Subgroups — C₅×#10624771 #16814280

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	#10624771	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	2	no	—
7	C₃×C₅×3₋¹⁺²	—	3	1	no	—
8	C₃×C₅×3₋¹⁺²	—	3	3	no	—
9	C₃×C₅×3₋¹⁺²	—	3	9	no	—
10	C₃²×C₅×C₉	C₃²	1	3	yes	—
11	C₅×C₃²⋊C₉	—	3	6	no	—
12	C₅×C₃²⋊C₉	C₃²	1	9	yes	—
13	C₃²×3₋¹⁺²	C₃×C₅	1	1	yes	characteristic
14	#6891	C₃×C₅	1	3	yes	—
15	#6867	C₃×C₅	1	9	yes	complemented
16	C₃³×C₅	—	3	3	no	—
17	C₃³×C₅	—	9	3	no	—
18	C₃³×C₅	H(3,𝔽₃)	1	2	yes	—
19	C₃³×C₅	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₅	C₃³	1	1	yes	characteristic
21	C₅×3₋¹⁺²	—	3	18	no	—
22	C₅×3₋¹⁺²	—	3	9	no	—
23	C₅×3₋¹⁺²	—	9	9	no	—
24	C₃×C₅×C₉	—	3	3	no	—
25	C₃×C₅×C₉	—	3	6	no	—
26	C₃×C₅×C₉	—	3	9	no	—
27	C₃×C₅×C₉	3₋¹⁺²	1	9	yes	complemented
28	C₃⁴	C₃²×C₅	1	1	yes	characteristic
29	C₃×3₋¹⁺²	—	3	2	no	—
30	C₃×3₋¹⁺²	—	3	1	no	—
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	3	9	no	—
33	C₃²×C₉	C₃²×C₅	1	3	yes	—
34	C₃²⋊C₉	—	3	6	no	—
35	C₃²⋊C₉	C₃²×C₅	1	9	yes	—
36	C₃²×C₅	—	3	3	no	—
37	C₃²×C₅	—	3	2	no	—
38	C₃²×C₅	—	3	1	no	—
39	C₃²×C₅	—	3	3	no	—
40	C₃²×C₅	—	3	6	no	—
41	C₃²×C₅	—	9	9	no	—
42	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₅	C₃×3₋¹⁺²	1	3	yes	—
44	C₅×C₉	—	3	9	no	—
45	C₅×C₉	—	9	9	no	—
46	C₃³	—	3	3	no	—
47	C₃³	—	9	3	no	—
48	C₃³	C₅×H(3,𝔽₃)	1	2	yes	—
49	C₃³	C₅×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃³	C₃³×C₅	1	1	yes	characteristic
51	3₋¹⁺²	—	3	18	no	—
52	3₋¹⁺²	—	3	9	no	—
53	3₋¹⁺²	—	9	9	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	—	3	6	no	—
56	C₃×C₉	—	3	9	no	—
57	C₃×C₉	C₅×3₋¹⁺²	1	9	yes	complemented
58	C₃×C₅	—	3	3	no	—
59	C₃×C₅	—	9	1	no	—
60	C₃×C₅	—	9	2	no	—
61	C₃×C₅	#6891	1	1	yes	characteristic
62	C₃×C₅	#6910	1	3	yes	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	2	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	3	no	—
67	C₃²	—	3	6	no	—
68	C₃²	—	9	9	no	—
69	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃²	C₃×C₅×3₋¹⁺²	1	3	yes	—
71	C₉	—	3	9	no	—
72	C₉	—	9	9	no	—
73	C₅	#10624771	1	1	yes	characteristic, direct product factor, Sylow
74	C₃	—	3	3	no	—
75	C₃	—	9	1	no	—
76	C₃	—	9	2	no	—
77	C₃	C₅×#6891	1	1	yes	characteristic
78	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. #10624771	C₅	direct (N × H)
15. #6867	C₃×C₅	semidirect
27. C₃×C₅×C₉	3₋¹⁺²	semidirect
57. C₃×C₉	C₅×3₋¹⁺²	semidirect
73. C₅	#10624771	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
#10624771	5

Subgroups — C5×#10624771 #16814280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624771 #16814280