Subgroups — C₅×#10624775 #16814284

Order 3645.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₅×#6891	C₃	1	3	yes	maximal
3	C₅×#6867	C₃	1	9	yes	maximal, complemented
4	#10624775	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃⁴×C₅	C₃²	1	1	yes	characteristic
6	C₃×C₅×3₋¹⁺²	—	3	3	no	—
7	C₃×C₅×3₋¹⁺²	—	3	3	no	—
8	C₃×C₅×3₋¹⁺²	—	3	9	no	—
9	C₃²×C₅×C₉	C₃²	1	3	yes	—
10	C₅×C₃²⋊C₉	—	3	6	no	—
11	C₅×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃²×3₋¹⁺²	C₃×C₅	1	1	yes	characteristic
13	#6891	C₃×C₅	1	3	yes	—
14	#6867	C₃×C₅	1	9	yes	complemented
15	C₃³×C₅	—	3	3	no	—
16	C₃³×C₅	—	9	3	no	—
17	C₃³×C₅	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₅	C₃³	1	1	yes	characteristic
19	C₅×3₋¹⁺²	—	3	27	no	—
20	C₅×3₋¹⁺²	—	9	9	no	—
21	C₃×C₅×C₉	—	3	3	no	—
22	C₃×C₅×C₉	—	3	6	no	—
23	C₃×C₅×C₉	—	3	9	no	—
24	C₃×C₅×C₉	3₋¹⁺²	1	9	yes	complemented
25	C₃⁴	C₃²×C₅	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	3	no	—
27	C₃×3₋¹⁺²	—	3	3	no	—
28	C₃×3₋¹⁺²	—	3	9	no	—
29	C₃²×C₉	C₃²×C₅	1	3	yes	—
30	C₃²⋊C₉	—	3	6	no	—
31	C₃²⋊C₉	C₃²×C₅	1	9	yes	—
32	C₃²×C₅	—	3	3	no	—
33	C₃²×C₅	—	3	3	no	—
34	C₃²×C₅	—	3	3	no	—
35	C₃²×C₅	—	3	6	no	—
36	C₃²×C₅	—	9	9	no	—
37	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₅	C₃×3₋¹⁺²	1	3	yes	—
39	C₅×C₉	—	3	9	no	—
40	C₅×C₉	—	9	9	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	9	3	no	—
43	C₃³	C₅×H(3,𝔽₃)	1	3	yes	—
44	C₃³	C₃³×C₅	1	1	yes	characteristic
45	3₋¹⁺²	—	3	27	no	—
46	3₋¹⁺²	—	9	9	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	6	no	—
49	C₃×C₉	—	3	9	no	—
50	C₃×C₉	C₅×3₋¹⁺²	1	9	yes	complemented
51	C₃×C₅	—	3	3	no	—
52	C₃×C₅	—	9	3	no	—
53	C₃×C₅	#6891	1	1	yes	characteristic
54	C₃×C₅	#6910	1	3	yes	—
55	C₃²	—	3	3	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	6	no	—
59	C₃²	—	9	9	no	—
60	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²	C₃×C₅×3₋¹⁺²	1	3	yes	—
62	C₉	—	3	9	no	—
63	C₉	—	9	9	no	—
64	C₅	#10624775	1	1	yes	characteristic, direct product factor, Sylow
65	C₃	—	3	3	no	—
66	C₃	—	9	3	no	—
67	C₃	C₅×#6891	1	1	yes	characteristic
68	C₃	C₅×#6910	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅
prime radical	C₃⁴×C₅	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅ ◁ C₃⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#6867	C₃	semidirect
4. #10624775	C₅	direct (N × H)
14. #6867	C₃×C₅	semidirect
24. C₃×C₅×C₉	3₋¹⁺²	semidirect
50. C₃×C₉	C₅×3₋¹⁺²	semidirect
64. C₅	#10624775	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×C₅×3₋¹⁺²	3
C₅×#6891	3
C₅×#6867	3
#10624775	5

Subgroups — C5×#10624775 #16814284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624775 #16814284