Subgroups — C₅×#10624761 #16814270

Order 3645.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅×#6910	C₃	1	8	yes	maximal, complemented
3	C₃×C₅×AΣL(1,𝔽₂₇)	C₃	1	4	yes	maximal, complemented
4	#10624761	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃⁴×C₅	C₃²	1	1	yes	characteristic, complemented
6	C₃×C₅×3₋¹⁺²	C₃²	1	8	yes	complemented
7	C₃×C₅×H(3,𝔽₃)	—	3	12	no	—
8	C₃×C₅×H(3,𝔽₃)	C₃²	1	4	yes	—
9	C₅×#731	—	3	24	no	—
10	C₅×AΣL(1,𝔽₂₇)	—	3	12	no	—
11	H(5,𝔽₃)	C₃×C₅	1	1	yes	characteristic, complemented
12	#6910	C₃×C₅	1	8	yes	complemented
13	C₃×AΣL(1,𝔽₂₇)	C₃×C₅	1	4	yes	complemented
14	C₃³×C₅	—	3	9	no	—
15	C₃³×C₅	—	3	4	no	—
16	C₃³×C₅	—	3	4	no	—
17	C₃³×C₅	—	9	3	no	—
18	C₃³×C₅	C₃³	1	1	yes	characteristic
19	C₅×3₋¹⁺²	—	3	8	no	—
20	C₅×3₋¹⁺²	—	3	16	no	—
21	C₅×H(3,𝔽₃)	—	3	18	no	—
22	C₅×H(3,𝔽₃)	—	3	8	no	—
23	C₅×H(3,𝔽₃)	H(3,𝔽₃)	1	12	yes	—
24	C₃×C₅×C₉	—	3	8	no	—
25	C₃⁴	C₃²×C₅	1	1	yes	characteristic, complemented
26	C₃×3₋¹⁺²	C₃²×C₅	1	8	yes	complemented
27	C₃×H(3,𝔽₃)	—	3	12	no	—
28	C₃×H(3,𝔽₃)	C₃²×C₅	1	4	yes	—
29	#731	—	3	24	no	—
30	AΣL(1,𝔽₂₇)	—	3	12	no	—
31	C₃²×C₅	—	3	12	no	—
32	C₃²×C₅	—	9	1	no	—
33	C₃²×C₅	—	9	12	no	—
34	C₃²×C₅	—	9	1	no	—
35	C₃²×C₅	—	27	4	no	—
36	C₃²×C₅	—	27	9	no	—
37	C₃²×C₅	C₃×H(3,𝔽₃)	1	4	yes	—
38	C₅×C₉	—	9	8	no	—
39	C₃³	—	3	9	no	—
40	C₃³	—	3	4	no	—
41	C₃³	—	3	4	no	—
42	C₃³	—	9	3	no	—
43	C₃³	C₃³×C₅	1	1	yes	characteristic
44	3₋¹⁺²	—	3	8	no	—
45	3₋¹⁺²	—	3	16	no	—
46	H(3,𝔽₃)	—	3	18	no	—
47	H(3,𝔽₃)	—	3	8	no	—
48	H(3,𝔽₃)	C₅×H(3,𝔽₃)	1	12	yes	—
49	C₃×C₉	—	3	8	no	—
50	C₃×C₅	—	3	4	no	—
51	C₃×C₅	—	9	12	no	—
52	C₃×C₅	—	9	3	no	—
53	C₃×C₅	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²	—	3	12	no	—
55	C₃²	—	9	1	no	—
56	C₃²	—	9	12	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	27	4	no	—
59	C₃²	—	27	9	no	—
60	C₃²	C₃×C₅×H(3,𝔽₃)	1	4	yes	—
61	C₉	—	9	8	no	—
62	C₅	#10624761	1	1	yes	characteristic, direct product factor, Sylow
63	C₃	—	3	4	no	—
64	C₃	—	9	12	no	—
65	C₃	—	9	3	no	—
66	C₃	C₅×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅×H(5,𝔽₃)	C₃	semidirect
2. C₅×#6910	C₃	semidirect
3. C₃×C₅×AΣL(1,𝔽₂₇)	C₃	semidirect
4. #10624761	C₅	direct (N × H)
5. C₃⁴×C₅	C₃²	semidirect
6. C₃×C₅×3₋¹⁺²	C₃²	semidirect
11. H(5,𝔽₃)	C₃×C₅	semidirect
12. #6910	C₃×C₅	semidirect
13. C₃×AΣL(1,𝔽₂₇)	C₃×C₅	semidirect
25. C₃⁴	C₃²×C₅	semidirect
26. C₃×3₋¹⁺²	C₃²×C₅	semidirect
62. C₅	#10624761	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×H(5,𝔽₃)	3
C₅×#6910	3
C₃×C₅×AΣL(1,𝔽₂₇)	3
#10624761	5

Subgroups — C5×#10624761 #16814270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624761 #16814270