Subgroups — C₅₃×#10624669 #50311554

Order 38637.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅₃×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₅₃×#6861	C₃	1	3	yes	maximal, complemented
4	C₅₃×#6859	C₃	1	3	yes	maximal, complemented
5	C₅₃×#6856	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×C₉⋊C₉	—	3	3	no	—
10	C₅₃×C₃²⋊C₉	—	3	6	no	—
11	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
12	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
13	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₃³×C₅₃	—	3	3	no	—
15	C₃³×C₅₃	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₅₃	—	9	9	no	—
17	H(3,𝔽₃)×C₅₃	—	9	3	no	—
18	C₃×C₉×C₅₃	—	3	3	no	—
19	C₃×C₉×C₅₃	—	3	3	no	—
20	C₃×C₉×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	—	9	1	no	—
22	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
23	#10624669	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃²×C₅₃	—	3	3	no	—
25	C₃²×C₅₃	—	3	1	no	—
26	C₃²×C₅₃	—	9	3	no	—
27	C₃²×C₅₃	—	9	6	no	—
28	C₃²×C₅₃	—	9	3	no	—
29	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₉×C₅₃	—	3	3	no	—
31	C₉×C₅₃	—	9	3	no	—
32	C₉×C₅₃	—	9	3	no	—
33	C₉×C₅₃	—	9	3	no	—
34	#6895	C₃×C₅₃	1	1	yes	characteristic, complemented
35	C₉⋊H(3,𝔽₃)	C₃×C₅₃	1	3	yes	complemented
36	#6861	C₃×C₅₃	1	3	yes	complemented
37	#6859	C₃×C₅₃	1	3	yes	complemented
38	#6856	C₃×C₅₃	1	3	yes	complemented
39	C₃×C₅₃	—	9	1	no	—
40	C₃×C₅₃	—	27	3	no	—
41	C₃×C₅₃	#6911	1	1	yes	characteristic
42	C₃×C₅₃	#6913	1	3	yes	—
43	C₃×3₋¹⁺²	—	3	3	no	—
44	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
45	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
46	C₉⋊C₉	—	3	3	no	—
47	C₃²⋊C₉	—	3	6	no	—
48	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
49	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
50	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
51	C₅₃	#10624669	1	1	yes	characteristic, direct product factor, Sylow
52	C₃³	—	3	3	no	—
53	C₃³	C₃³×C₅₃	1	1	yes	characteristic
54	3₋¹⁺²	—	9	9	no	—
55	H(3,𝔽₃)	—	9	3	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	—	9	1	no	—
60	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	9	3	no	—
64	C₃²	—	9	3	no	—
65	C₃²	—	9	6	no	—
66	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
67	C₉	—	3	3	no	—
68	C₉	—	9	3	no	—
69	C₉	—	9	3	no	—
70	C₉	—	9	3	no	—
71	C₃	—	9	1	no	—
72	C₃	—	27	3	no	—
73	C₃	C₅₃×#6911	1	1	yes	characteristic
74	C₃	C₅₃×#6913	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#6895	C₃	semidirect
2. C₅₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₅₃×#6861	C₃	semidirect
4. C₅₃×#6859	C₃	semidirect
5. C₅₃×#6856	C₃	semidirect
23. #10624669	C₅₃	direct (N × H)
34. #6895	C₃×C₅₃	semidirect
35. C₉⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
36. #6861	C₃×C₅₃	semidirect
37. #6859	C₃×C₅₃	semidirect
38. #6856	C₃×C₅₃	semidirect
51. C₅₃	#10624669	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#6895	3
C₅₃×C₉⋊H(3,𝔽₃)	3
C₅₃×#6861	3
C₅₃×#6859	3
C₅₃×#6856	3
#10624669	53

Subgroups — C53×#10624669 #50311554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624669 #50311554