Subgroups — C₃²×#16810011 #44478232

Order 30420.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₁₃⋊D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#16810011	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×#10621729	—	5	1	no	maximal, Hall
5	C₃×C₅×C₁₃⋊D₁₃	C₂×C₃	1	4	yes	—
6	#16810011	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₁₃⋊D₁₃	D₅	1	1	yes	characteristic
8	C₃×C₅×C₁₃²	C₃×C₄	1	4	yes	complemented
9	C₃²×#63118	—	13	2	no	maximal
10	C₃×#10621729	—	5	4	no	—
11	C₅×C₁₃⋊D₁₃	C₂×C₃²	1	1	yes	characteristic
12	C₃²×C₁₃²	Dic₅	1	1	yes	characteristic, complemented, Hall
13	C₃²×C₅×D₁₃	—	13	2	no	—
14	C₃²×C₅×D₁₃	—	26	6	no	—
15	C₃×C₁₃⋊D₁₃	C₃×D₅	1	4	yes	—
16	C₅×C₁₃²	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
17	C₃×#63118	—	13	8	no	—
18	#10621729	—	5	1	no	Hall
19	C₃²×C₅×C₁₃	—	2	6	no	—
20	C₃²×C₅×C₁₃	#261	1	2	yes	complemented
21	C₃×C₁₃²	C₃×Dic₅	1	4	yes	complemented
22	C₃²×#261	—	65	2	no	—
23	C₃×C₅×D₁₃	—	13	8	no	—
24	C₃×C₅×D₁₃	—	26	24	no	—
25	C₁₃⋊D₁₃	C₃²×D₅	1	1	yes	characteristic
26	#63118	—	13	2	no	—
27	C₃²×D₁₃	—	13	2	no	—
28	C₃²×D₁₃	—	26	6	no	—
29	C₃×C₅×C₁₃	—	2	24	no	—
30	C₃×C₅×C₁₃	C₃×#261	1	8	yes	complemented
31	C₃²×Dic₅	—	169	1	no	Hall
32	C₁₃²	C₃²×Dic₅	1	1	yes	characteristic, complemented, Sylow
33	C₃×#261	—	65	8	no	—
34	C₅×D₁₃	—	13	2	no	—
35	C₅×D₁₃	—	26	6	no	—
36	C₃²×C₁₃	—	2	6	no	—
37	C₃²×C₁₃	#63118	1	2	yes	complemented
38	C₂×C₃²×C₅	—	169	1	no	—
39	C₃×D₁₃	—	13	8	no	—
40	C₃×D₁₃	—	26	24	no	—
41	C₅×C₁₃	—	2	6	no	—
42	C₅×C₁₃	C₃²×#261	1	2	yes	complemented
43	C₃×Dic₅	—	169	4	no	—
44	#261	—	65	2	no	—
45	C₃²×C₅	#10621729	1	1	yes	characteristic, complemented, Hall
46	C₃×C₁₃	—	2	24	no	—
47	C₃×C₁₃	C₃×#63118	1	8	yes	complemented
48	C₃²×C₄	—	845	1	no	Hall
49	C₂×C₃×C₅	—	169	4	no	—
50	D₁₃	—	13	2	no	—
51	D₁₃	—	26	6	no	—
52	Dic₅	—	169	1	no	Hall
53	C₂×C₃²	—	169	1	no	—
54	C₃×C₅	C₃×#10621729	1	4	yes	complemented
55	C₁₃	—	2	6	no	—
56	C₁₃	C₃²×#63118	1	2	yes	complemented
57	C₃×C₄	—	845	4	no	—
58	C₂×C₅	—	169	1	no	—
59	C₃²	#16810011	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₃	—	169	4	no	—
61	C₅	C₃²×#10621729	1	1	yes	characteristic, complemented, Sylow
62	C₄	—	845	1	no	Sylow
63	C₃	C₃×#16810011	1	4	yes	direct product factor
64	C₂	—	169	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	#16810011
commutator, hypocenter	C₅×C₁₃²	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃²	C₄
prime radical	C₃²×C₅×C₁₃⋊D₁₃	C₂
cosocle	C₅×C₁₃⋊D₁₃	C₂×C₃²
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×#16810011	C₃	direct (N × H)
3. C₃²×C₅×C₁₃²	C₄	semidirect
6. #16810011	C₃²	direct (N × H)
8. C₃×C₅×C₁₃²	C₃×C₄	semidirect
12. C₃²×C₁₃²	Dic₅	semidirect
16. C₅×C₁₃²	C₃²×C₄	semidirect
20. C₃²×C₅×C₁₃	#261	semidirect
21. C₃×C₁₃²	C₃×Dic₅	semidirect
30. C₃×C₅×C₁₃	C₃×#261	semidirect
32. C₁₃²	C₃²×Dic₅	semidirect
37. C₃²×C₁₃	#63118	semidirect
42. C₅×C₁₃	C₃²×#261	semidirect
45. C₃²×C₅	#10621729	semidirect
47. C₃×C₁₃	C₃×#63118	semidirect
54. C₃×C₅	C₃×#10621729	semidirect
56. C₁₃	C₃²×#63118	semidirect
59. C₃²	#16810011	direct (N × H)
61. C₅	C₃²×#10621729	semidirect
63. C₃	C₃×#16810011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₁₃⋊D₁₃	2
C₃×#16810011	3
C₃²×#10621729	5
C₃²×#63118	13

Subgroups — C32×#16810011 #44478232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#16810011 #44478232