Subgroups — C₅₃×#10624775 #50311660

Order 38637.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6891	C₃	1	3	yes	maximal
3	C₅₃×#6867	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×3₋¹⁺²×C₅₃	—	3	9	no	—
8	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
9	C₅₃×C₃²⋊C₉	—	3	6	no	—
10	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
11	C₃³×C₅₃	—	3	3	no	—
12	C₃³×C₅₃	—	9	3	no	—
13	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
14	C₃³×C₅₃	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₅₃	—	3	27	no	—
16	3₋¹⁺²×C₅₃	—	9	9	no	—
17	C₃×C₉×C₅₃	—	3	3	no	—
18	C₃×C₉×C₅₃	—	3	6	no	—
19	C₃×C₉×C₅₃	—	3	9	no	—
20	C₃×C₉×C₅₃	3₋¹⁺²	1	9	yes	complemented
21	#10624775	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₅₃	—	3	3	no	—
23	C₃²×C₅₃	—	3	3	no	—
24	C₃²×C₅₃	—	3	3	no	—
25	C₃²×C₅₃	—	3	6	no	—
26	C₃²×C₅₃	—	9	9	no	—
27	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
29	C₉×C₅₃	—	3	9	no	—
30	C₉×C₅₃	—	9	9	no	—
31	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
32	#6891	C₃×C₅₃	1	3	yes	—
33	#6867	C₃×C₅₃	1	9	yes	complemented
34	C₃×C₅₃	—	3	3	no	—
35	C₃×C₅₃	—	9	3	no	—
36	C₃×C₅₃	#6891	1	1	yes	characteristic
37	C₃×C₅₃	#6910	1	3	yes	—
38	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	3	no	—
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	—	3	9	no	—
42	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
43	C₃²⋊C₉	—	3	6	no	—
44	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
45	C₅₃	#10624775	1	1	yes	characteristic, direct product factor, Sylow
46	C₃³	—	3	3	no	—
47	C₃³	—	9	3	no	—
48	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
49	C₃³	C₃³×C₅₃	1	1	yes	characteristic
50	3₋¹⁺²	—	3	27	no	—
51	3₋¹⁺²	—	9	9	no	—
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	—	3	6	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	3₋¹⁺²×C₅₃	1	9	yes	complemented
56	C₃²	—	3	3	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	3	no	—
59	C₃²	—	3	6	no	—
60	C₃²	—	9	9	no	—
61	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
62	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
63	C₉	—	3	9	no	—
64	C₉	—	9	9	no	—
65	C₃	—	3	3	no	—
66	C₃	—	9	3	no	—
67	C₃	C₅₃×#6891	1	1	yes	characteristic
68	C₃	C₅₃×#6910	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#6867	C₃	semidirect
20. C₃×C₉×C₅₃	3₋¹⁺²	semidirect
21. #10624775	C₅₃	direct (N × H)
33. #6867	C₃×C₅₃	semidirect
45. C₅₃	#10624775	direct (N × H)
55. C₃×C₉	3₋¹⁺²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₅₃	3
C₅₃×#6891	3
C₅₃×#6867	3
#10624775	53

Subgroups — C53×#10624775 #50311660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624775 #50311660