Subgroups — C₃⋊(C₃⋊#16810004) #44478309

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⋊#16810004	—	3	4	no	maximal
3	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊(C₃⋊#10621729)	—	5	1	no	maximal, Hall
5	C₃×C₅⋊(C₁₃⋊D₁₃)	D₃	1	4	yes	—
6	#16810004	—	9	1	no	Hall
7	C₃²×C₁₃⋊D₁₃	—	5	1	no	—
8	C₃×C₅×C₁₃²	Dic₃	1	4	yes	complemented
9	C₃⋊(C₃⋊#63121)	—	13	1	no	maximal
10	C₃⋊(C₃⋊#63122)	—	13	1	no	maximal
11	C₃⋊#10621729	—	15	4	no	—
12	C₅⋊(C₁₃⋊D₁₃)	C₃⋊D₃	1	1	yes	characteristic
13	C₃²×C₁₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
14	C₃²×D₆₅	—	13	1	no	—
15	C₃²×D₆₅	—	13	1	no	—
16	C₃²×D₆₅	—	26	6	no	—
17	C₃×C₁₃⋊D₁₃	—	5	4	no	—
18	C₅×C₁₃²	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₃⋊#63122	—	39	4	no	—
20	C₃⋊#63121	—	39	4	no	—
21	#10621729	—	45	1	no	Hall
22	C₃²×C₅×C₁₃	—	2	6	no	—
23	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, complemented
24	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, complemented
25	C₃×C₁₃²	ΓL(1,𝔽₁₆)	1	4	yes	complemented
26	C₃⋊(C₃⋊#261)	—	65	1	no	—
27	C₃⋊(C₃⋊#261)	—	65	1	no	—
28	C₃×D₆₅	—	13	4	no	—
29	C₃×D₆₅	—	13	4	no	—
30	C₃×D₆₅	—	26	24	no	—
31	C₁₃⋊D₁₃	—	5	1	no	—
32	#63122	—	117	1	no	—
33	#63121	—	117	1	no	—
34	C₃²×D₁₃	—	65	1	no	—
35	C₃²×D₁₃	—	65	1	no	—
36	C₃²×D₁₃	—	130	6	no	—
37	C₃×C₅×C₁₃	—	2	24	no	—
38	C₃×C₅×C₁₃	C₃⋊#261	1	4	yes	complemented
39	C₃×C₅×C₁₃	C₃⋊#261	1	4	yes	complemented
40	C₃⋊ΓL(1,𝔽₁₆)	—	169	1	no	Hall
41	C₁₃²	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
42	C₃⋊#261	—	195	4	no	—
43	C₃⋊#261	—	195	4	no	—
44	D₆₅	—	13	1	no	—
45	D₆₅	—	13	1	no	—
46	D₆₅	—	26	6	no	—
47	C₃²×C₁₃	—	2	6	no	—
48	C₃²×C₁₃	#63121	1	1	yes	characteristic, complemented
49	C₃²×C₁₃	#63122	1	1	yes	characteristic, complemented
50	C₃²×D₅	—	169	1	no	—
51	C₃×D₁₃	—	65	4	no	—
52	C₃×D₁₃	—	65	4	no	—
53	C₃×D₁₃	—	130	24	no	—
54	C₅×C₁₃	—	2	6	no	—
55	C₅×C₁₃	C₃⋊(C₃⋊#261)	1	1	yes	characteristic, complemented
56	C₅×C₁₃	C₃⋊(C₃⋊#261)	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₁₆)	—	507	4	no	—
58	#261	—	585	1	no	—
59	#261	—	585	1	no	—
60	C₃²×C₅	#10621729	1	1	yes	characteristic, complemented, Hall
61	C₃×C₁₃	—	2	24	no	—
62	C₃×C₁₃	C₃⋊#63121	1	4	yes	complemented
63	C₃×C₁₃	C₃⋊#63122	1	4	yes	complemented
64	C₃⋊Dic₃	—	845	1	no	Hall
65	C₃×D₅	—	169	4	no	—
66	D₁₃	—	65	1	no	—
67	D₁₃	—	65	1	no	—
68	D₁₃	—	130	6	no	—
69	AGL(1,𝔽₅)	—	1521	1	no	Hall
70	C₂×C₃²	—	845	1	no	—
71	C₃×C₅	C₃⋊#10621729	1	4	yes	complemented
72	C₁₃	—	2	6	no	—
73	C₁₃	C₃⋊(C₃⋊#63122)	1	1	yes	characteristic, complemented
74	C₁₃	C₃⋊(C₃⋊#63121)	1	1	yes	characteristic, complemented
75	Dic₃	—	2535	4	no	—
76	D₅	—	169	1	no	—
77	C₃²	#16810004	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	—	845	4	no	—
79	C₅	C₃⋊(C₃⋊#10621729)	1	1	yes	characteristic, complemented, Sylow
80	C₄	—	7605	1	no	Sylow
81	C₃	C₃⋊#16810004	1	4	yes	complemented
82	C₂	—	845	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₁₃²	C₄
prime radical, cosocle	C₃²×C₅⋊(C₁₃⋊D₁₃)	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
18. C₅×C₁₃²	C₃⋊Dic₃	semidirect
23. C₃²×C₅×C₁₃	#261	semidirect
24. C₃²×C₅×C₁₃	#261	semidirect
25. C₃×C₁₃²	ΓL(1,𝔽₁₆)	semidirect
38. C₃×C₅×C₁₃	C₃⋊#261	semidirect
39. C₃×C₅×C₁₃	C₃⋊#261	semidirect
41. C₁₃²	C₃⋊ΓL(1,𝔽₁₆)	semidirect
48. C₃²×C₁₃	#63121	semidirect
49. C₃²×C₁₃	#63122	semidirect
55. C₅×C₁₃	C₃⋊(C₃⋊#261)	semidirect
56. C₅×C₁₃	C₃⋊(C₃⋊#261)	semidirect
60. C₃²×C₅	#10621729	semidirect
62. C₃×C₁₃	C₃⋊#63121	semidirect
63. C₃×C₁₃	C₃⋊#63122	semidirect
71. C₃×C₅	C₃⋊#10621729	semidirect
73. C₁₃	C₃⋊(C₃⋊#63122)	semidirect
74. C₁₃	C₃⋊(C₃⋊#63121)	semidirect
77. C₃²	#16810004	semidirect
79. C₅	C₃⋊(C₃⋊#10621729)	semidirect
81. C₃	C₃⋊#16810004	semidirect

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⋊#16810004	3
C₃⋊(C₃⋊#10621729)	5
C₃⋊(C₃⋊#63121)	13
C₃⋊(C₃⋊#63122)	13

Subgroups — C3⋊(C3⋊#16810004) #44478309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊#16810004) #44478309