Subgroups — C₅₃×#10624772 #50311657

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×#6868	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	2	no	—
13	C₃³×C₅₃	—	9	1	no	—
14	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
15	C₃³×C₅₃	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₅₃	—	3	27	no	—
17	3₋¹⁺²×C₅₃	—	9	9	no	—
18	C₃×C₉×C₅₃	—	3	3	no	—
19	C₃×C₉×C₅₃	—	3	6	no	—
20	C₃×C₉×C₅₃	—	3	9	no	—
21	C₃×C₉×C₅₃	3₋¹⁺²	1	9	yes	complemented
22	#10624772	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₅₃	—	3	2	no	—
24	C₃²×C₅₃	—	3	1	no	—
25	C₃²×C₅₃	—	3	6	no	—
26	C₃²×C₅₃	—	3	3	no	—
27	C₃²×C₅₃	—	3	3	no	—
28	C₃²×C₅₃	—	9	9	no	—
29	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
31	C₉×C₅₃	—	3	9	no	—
32	C₉×C₅₃	—	9	9	no	—
33	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
34	#6891	C₃×C₅₃	1	3	yes	—
35	#6868	C₃×C₅₃	1	9	yes	complemented
36	C₃×C₅₃	—	3	3	no	—
37	C₃×C₅₃	—	9	3	no	—
38	C₃×C₅₃	#6891	1	1	yes	characteristic
39	C₃×C₅₃	#6911	1	1	yes	characteristic
40	C₃×C₅₃	#6912	1	2	yes	—
41	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	3	no	—
43	C₃×3₋¹⁺²	—	3	3	no	—
44	C₃×3₋¹⁺²	—	3	9	no	—
45	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
46	C₃²⋊C₉	—	3	6	no	—
47	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
48	C₅₃	#10624772	1	1	yes	characteristic, direct product factor, Sylow
49	C₃³	—	3	3	no	—
50	C₃³	—	9	1	no	—
51	C₃³	—	9	2	no	—
52	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
53	C₃³	C₃³×C₅₃	1	1	yes	characteristic
54	3₋¹⁺²	—	3	27	no	—
55	3₋¹⁺²	—	9	9	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	9	no	—
59	C₃×C₉	3₋¹⁺²×C₅₃	1	9	yes	complemented
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	9	9	no	—
66	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
67	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
68	C₉	—	3	9	no	—
69	C₉	—	9	9	no	—
70	C₃	—	3	3	no	—
71	C₃	—	9	3	no	—
72	C₃	C₅₃×#6891	1	1	yes	characteristic
73	C₃	C₅₃×#6911	1	1	yes	characteristic
74	C₃	C₅₃×#6912	1	2	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×#6868	C₃	semidirect
21. C₃×C₉×C₅₃	3₋¹⁺²	semidirect
22. #10624772	C₅₃	direct (N × H)
35. #6868	C₃×C₅₃	semidirect
48. C₅₃	#10624772	direct (N × H)
59. 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×#6868	3
#10624772	53

Subgroups — C53×#10624772 #50311657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624772 #50311657