Subgroups — C₅₃×#10624771 #50311656

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	2	no	—
6	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
7	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
8	C₃×3₋¹⁺²×C₅₃	—	3	9	no	—
9	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
10	C₅₃×C₃²⋊C₉	—	3	6	no	—
11	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃³×C₅₃	—	3	3	no	—
13	C₃³×C₅₃	—	9	3	no	—
14	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
15	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₅₃	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₅₃	—	3	18	no	—
18	3₋¹⁺²×C₅₃	—	3	9	no	—
19	3₋¹⁺²×C₅₃	—	9	9	no	—
20	C₃×C₉×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	—	3	6	no	—
22	C₃×C₉×C₅₃	—	3	9	no	—
23	C₃×C₉×C₅₃	3₋¹⁺²	1	9	yes	complemented
24	#10624771	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₅₃	—	3	3	no	—
26	C₃²×C₅₃	—	3	2	no	—
27	C₃²×C₅₃	—	3	1	no	—
28	C₃²×C₅₃	—	3	3	no	—
29	C₃²×C₅₃	—	3	6	no	—
30	C₃²×C₅₃	—	9	9	no	—
31	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
33	C₉×C₅₃	—	3	9	no	—
34	C₉×C₅₃	—	9	9	no	—
35	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
36	#6891	C₃×C₅₃	1	3	yes	—
37	#6867	C₃×C₅₃	1	9	yes	complemented
38	C₃×C₅₃	—	3	3	no	—
39	C₃×C₅₃	—	9	1	no	—
40	C₃×C₅₃	—	9	2	no	—
41	C₃×C₅₃	#6891	1	1	yes	characteristic
42	C₃×C₅₃	#6910	1	3	yes	—
43	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃×3₋¹⁺²	—	3	9	no	—
48	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
49	C₃²⋊C₉	—	3	6	no	—
50	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
51	C₅₃	#10624771	1	1	yes	characteristic, direct product factor, Sylow
52	C₃³	—	3	3	no	—
53	C₃³	—	9	3	no	—
54	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
55	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
56	C₃³	C₃³×C₅₃	1	1	yes	characteristic
57	3₋¹⁺²	—	3	18	no	—
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	9	9	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	—	3	9	no	—
63	C₃×C₉	3₋¹⁺²×C₅₃	1	9	yes	complemented
64	C₃²	—	3	3	no	—
65	C₃²	—	3	2	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	9	9	no	—
70	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
71	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
72	C₉	—	3	9	no	—
73	C₉	—	9	9	no	—
74	C₃	—	3	3	no	—
75	C₃	—	9	1	no	—
76	C₃	—	9	2	no	—
77	C₃	C₅₃×#6891	1	1	yes	characteristic
78	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
23. C₃×C₉×C₅₃	3₋¹⁺²	semidirect
24. #10624771	C₅₃	direct (N × H)
37. #6867	C₃×C₅₃	semidirect
51. C₅₃	#10624771	direct (N × H)
63. 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
#10624771	53

Subgroups — C53×#10624771 #50311656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624771 #50311656