Subgroups — C₅×D₅×C₃⋊(C₃⋊D₃) #72247643

Order 2700.

672 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊(C₃⋊D₁₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅×C₃⋊D₃	—	3	13	no	maximal
5	C₃³×C₅²	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₅×C₃⋊(C₃⋊D₃)	—	5	1	no	maximal
7	D₅×C₃⋊(C₃⋊D₃)	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₅×C₃⋊D₁₅	—	3	13	no	—
9	C₅²×C₃⋊D₃	—	3	13	no	—
10	C₃²×C₅×D₅	D₃	1	13	yes	complemented
11	C₅×D₃×D₅	—	9	13	no	—
12	C₂×C₃³×C₅	—	5	1	no	—
13	C₃⋊(C₃⋊D₁₅)	C₂×C₅	1	1	yes	characteristic, complemented
14	C₅×C₃⋊(C₃⋊D₃)	—	2	2	no	—
15	C₅×C₃⋊(C₃⋊D₃)	—	5	1	no	—
16	C₅×C₃⋊(C₃⋊D₃)	D₅	1	1	yes	characteristic, direct product factor
17	C₅×C₃⋊(C₃⋊D₃)	C₂×C₅	1	1	yes	characteristic, complemented
18	C₃³×D₅	C₂×C₅	1	1	yes	characteristic, complemented
19	C₃²×C₅²	C₂×D₃	1	13	yes	complemented
20	C₂×C₅×C₃⋊D₃	—	15	13	no	—
21	D₅×C₃⋊D₃	—	3	13	no	—
22	C₅×D₁₅	—	9	13	no	—
23	C₅²×D₃	—	9	13	no	—
24	C₃×C₅×D₅	C₃⋊D₃	1	13	yes	complemented
25	C₃³×C₅	—	2	2	no	—
26	C₃³×C₅	C₂×D₅	1	1	yes	characteristic, complemented
27	C₃³×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
28	C₂×C₃⋊(C₃⋊D₃)	—	5	1	no	Hall
29	C₂×C₅×D₅	—	27	1	no	Hall
30	C₂×C₃²×C₅	—	5	13	no	—
31	C₃⋊D₁₅	—	3	13	no	—
32	C₅×C₃⋊D₃	—	3	13	no	—
33	C₅×C₃⋊D₃	—	3	13	no	—
34	C₅×C₃⋊D₃	—	6	26	no	—
35	C₅×C₃⋊D₃	—	15	13	no	—
36	C₃²×D₅	C₅×D₃	1	13	yes	complemented
37	C₃×C₅²	C₂×C₃⋊D₃	1	13	yes	complemented
38	C₂×C₅×D₃	—	45	13	no	—
39	D₃×D₅	—	9	13	no	—
40	C₂×C₃³	—	5	1	no	—
41	C₃⋊(C₃⋊D₃)	—	5	1	no	—
42	C₃⋊(C₃⋊D₃)	C₅×D₅	1	1	yes	characteristic, direct product factor
43	C₂×C₅²	—	27	1	no	—
44	C₅×D₅	—	27	1	no	—
45	C₅×D₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
46	C₃²×C₅	—	2	26	no	—
47	C₃²×C₅	D₃×D₅	1	13	yes	complemented
48	C₃²×C₅	C₂×C₅×D₃	1	13	yes	complemented
49	C₂×C₃⋊D₃	—	15	13	no	—
50	C₂×C₃×C₅	—	5	13	no	—
51	D₁₅	—	9	13	no	—
52	C₃×D₅	C₅×C₃⋊D₃	1	13	yes	complemented
53	C₅×D₃	—	9	13	no	—
54	C₅×D₃	—	9	13	no	—
55	C₅×D₃	—	18	26	no	—
56	C₅×D₃	—	45	13	no	—
57	C₃³	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
58	C₅²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	—	135	1	no	—
60	C₂×D₅	—	27	1	no	—
61	C₂×C₃²	—	5	13	no	—
