Subgroups — D₁₅⋊(C₃⋊(C₃⋊D₃)) #13108554

Order 1620.

872 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃⋊(C₃⋊D₁₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₅⋊(C₃⋊D₃)	—	3	13	no	maximal
5	D₅×C₃⋊(C₃⋊D₃)	—	3	1	no	maximal
6	C₃⁴×C₅	C₂²	1	1	yes	characteristic, complemented, Hall
7	D₃×C₃⋊(C₃⋊D₃)	—	5	1	no	maximal, Hall
8	C₃⋊(C₃⋊D₁₅)	D₃	1	1	yes	characteristic, complemented
9	C₅×C₃⋊(C₃⋊D₃)	—	3	1	no	—
10	C₅×C₃⋊(C₃⋊D₃)	—	3	13	no	—
11	C₅×C₃⋊(C₃⋊D₃)	—	6	13	no	—
12	C₃×C₃⋊D₁₅	—	3	13	no	—
13	C₃²×D₁₅	D₃	1	13	yes	complemented
14	C₃³×D₅	—	3	1	no	—
15	(C₃⋊D₃)⋊D₅	—	9	13	no	—
16	D₅×C₃⋊D₃	—	9	13	no	—
17	C₃⋊(C₃⋊(C₃⋊D₃))	D₅	1	1	yes	characteristic, complemented
18	C₃×C₃⋊(C₃⋊D₃)	—	5	1	no	—
19	C₃³×D₃	—	5	1	no	—
20	C₃³×C₅	—	2	13	no	—
21	C₃³×C₅	C₂×D₃	1	13	yes	complemented
22	C₃³×C₅	C₂×D₃	1	1	yes	characteristic, complemented
23	C₂×C₃⋊(C₃⋊D₃)	—	15	1	no	—
24	D₃×C₃⋊D₃	—	15	13	no	—
25	C₃⋊D₁₅	—	3	13	no	—
26	C₅×C₃⋊D₃	—	9	13	no	—
27	C₅×C₃⋊D₃	—	9	13	no	—
28	C₅×C₃⋊D₃	—	18	52	no	—
29	C₃×D₁₅	—	9	13	no	—
30	C₃×D₁₅	C₃⋊D₃	1	13	yes	complemented
31	C₃²×D₅	—	3	13	no	—
32	C₃⁴	C₂×D₅	1	1	yes	characteristic, complemented, Sylow
33	D₃×D₅	—	27	13	no	—
34	D₃×D₅	—	27	1	no	—
35	C₂×C₃³	—	15	1	no	—
36	C₃⋊(C₃⋊D₃)	—	3	1	no	—
37	C₃⋊(C₃⋊D₃)	—	3	13	no	—
38	C₃⋊(C₃⋊D₃)	—	5	1	no	—
39	C₃⋊(C₃⋊D₃)	—	6	13	no	—
40	C₃×C₃⋊D₃	—	15	13	no	—
41	C₃²×D₃	—	5	13	no	—
42	C₃²×C₅	—	2	52	no	—
43	C₃²×C₅	D₃²	1	13	yes	complemented
44	C₃²×C₅	C₂×C₃⋊D₃	1	13	yes	complemented
45	C₂×C₃⋊D₃	—	45	13	no	—
46	D₃²	—	45	13	no	—
47	D₁₅	—	9	13	no	—
48	D₁₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
49	C₃×D₅	—	3	13	no	—
50	C₃×D₅	—	27	1	no	—
51	C₅×D₃	—	27	13	no	—
52	C₅×D₃	—	27	1	no	—
53	C₅×D₃	—	54	13	no	—
54	C₃³	—	2	13	no	—
55	C₃³	D₃×D₅	1	13	yes	complemented
56	C₃³	D₃×D₅	1	1	yes	characteristic, complemented
57	C₂×D₅	—	81	1	no	Hall
58	C₂×C₃²	—	15	13	no	—
59	C₃⋊D₃	—	9	13	no	—
60	C₃⋊D₃	—	9	13	no	—
61	C₃⋊D₃	—	15	13	no	—
62	C₃⋊D₃	—	18	52	no	—
63	C₃×D₃	—	5	13	no	—
64	C₃×D₃	—	45	13	no	—
65	C₃×C₅	—	2	13	no	—
66	C₃×C₅	D₃×C₃⋊D₃	1	13	yes	complemented
67	C₃×C₅	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
68	C₂×D₃	—	135	1	no	—
69	C₂×D₃	—	135	13	no	—
70	C₂×C₅	—	81	1	no	—
71	D₅	—	3	1	no	—
72	D₅	—	27	1	no	—
73	C₃²	—	2	52	no	—
74	C₃²	D₅×C₃⋊D₃	1	13	yes	complemented
75	C₃²	(C₃⋊D₃)⋊D₅	1	13	yes	complemented
76	C₂×C₃	—	15	13	no	—
77	C₂×C₃	—	135	1	no	—
78	D₃	—	5	1	no	—
79	D₃	—	27	1	no	—
80	D₃	—	27	13	no	—
81	D₃	—	45	13	no	—
82	D₃	—	54	13	no	—
83	C₅	D₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	405	1	no	Sylow
85	C₃	—	2	13	no	—
86	C₃	D₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
87	C₃	D₁₅⋊(C₃⋊D₃)	1	13	yes	complemented
88	C₂	—	15	1	no	—
89	C₂	—	81	1	no	—
90	C₂	—	135	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃⁴×C₅	C₂²
LCS	G ▷ C₃⁴×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
2. C₃×C₃⋊(C₃⋊D₁₅)	C₂	semidirect
3. C₃³×D₁₅	C₂	semidirect
6. C₃⁴×C₅	C₂²	semidirect
8. C₃⋊(C₃⋊D₁₅)	D₃	semidirect
13. C₃²×D₁₅	D₃	semidirect
17. C₃⋊(C₃⋊(C₃⋊D₃))	D₅	semidirect
21. C₃³×C₅	C₂×D₃	semidirect
22. C₃³×C₅	C₂×D₃	semidirect
30. C₃×D₁₅	C₃⋊D₃	semidirect
32. C₃⁴	C₂×D₅	semidirect
43. C₃²×C₅	D₃²	semidirect
44. C₃²×C₅	C₂×C₃⋊D₃	semidirect
48. D₁₅	C₃⋊(C₃⋊D₃)	semidirect
55. C₃³	D₃×D₅	semidirect
56. C₃³	D₃×D₅	semidirect
66. C₃×C₅	D₃×C₃⋊D₃	semidirect
67. C₃×C₅	C₂×C₃⋊(C₃⋊D₃)	semidirect
74. C₃²	D₅×C₃⋊D₃	semidirect
75. C₃²	(C₃⋊D₃)⋊D₅	semidirect
83. C₅	D₃×C₃⋊(C₃⋊D₃)	semidirect
86. C₃	D₅×C₃⋊(C₃⋊D₃)	semidirect
87. C₃	D₁₅⋊(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⋊(C₃⋊D₃))	2
C₃×C₃⋊(C₃⋊D₁₅)	2
C₃³×D₁₅	2
D₁₅⋊(C₃⋊D₃)	3
D₅×C₃⋊(C₃⋊D₃)	3
D₃×C₃⋊(C₃⋊D₃)	5

Subgroups — D15⋊(C3⋊(C3⋊D3)) #13108554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₅⋊(C₃⋊(C₃⋊D₃)) #13108554