Subgroups — C₃²×#16810012 #44478228

Order 30420.

180 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×#16810012	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×#10621730	—	5	1	no	maximal, Hall
5	C₃×C₅×C₁₃×D₁₃	C₂×C₃	1	4	yes	—
6	#16810012	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	1	no	maximal
10	C₃²×Dic₆₅	—	13	1	no	maximal
11	C₃×#10621730	—	5	4	no	—
12	C₅×C₁₃×D₁₃	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₁₃²	Dic₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃²×C₅×C₁₃	—	13	1	no	—
15	C₃²×C₅×D₁₃	D₁₃	1	1	yes	characteristic
16	C₃×C₁₃×D₁₃	C₃×D₅	1	4	yes	—
17	C₅×C₁₃²	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
18	C₃×#63118	—	13	4	no	—
19	C₃×Dic₆₅	—	13	4	no	—
20	#10621730	—	5	1	no	Hall
21	C₃²×C₅×C₁₃	—	4	3	no	—
22	C₃²×C₅×C₁₃	Dic₁₃	1	1	yes	characteristic, complemented
23	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, complemented
24	C₃×C₁₃²	C₃×Dic₅	1	4	yes	complemented
25	C₃²×#261	—	65	1	no	—
26	C₃²×Dic₁₃	—	65	1	no	—
27	C₂×C₃×C₅×C₁₃	—	13	4	no	—
28	C₃×C₅×D₁₃	C₃×D₁₃	1	4	yes	—
29	C₁₃×D₁₃	C₃²×D₅	1	1	yes	characteristic
30	#63118	—	13	1	no	—
31	Dic₆₅	—	13	1	no	—
32	C₂×C₃²×C₁₃	—	13	1	no	—
33	C₃²×D₁₃	D₆₅	1	1	yes	characteristic
34	C₃×C₅×C₁₃	—	4	12	no	—
35	C₃×C₅×C₁₃	C₃×Dic₁₃	1	4	yes	complemented
36	C₃×C₅×C₁₃	C₃×#261	1	4	yes	complemented
37	C₃²×Dic₅	—	169	1	no	Hall
38	C₁₃²	C₃²×Dic₅	1	1	yes	characteristic, complemented, Sylow
39	C₃×#261	—	65	4	no	—
40	C₃×Dic₁₃	—	65	4	no	—
41	C₂×C₅×C₁₃	—	13	1	no	—
42	C₅×D₁₃	C₃²×D₁₃	1	1	yes	characteristic
43	C₃²×C₁₃	—	4	3	no	—
44	C₃²×C₁₃	Dic₆₅	1	1	yes	characteristic, complemented
45	C₃²×C₁₃	#63118	1	1	yes	characteristic, complemented
46	C₂×C₃²×C₅	—	13	1	no	—
47	C₂×C₃×C₁₃	—	13	4	no	—
48	C₃×D₁₃	C₃×D₆₅	1	4	yes	—
49	C₅×C₁₃	—	4	3	no	—
50	C₅×C₁₃	C₃²×Dic₁₃	1	1	yes	characteristic, complemented
51	C₅×C₁₃	C₃²×#261	1	1	yes	characteristic, complemented
52	C₃×Dic₅	—	169	4	no	—
53	#261	—	65	1	no	—
54	Dic₁₃	—	65	1	no	—
55	C₃²×C₅	#10621730	1	1	yes	characteristic, complemented, Hall
56	C₃×C₁₃	—	4	12	no	—
57	C₃×C₁₃	C₃×Dic₆₅	1	4	yes	complemented
58	C₃×C₁₃	C₃×#63118	1	4	yes	complemented
59	C₃²×C₄	—	845	1	no	Hall
60	C₂×C₃×C₅	—	13	4	no	—
61	C₂×C₁₃	—	13	1	no	—
62	D₁₃	C₃²×D₆₅	1	1	yes	characteristic
63	Dic₅	—	169	1	no	Hall
64	C₂×C₃²	—	13	1	no	—
65	C₃×C₅	C₃×#10621730	1	4	yes	complemented
66	C₁₃	—	4	3	no	—
67	C₁₃	C₃²×Dic₆₅	1	1	yes	characteristic, complemented
68	C₁₃	C₃²×#63118	1	1	yes	characteristic, complemented
69	C₃×C₄	—	845	4	no	—
70	C₂×C₅	—	13	1	no	—
71	C₃²	#16810012	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×C₃	—	13	4	no	—
73	C₅	C₃²×#10621730	1	1	yes	characteristic, complemented, Sylow
74	C₄	—	845	1	no	Sylow
75	C₃	C₃×#16810012	1	4	yes	direct product factor
76	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	#16810012
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×#16810012	C₃	direct (N × H)
3. C₃²×C₅×C₁₃²	C₄	semidirect
6. #16810012	C₃²	direct (N × H)
8. C₃×C₅×C₁₃²	C₃×C₄	semidirect
13. C₃²×C₁₃²	Dic₅	semidirect
17. C₅×C₁₃²	C₃²×C₄	semidirect
22. C₃²×C₅×C₁₃	Dic₁₃	semidirect
23. C₃²×C₅×C₁₃	#261	semidirect
24. C₃×C₁₃²	C₃×Dic₅	semidirect
35. C₃×C₅×C₁₃	C₃×Dic₁₃	semidirect
36. C₃×C₅×C₁₃	C₃×#261	semidirect
38. C₁₃²	C₃²×Dic₅	semidirect
44. C₃²×C₁₃	Dic₆₅	semidirect
45. C₃²×C₁₃	#63118	semidirect
50. C₅×C₁₃	C₃²×Dic₁₃	semidirect
51. C₅×C₁₃	C₃²×#261	semidirect
55. C₃²×C₅	#10621730	semidirect
57. C₃×C₁₃	C₃×Dic₆₅	semidirect
58. C₃×C₁₃	C₃×#63118	semidirect
65. C₃×C₅	C₃×#10621730	semidirect
67. C₁₃	C₃²×Dic₆₅	semidirect
68. C₁₃	C₃²×#63118	semidirect
71. C₃²	#16810012	direct (N × H)
73. C₅	C₃²×#10621730	semidirect
75. C₃	C₃×#16810012	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×#16810012	3
C₃²×#10621730	5
C₃²×#63118	13
C₃²×Dic₆₅	13

Subgroups — C32×#16810012 #44478228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#16810012 #44478228