Subgroups — C₃⋊(C₃⋊#16810006) #44478307

Order 30420.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₃×D₆₅	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#16810006	—	3	4	no	maximal
3	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊(C₃⋊#10621730)	—	5	1	no	maximal, Hall
5	C₃×C₁₃×D₆₅	D₃	1	4	yes	—
6	#16810006	—	9	1	no	Hall
7	C₃²×C₁₃×D₁₃	—	5	1	no	—
8	C₃×C₅×C₁₃²	Dic₃	1	4	yes	complemented
9	C₃⋊(C₃⋊#63120)	—	13	1	no	maximal
10	C₃⋊(C₃⋊#63121)	—	13	1	no	maximal
11	C₃⋊#10621730	—	15	4	no	—
12	C₁₃×D₆₅	C₃⋊D₃	1	1	yes	characteristic
13	C₃²×C₁₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
14	C₃²×D₆₅	D₁₃	1	1	yes	characteristic
15	C₃²×D₅×C₁₃	—	13	1	no	—
16	C₃×C₁₃×D₁₃	—	5	4	no	—
17	C₅×C₁₃²	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
18	C₃⋊#63120	—	39	4	no	—
19	C₃⋊#63121	—	39	4	no	—
20	#10621730	—	45	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	ΓL(1,𝔽₁₆)	1	4	yes	complemented
25	C₃⋊(C₃⋊#261)	—	65	1	no	—
26	C₃⋊Dic₃₉	—	65	1	no	—
27	C₃×D₆₅	D₃₉	1	4	yes	—
28	C₃×D₅×C₁₃	—	13	4	no	—
29	C₁₃×D₁₃	—	5	1	no	—
30	#63121	—	117	1	no	—
31	#63120	—	117	1	no	—
32	C₂×C₃²×C₁₃	—	65	1	no	—
33	C₃²×D₁₃	—	5	1	no	—
34	C₃×C₅×C₁₃	—	4	12	no	—
35	C₃×C₅×C₁₃	Dic₃₉	1	4	yes	complemented
36	C₃×C₅×C₁₃	C₃⋊#261	1	4	yes	complemented
37	C₃⋊ΓL(1,𝔽₁₆)	—	169	1	no	Hall
38	C₁₃²	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
39	C₃⋊#261	—	195	4	no	—
40	Dic₃₉	—	195	4	no	—
41	D₆₅	C₃⋊D₃₉	1	1	yes	characteristic
42	D₅×C₁₃	—	13	1	no	—
43	C₃²×C₁₃	—	4	3	no	—
44	C₃²×C₁₃	#63120	1	1	yes	characteristic, complemented
45	C₃²×C₁₃	#63121	1	1	yes	characteristic, complemented
46	C₃²×D₅	—	13	1	no	—
47	C₂×C₃×C₁₃	—	65	4	no	—
48	C₃×D₁₃	—	5	4	no	—
49	C₅×C₁₃	—	4	3	no	—
50	C₅×C₁₃	C₃⋊Dic₃₉	1	1	yes	characteristic, complemented
51	C₅×C₁₃	C₃⋊(C₃⋊#261)	1	1	yes	characteristic, complemented
52	ΓL(1,𝔽₁₆)	—	507	4	no	—
53	#261	—	585	1	no	—
54	Dic₁₃	—	585	1	no	—
55	C₃²×C₅	#10621730	1	1	yes	characteristic, complemented, Hall
56	C₃×C₁₃	—	4	12	no	—
57	C₃×C₁₃	C₃⋊#63121	1	4	yes	complemented
58	C₃×C₁₃	C₃⋊#63120	1	4	yes	complemented
59	C₃⋊Dic₃	—	845	1	no	Hall
60	C₃×D₅	—	13	4	no	—
61	C₂×C₁₃	—	65	1	no	—
62	D₁₃	—	5	1	no	—
63	AGL(1,𝔽₅)	—	1521	1	no	Hall
64	C₂×C₃²	—	65	1	no	—
65	C₃×C₅	C₃⋊#10621730	1	4	yes	complemented
66	C₁₃	—	4	3	no	—
67	C₁₃	C₃⋊(C₃⋊#63121)	1	1	yes	characteristic, complemented
68	C₁₃	C₃⋊(C₃⋊#63120)	1	1	yes	characteristic, complemented
69	Dic₃	—	2535	4	no	—
70	D₅	—	13	1	no	—
71	C₃²	#16810006	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃	—	65	4	no	—
73	C₅	C₃⋊(C₃⋊#10621730)	1	1	yes	characteristic, complemented, Sylow
74	C₄	—	7605	1	no	Sylow
75	C₃	C₃⋊#16810006	1	4	yes	complemented
76	C₂	—	65	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₁₃²	C₄
prime radical, cosocle	C₃²×C₁₃×D₆₅	C₂
LCS	G ▷ C₃²×C₅×C₁₃²
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₁₃²	C₄	semidirect
8. C₃×C₅×C₁₃²	Dic₃	semidirect
13. C₃²×C₁₃²	AGL(1,𝔽₅)	semidirect
17. C₅×C₁₃²	C₃⋊Dic₃	semidirect
22. C₃²×C₅×C₁₃	Dic₁₃	semidirect
23. C₃²×C₅×C₁₃	#261	semidirect
24. C₃×C₁₃²	ΓL(1,𝔽₁₆)	semidirect
35. C₃×C₅×C₁₃	Dic₃₉	semidirect
36. C₃×C₅×C₁₃	C₃⋊#261	semidirect
38. C₁₃²	C₃⋊ΓL(1,𝔽₁₆)	semidirect
44. C₃²×C₁₃	#63120	semidirect
45. C₃²×C₁₃	#63121	semidirect
50. C₅×C₁₃	C₃⋊Dic₃₉	semidirect
51. C₅×C₁₃	C₃⋊(C₃⋊#261)	semidirect
55. C₃²×C₅	#10621730	semidirect
57. C₃×C₁₃	C₃⋊#63121	semidirect
58. C₃×C₁₃	C₃⋊#63120	semidirect
65. C₃×C₅	C₃⋊#10621730	semidirect
67. C₁₃	C₃⋊(C₃⋊#63121)	semidirect
68. C₁₃	C₃⋊(C₃⋊#63120)	semidirect
71. C₃²	#16810006	semidirect
73. C₅	C₃⋊(C₃⋊#10621730)	semidirect
75. C₃	C₃⋊#16810006	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₁₃×D₆₅	2
C₃⋊#16810006	3
C₃⋊(C₃⋊#10621730)	5
C₃⋊(C₃⋊#63120)	13
C₃⋊(C₃⋊#63121)	13

Subgroups — C3⋊(C3⋊#16810006) #44478307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊#16810006) #44478307