Subgroups — C₅×#10624776 #16814285

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×#6868	C₃	1	9	yes	maximal, complemented
4	#10624776	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	#6868	C₃×C₅	1	9	yes	complemented
15	C₃³×C₅	—	3	3	no	—
16	C₃³×C₅	—	9	2	no	—
17	C₃³×C₅	—	9	1	no	—
18	C₃³×C₅	H(3,𝔽₃)	1	3	yes	—
19	C₃³×C₅	C₃³	1	1	yes	characteristic
20	C₅×3₋¹⁺²	—	3	27	no	—
21	C₅×3₋¹⁺²	—	9	9	no	—
22	C₃×C₅×C₉	—	3	3	no	—
23	C₃×C₅×C₉	—	3	6	no	—
24	C₃×C₅×C₉	—	3	9	no	—
25	C₃×C₅×C₉	3₋¹⁺²	1	9	yes	complemented
26	C₃⁴	C₃²×C₅	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	3	no	—
28	C₃×3₋¹⁺²	—	3	3	no	—
29	C₃×3₋¹⁺²	—	3	9	no	—
30	C₃²×C₉	C₃²×C₅	1	3	yes	—
31	C₃²⋊C₉	—	3	6	no	—
32	C₃²⋊C₉	C₃²×C₅	1	9	yes	—
33	C₃²×C₅	—	3	2	no	—
34	C₃²×C₅	—	3	1	no	—
35	C₃²×C₅	—	3	6	no	—
36	C₃²×C₅	—	3	3	no	—
37	C₃²×C₅	—	3	3	no	—
38	C₃²×C₅	—	9	9	no	—
39	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃²×C₅	C₃×3₋¹⁺²	1	3	yes	—
41	C₅×C₉	—	3	9	no	—
42	C₅×C₉	—	9	9	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	9	1	no	—
45	C₃³	—	9	2	no	—
46	C₃³	C₅×H(3,𝔽₃)	1	3	yes	—
47	C₃³	C₃³×C₅	1	1	yes	characteristic
48	3₋¹⁺²	—	3	27	no	—
49	3₋¹⁺²	—	9	9	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	3	6	no	—
52	C₃×C₉	—	3	9	no	—
53	C₃×C₉	C₅×3₋¹⁺²	1	9	yes	complemented
54	C₃×C₅	—	3	3	no	—
55	C₃×C₅	—	9	3	no	—
56	C₃×C₅	#6891	1	1	yes	characteristic
57	C₃×C₅	#6911	1	1	yes	characteristic
58	C₃×C₅	#6912	1	2	yes	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	2	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	6	no	—
64	C₃²	—	9	9	no	—
65	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²	C₃×C₅×3₋¹⁺²	1	3	yes	—
67	C₉	—	3	9	no	—
68	C₉	—	9	9	no	—
69	C₅	#10624776	1	1	yes	characteristic, direct product factor, Sylow
70	C₃	—	3	3	no	—
71	C₃	—	9	3	no	—
72	C₃	C₅×#6891	1	1	yes	characteristic
73	C₃	C₅×#6911	1	1	yes	characteristic
74	C₃	C₅×#6912	1	2	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×#6868	C₃	semidirect
4. #10624776	C₅	direct (N × H)
14. #6868	C₃×C₅	semidirect
25. C₃×C₅×C₉	3₋¹⁺²	semidirect
53. C₃×C₉	C₅×3₋¹⁺²	semidirect
69. C₅	#10624776	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×#6868	3
#10624776	5

Subgroups — C5×#10624776 #16814285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624776 #16814285