Subgroups — C₃×C₅₃×#6891 #50311751

Order 38637.

2636 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×3₋¹⁺²×C₅₃	C₃	1	3	yes	maximal, complemented
3	C₅₃×#6891	C₃	1	27	yes	maximal, direct product factor
4	C₃×C₅₃×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₅₃	—	3	9	no	—
6	C₃⁴×C₅₃	—	3	27	no	—
7	C₃⁴×C₅₃	C₃²	1	9	yes	—
8	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
9	C₃⁴×C₅₃	C₃²	1	3	yes	—
10	C₃×3₋¹⁺²×C₅₃	—	3	27	no	—
11	C₃×3₋¹⁺²×C₅₃	C₃²	1	27	yes	complemented
12	C₃²×C₉×C₅₃	C₃²	1	9	yes	complemented
13	C₅₃×C₃²⋊C₉	C₃²	1	81	yes	complemented
14	C₃³×C₅₃	—	3	243	no	—
15	C₃³×C₅₃	—	3	27	no	—
16	C₃³×C₅₃	—	3	9	no	—
17	C₃³×C₅₃	—	3	9	no	—
18	C₃³×C₅₃	—	3	18	no	—
19	C₃³×C₅₃	—	3	81	no	—
20	C₃³×C₅₃	H(3,𝔽₃)	1	9	yes	—
21	C₃³×C₅₃	3₋¹⁺²	1	27	yes	complemented
22	C₃³×C₅₃	C₃³	1	3	yes	—
23	C₃³×C₅₃	C₃³	1	9	yes	—
24	C₃³×C₅₃	C₃³	1	1	yes	characteristic
25	3₋¹⁺²×C₅₃	—	3	81	no	—
26	C₃×C₉×C₅₃	—	3	27	no	—
27	C₃×C₉×C₅₃	C₃³	1	27	yes	complemented
28	C₃×#6891	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	3	243	no	—
30	C₃²×C₅₃	—	3	9	no	—
31	C₃²×C₅₃	—	3	9	no	—
32	C₃²×C₅₃	—	3	18	no	—
33	C₃²×C₅₃	—	3	27	no	—
34	C₃²×C₅₃	—	3	81	no	—
35	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₅₃	C₃×3₋¹⁺²	1	9	yes	—
38	C₃²×C₅₃	C₃×3₋¹⁺²	1	27	yes	complemented
39	C₃²×C₅₃	C₃⁴	1	1	yes	characteristic
40	C₉×C₅₃	—	3	27	no	—
41	C₃⁵	C₃×C₅₃	1	1	yes	characteristic
42	C₃²×3₋¹⁺²	C₃×C₅₃	1	3	yes	complemented
43	#6891	C₃×C₅₃	1	27	yes	direct product factor
44	C₃×C₃²⋊C₉	C₃×C₅₃	1	9	yes	complemented
45	C₃×C₅₃	—	3	9	no	—
46	C₃×C₅₃	—	3	27	no	—
47	C₃×C₅₃	#6891	1	9	yes	direct product factor
48	C₃×C₅₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₅₃	C₃²×3₋¹⁺²	1	3	yes	—
50	C₃⁴	—	3	9	no	—
51	C₃⁴	—	3	27	no	—
52	C₃⁴	C₃²×C₅₃	1	9	yes	—
53	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
54	C₃⁴	C₃²×C₅₃	1	3	yes	—
55	C₃×3₋¹⁺²	—	3	27	no	—
56	C₃×3₋¹⁺²	C₃²×C₅₃	1	27	yes	complemented
57	C₃²×C₉	C₃²×C₅₃	1	9	yes	complemented
58	C₃²⋊C₉	C₃²×C₅₃	1	81	yes	complemented
59	C₅₃	C₃×#6891	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	—	3	243	no	—
61	C₃³	—	3	27	no	—
62	C₃³	—	3	9	no	—
63	C₃³	—	3	9	no	—
64	C₃³	—	3	18	no	—
65	C₃³	—	3	81	no	—
66	C₃³	H(3,𝔽₃)×C₅₃	1	9	yes	—
67	C₃³	3₋¹⁺²×C₅₃	1	27	yes	complemented
68	C₃³	C₃³×C₅₃	1	3	yes	—
69	C₃³	C₃³×C₅₃	1	9	yes	—
70	C₃³	C₃³×C₅₃	1	1	yes	characteristic
71	3₋¹⁺²	—	3	81	no	—
72	C₃×C₉	—	3	27	no	—
73	C₃×C₉	C₃³×C₅₃	1	27	yes	complemented
74	C₃²	—	3	243	no	—
75	C₃²	—	3	27	no	—
76	C₃²	—	3	81	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	3	18	no	—
80	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	9	yes	—
81	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	3	yes	—
82	C₃²	C₃×3₋¹⁺²×C₅₃	1	9	yes	—
83	C₃²	C₃×3₋¹⁺²×C₅₃	1	27	yes	complemented
84	C₃²	C₃⁴×C₅₃	1	1	yes	characteristic
85	C₉	—	3	27	no	—
86	C₃	—	3	9	no	—
87	C₃	—	3	27	no	—
88	C₃	C₅₃×#6891	1	9	yes	direct product factor
89	C₃	C₃²×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
90	C₃	C₃²×3₋¹⁺²×C₅₃	1	3	yes	—

	subgroup	G/H
center, socle	C₃³×C₅₃	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₅₃
prime radical	C₃⁵×C₅₃	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×3₋¹⁺²×C₅₃	C₃	semidirect
3. C₅₃×#6891	C₃	direct (N × H)
4. C₃×C₅₃×C₃²⋊C₉	C₃	semidirect
11. C₃×3₋¹⁺²×C₅₃	C₃²	semidirect
12. C₃²×C₉×C₅₃	C₃²	semidirect
13. C₅₃×C₃²⋊C₉	C₃²	semidirect
21. C₃³×C₅₃	3₋¹⁺²	semidirect
27. C₃×C₉×C₅₃	C₃³	semidirect
28. C₃×#6891	C₅₃	direct (N × H)
38. C₃²×C₅₃	C₃×3₋¹⁺²	semidirect
42. C₃²×3₋¹⁺²	C₃×C₅₃	semidirect
43. #6891	C₃×C₅₃	direct (N × H)
44. C₃×C₃²⋊C₉	C₃×C₅₃	semidirect
47. C₃×C₅₃	#6891	direct (N × H)
56. C₃×3₋¹⁺²	C₃²×C₅₃	semidirect
57. C₃²×C₉	C₃²×C₅₃	semidirect
58. C₃²⋊C₉	C₃²×C₅₃	semidirect
59. C₅₃	C₃×#6891	direct (N × H)
67. C₃³	3₋¹⁺²×C₅₃	semidirect
73. C₃×C₉	C₃³×C₅₃	semidirect
83. C₃²	C₃×3₋¹⁺²×C₅₃	semidirect
88. C₃	C₅₃×#6891	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁵×C₅₃	3
C₃²×3₋¹⁺²×C₅₃	3
C₅₃×#6891	3
C₃×C₅₃×C₃²⋊C₉	3
C₃×#6891	53

Subgroups — C3×C53×#6891 #50311751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×#6891 #50311751