Subgroups — C₃²×#16861522 #49147103

Order 36900.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅⋊D₂₀₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#16861522	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×#11717962	—	5	1	no	maximal
5	C₃²×#11717961	—	5	1	no	maximal
6	C₃×C₅⋊D₂₀₅	C₂×C₃	1	4	yes	—
7	#16861522	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×D₂₀₅	—	5	1	no	—
9	C₃²×D₂₀₅	—	5	1	no	—
10	C₃²×D₂₀₅	—	10	2	no	—
11	C₃×C₅²×C₄₁	C₃×C₄	1	4	yes	complemented
12	C₃×#11717962	—	5	4	no	—
13	C₃×#11717961	—	5	4	no	—
14	C₅⋊D₂₀₅	C₂×C₃²	1	1	yes	characteristic
15	C₃²×C₅×C₄₁	—	2	2	no	—
16	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
18	C₃²×#4223	—	25	1	no	Hall
19	C₃×D₂₀₅	—	5	4	no	—
20	C₃×D₂₀₅	—	5	4	no	—
21	C₃×D₂₀₅	—	10	8	no	—
22	C₅²×C₄₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
23	C₃²×#1044	—	41	1	no	maximal, Hall
24	#11717962	—	5	1	no	—
25	#11717961	—	5	1	no	—
26	C₃²×D₄₁	—	25	1	no	—
27	C₃×C₅×C₄₁	—	2	8	no	—
28	C₃×C₅×C₄₁	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
29	C₃×C₅×C₄₁	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
30	C₃×#4223	—	25	4	no	—
31	C₃²×C₅⋊D₅	—	41	1	no	—
32	D₂₀₅	—	5	1	no	—
33	D₂₀₅	—	5	1	no	—
34	D₂₀₅	—	10	2	no	—
35	C₃²×C₄₁	#1044	1	1	yes	characteristic, complemented, Hall
36	C₃×#1044	—	41	4	no	—
37	C₃×D₄₁	—	25	4	no	—
38	C₃²×C₅²	#4223	1	1	yes	characteristic, complemented, Hall
39	C₅×C₄₁	—	2	2	no	—
40	C₅×C₄₁	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
41	C₅×C₄₁	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	C₃²×AGL(1,𝔽₅)	—	205	1	no	—
43	C₃²×AGL(1,𝔽₅)	—	205	1	no	—
44	#4223	—	25	1	no	Hall
45	C₃×C₅⋊D₅	—	41	4	no	—
46	C₃×C₄₁	C₃×#1044	1	4	yes	complemented
47	#1044	—	41	1	no	Hall
48	C₃²×D₅	—	410	2	no	—
49	C₃²×D₅	—	205	1	no	—
50	C₃²×D₅	—	205	1	no	—
51	D₄₁	—	25	1	no	—
52	C₃×C₅²	C₃×#4223	1	4	yes	complemented
53	C₃×AGL(1,𝔽₅)	—	205	4	no	—
54	C₃×AGL(1,𝔽₅)	—	205	4	no	—
55	C₅⋊D₅	—	41	1	no	—
56	C₃²×C₅	—	2	2	no	—
57	C₃²×C₅	#11717961	1	1	yes	characteristic, complemented
58	C₃²×C₅	#11717962	1	1	yes	characteristic, complemented
59	C₄₁	C₃²×#1044	1	1	yes	characteristic, complemented, Sylow
60	C₃²×C₄	—	1025	1	no	Hall
61	C₃×D₅	—	410	8	no	—
62	C₃×D₅	—	205	4	no	—
63	C₃×D₅	—	205	4	no	—
64	C₅²	C₃²×#4223	1	1	yes	characteristic, complemented, Sylow
65	AGL(1,𝔽₅)	—	205	1	no	—
66	AGL(1,𝔽₅)	—	205	1	no	—
67	C₂×C₃²	—	1025	1	no	—
68	C₃×C₅	—	2	8	no	—
69	C₃×C₅	C₃×#11717961	1	4	yes	complemented
70	C₃×C₅	C₃×#11717962	1	4	yes	complemented
71	C₃×C₄	—	1025	4	no	—
72	D₅	—	410	2	no	—
73	D₅	—	205	1	no	—
74	D₅	—	205	1	no	—
75	C₃²	#16861522	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₃	—	1025	4	no	—
77	C₅	—	2	2	no	—
78	C₅	C₃²×#11717961	1	1	yes	characteristic, complemented
79	C₅	C₃²×#11717962	1	1	yes	characteristic, complemented
80	C₄	—	1025	1	no	Sylow
81	C₃	C₃×#16861522	1	4	yes	direct product factor
82	C₂	—	1025	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	#16861522
commutator, hypocenter	C₅²×C₄₁	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₄₁	C₄
prime radical	C₃²×C₅⋊D₂₀₅	C₂
cosocle	C₅⋊D₂₀₅	C₂×C₃²
LCS	G ▷ C₅²×C₄₁
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×#16861522	C₃	direct (N × H)
3. C₃²×C₅²×C₄₁	C₄	semidirect
7. #16861522	C₃²	direct (N × H)
11. C₃×C₅²×C₄₁	C₃×C₄	semidirect
16. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
17. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
22. C₅²×C₄₁	C₃²×C₄	semidirect
28. C₃×C₅×C₄₁	C₃×AGL(1,𝔽₅)	semidirect
29. C₃×C₅×C₄₁	C₃×AGL(1,𝔽₅)	semidirect
35. C₃²×C₄₁	#1044	semidirect
38. C₃²×C₅²	#4223	semidirect
40. C₅×C₄₁	C₃²×AGL(1,𝔽₅)	semidirect
41. C₅×C₄₁	C₃²×AGL(1,𝔽₅)	semidirect
46. C₃×C₄₁	C₃×#1044	semidirect
52. C₃×C₅²	C₃×#4223	semidirect
57. C₃²×C₅	#11717961	semidirect
58. C₃²×C₅	#11717962	semidirect
59. C₄₁	C₃²×#1044	semidirect
64. C₅²	C₃²×#4223	semidirect
69. C₃×C₅	C₃×#11717961	semidirect
70. C₃×C₅	C₃×#11717962	semidirect
75. C₃²	#16861522	direct (N × H)
78. C₅	C₃²×#11717961	semidirect
79. C₅	C₃²×#11717962	semidirect
81. C₃	C₃×#16861522	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⋊D₂₀₅	2
C₃×#16861522	3
C₃²×#11717962	5
C₃²×#11717961	5
C₃²×#1044	41

Subgroups — C32×#16861522 #49147103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#16861522 #49147103