62	C₃⋊D₃	—	3	13	no	—
63	C₃⋊D₃	—	15	13	no	—
64	C₃×C₅	—	2	26	no	—
65	C₃×C₅	D₅×C₃⋊D₃	1	13	yes	complemented
66	C₃×C₅	C₂×C₅×C₃⋊D₃	1	13	yes	complemented
67	C₂×D₃	—	45	13	no	—
68	C₂×C₅	—	5	1	no	—
69	C₂×C₅	—	27	1	no	—
70	C₂×C₅	—	27	1	no	—
71	C₂×C₅	—	54	2	no	—
72	C₂×C₅	—	135	1	no	—
73	D₅	—	27	1	no	—
74	D₅	C₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
75	C₃²	C₅×D₃×D₅	1	13	yes	complemented
76	C₂×C₃	—	5	13	no	—
77	D₃	—	9	13	no	—
78	D₃	—	45	13	no	—
79	C₅	—	2	2	no	—
80	C₅	D₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
81	C₅	C₂×C₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
82	C₂²	—	135	1	no	Sylow
83	C₃	C₅×D₅×C₃⋊D₃	1	13	yes	complemented
84	C₂	—	5	1	no	—
85	C₂	—	27	1	no	—
86	C₂	—	135	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₅×C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³×C₅	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃³×C₅²	C₂²
LCS	G ▷ C₃³×C₅
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₃⋊(C₃⋊D₁₅)	C₂	semidirect
2. C₅²×C₃⋊(C₃⋊D₃)	C₂	semidirect
3. C₃³×C₅×D₅	C₂	semidirect
5. C₃³×C₅²	C₂²	semidirect
7. D₅×C₃⋊(C₃⋊D₃)	C₅	direct (N × H)
10. C₃²×C₅×D₅	D₃	semidirect
13. C₃⋊(C₃⋊D₁₅)	C₂×C₅	semidirect
16. C₅×C₃⋊(C₃⋊D₃)	D₅	direct (N × H)
17. C₅×C₃⋊(C₃⋊D₃)	C₂×C₅	semidirect
18. C₃³×D₅	C₂×C₅	semidirect
19. C₃²×C₅²	C₂×D₃	semidirect
24. C₃×C₅×D₅	C₃⋊D₃	semidirect
26. C₃³×C₅	C₂×D₅	semidirect
27. C₃³×C₅	C₂²×C₅	semidirect
36. C₃²×D₅	C₅×D₃	semidirect
37. C₃×C₅²	C₂×C₃⋊D₃	semidirect
42. C₃⋊(C₃⋊D₃)	C₅×D₅	direct (N × H)
45. C₅×D₅	C₃⋊(C₃⋊D₃)	direct (N × H)
47. C₃²×C₅	D₃×D₅	semidirect
48. C₃²×C₅	C₂×C₅×D₃	semidirect
52. C₃×D₅	C₅×C₃⋊D₃	semidirect
57. C₃³	C₂×C₅×D₅	semidirect
58. C₅²	C₂×C₃⋊(C₃⋊D₃)	semidirect
65. C₃×C₅	D₅×C₃⋊D₃	semidirect
66. C₃×C₅	C₂×C₅×C₃⋊D₃	semidirect
74. D₅	C₅×C₃⋊(C₃⋊D₃)	direct (N × H)
75. C₃²	C₅×D₃×D₅	semidirect
80. C₅	D₅×C₃⋊(C₃⋊D₃)	direct (N × H)
81. C₅	C₂×C₅×C₃⋊(C₃⋊D₃)	semidirect
83. C₃	C₅×D₅×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊(C₃⋊D₁₅)	2
C₅²×C₃⋊(C₃⋊D₃)	2
C₃³×C₅×D₅	2
C₅×D₅×C₃⋊D₃	3
C₂×C₅×C₃⋊(C₃⋊D₃)	5
D₅×C₃⋊(C₃⋊D₃)	5

Subgroups — C5×D5×C3⋊(C3⋊D3) #72247643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×C₃⋊(C₃⋊D₃) #72247643