₃⋊Dic₃	—	85	1	no	Hall
59	D₁₇	C₅×C₃⋊D₁₅	1	1	yes	characteristic
60	C₂×C₃×C₅	—	17	4	no	—
61	C₂×C₃×C₅	—	17	4	no	—
62	C₂×C₃×C₅	—	34	8	no	—
63	C₅²	C₃⋊(C₃⋊#595)	1	1	yes	characteristic, complemented, Sylow
64	C₄×C₅	—	765	1	no	—
65	Dic₅	—	153	1	no	—
66	C₂×C₃²	—	17	1	no	—
67	C₁₇	C₅×C₃⋊Dic₁₅	1	1	yes	characteristic, complemented, Sylow
68	C₃×C₅	—	2	8	no	—
69	C₃×C₅	C₅×C₃⋊#595	1	4	yes	complemented
70	C₃×C₅	C₃⋊#66772	1	4	yes	complemented
71	Dic₃	—	255	4	no	—
72	C₂×C₅	—	17	1	no	—
73	C₂×C₅	—	17	1	no	—
74	C₂×C₅	—	34	2	no	—
75	C₃²	C₅×#66772	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃	—	17	4	no	—
77	C₅	—	2	2	no	—
78	C₅	C₅×C₃⋊(C₃⋊#595)	1	1	yes	characteristic, complemented
79	C₅	C₃⋊(C₃⋊#66772)	1	1	yes	characteristic, direct product factor
80	C₄	—	765	1	no	Sylow
81	C₃	C₅×C₃⋊#66772	1	4	yes	complemented
82	C₂	—	17	1	no	—

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(C₃⋊#66772)
commutator, hypocenter	C₃²×C₅×C₁₇	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₇	C₄
prime radical	C₃²×C₅²×D₁₇	C₂
cosocle	C₃²×C₅×D₁₇	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
3. C₃²×C₅²×C₁₇	C₄	semidirect
4. C₃⋊(C₃⋊#66772)	C₅	direct (N × H)
11. C₃×C₅²×C₁₇	Dic₃	semidirect
17. C₃²×C₅×C₁₇	Dic₅	semidirect
18. C₃²×C₅×C₁₇	C₄×C₅	semidirect
24. C₅²×C₁₇	C₃⋊Dic₃	semidirect
30. C₃×C₅×C₁₇	C₅×Dic₃	semidirect
31. C₃×C₅×C₁₇	Dic₁₅	semidirect
32. C₃²×C₅²	#595	semidirect
39. C₃²×C₁₇	C₅×Dic₅	semidirect
47. C₅×C₁₇	C₅×C₃⋊Dic₃	semidirect
48. C₅×C₁₇	C₃⋊Dic₁₅	semidirect
49. C₃×C₅²	C₃⋊#595	semidirect
53. C₃×C₁₇	C₅×Dic₁₅	semidirect
56. C₃²×C₅	C₅×#595	semidirect
57. C₃²×C₅	#66772	semidirect
63. C₅²	C₃⋊(C₃⋊#595)	semidirect
67. C₁₇	C₅×C₃⋊Dic₁₅	semidirect
69. C₃×C₅	C₅×C₃⋊#595	semidirect
70. C₃×C₅	C₃⋊#66772	semidirect
75. C₃²	C₅×#66772	semidirect
78. C₅	C₅×C₃⋊(C₃⋊#595)	semidirect
79. C₅	C₃⋊(C₃⋊#66772)	direct (N × H)
81. C₃	C₅×C₃⋊#66772	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅²×D₁₇	2
C₅×C₃⋊#66772	3
C₃⋊(C₃⋊#66772)	5
C₅×C₃⋊(C₃⋊#595)	5
C₅×C₃⋊Dic₁₅	17

Subgroups — C5×C3⋊(C3⋊#66772) #29505990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₃⋊(C₃⋊#66772) #29505990