Subgroups — C₅₃×#10624737 #50311622

Order 38637.

572 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, complemented
2	C₃×C₅₃×#732	C₃	1	9	yes	maximal, complemented
3	C₅₃×C₃²⋊H(3,𝔽₃)	C₃	1	3	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	9	no	—
7	C₃×H(3,𝔽₃)×C₅₃	—	3	9	no	—
8	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	9	yes	—
9	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
10	C₅₃×#732	—	3	27	no	—
11	C₃³×C₅₃	—	3	3	no	—
12	C₃³×C₅₃	—	3	9	no	—
13	C₃³×C₅₃	—	9	3	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	H(3,𝔽₃)×C₅₃	—	3	27	no	—
18	H(3,𝔽₃)×C₅₃	—	9	27	no	—
19	C₃×C₉×C₅₃	—	3	6	no	—
20	C₃×C₉×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	H(3,𝔽₃)	1	9	yes	complemented
22	#10624737	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₅₃	—	3	3	no	—
24	C₃²×C₅₃	—	3	3	no	—
25	C₃²×C₅₃	—	3	9	no	—
26	C₃²×C₅₃	—	9	9	no	—
27	C₃²×C₅₃	—	9	9	no	—
28	C₃²×C₅₃	—	9	27	no	—
29	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	3	yes	—
30	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₉×C₅₃	—	3	9	no	—
32	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic, complemented
33	C₃×#732	C₃×C₅₃	1	9	yes	complemented
34	C₃²⋊H(3,𝔽₃)	C₃×C₅₃	1	3	yes	complemented
35	C₃×C₅₃	—	3	3	no	—
36	C₃×C₅₃	—	9	3	no	—
37	C₃×C₅₃	—	27	9	no	—
38	C₃×C₅₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₅₃	#6911	1	3	yes	—
40	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	3	3	no	—
42	C₃×3₋¹⁺²	—	3	9	no	—
43	C₃×H(3,𝔽₃)	—	3	9	no	—
44	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	9	yes	—
45	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
46	#732	—	3	27	no	—
47	C₅₃	#10624737	1	1	yes	characteristic, direct product factor, Sylow
48	C₃³	—	3	3	no	—
49	C₃³	—	3	9	no	—
50	C₃³	—	9	3	no	—
51	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
52	C₃³	C₃³×C₅₃	1	1	yes	characteristic
53	3₋¹⁺²	—	3	27	no	—
54	H(3,𝔽₃)	—	3	27	no	—
55	H(3,𝔽₃)	—	9	27	no	—
56	C₃×C₉	—	3	6	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	H(3,𝔽₃)×C₅₃	1	9	yes	complemented
59	C₃²	—	3	3	no	—
60	C₃²	—	3	3	no	—
61	C₃²	—	3	9	no	—
62	C₃²	—	9	9	no	—
63	C₃²	—	9	9	no	—
64	C₃²	—	9	27	no	—
65	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	3	yes	—
66	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
67	C₉	—	3	9	no	—
68	C₃	—	3	3	no	—
69	C₃	—	9	3	no	—
70	C₃	—	27	9	no	—
71	C₃	C₅₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
72	C₃	C₅₃×#6911	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
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
1. C₃²×3₋¹⁺²×C₅₃	C₃	semidirect
2. C₃×C₅₃×#732	C₃	semidirect
3. C₅₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
21. C₃×C₉×C₅₃	H(3,𝔽₃)	semidirect
22. #10624737	C₅₃	direct (N × H)
32. C₃²×3₋¹⁺²	C₃×C₅₃	semidirect
33. C₃×#732	C₃×C₅₃	semidirect
34. C₃²⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
47. C₅₃	#10624737	direct (N × H)
58. C₃×C₉	H(3,𝔽₃)×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
C₃×C₅₃×#732	3
C₅₃×C₃²⋊H(3,𝔽₃)	3
#10624737	53

Subgroups — C53×#10624737 #50311622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624737 #50311622