Subgroups — C₅₃×#10624570 #50311455

Order 38637.

428 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	C₃×C₅₃×C₃²⋊C₉	C₃	1	3	yes	maximal
3	C₅₃×#6855	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	9	no	—
7	C₃²×C₉×C₅₃	C₃²	1	9	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
9	C₅₃×C₃²⋊C₉	—	3	9	no	—
10	C₃³×C₅₃	—	3	3	no	—
11	C₃³×C₅₃	—	3	9	no	—
12	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
13	C₃³×C₅₃	C₃³	1	1	yes	characteristic
14	3₋¹⁺²×C₅₃	—	3	27	no	—
15	C₃×C₉×C₅₃	—	3	3	no	—
16	C₃×C₉×C₅₃	—	3	6	no	—
17	C₃×C₉×C₅₃	—	3	9	no	—
18	C₃×C₉×C₅₃	—	3	27	no	—
19	C₃×C₉×C₅₃	3₋¹⁺²	1	9	yes	—
20	#10624570	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₅₃	—	3	3	no	—
22	C₃²×C₅₃	—	3	9	no	—
23	C₃²×C₅₃	—	9	9	no	—
24	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
26	C₃²×C₅₃	#737	1	9	yes	—
27	C₉×C₅₃	—	3	9	no	—
28	C₉×C₅₃	—	9	9	no	—
29	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
30	C₃×C₃²⋊C₉	C₃×C₅₃	1	3	yes	—
31	#6855	C₃×C₅₃	1	9	yes	complemented
32	C₃×C₅₃	—	9	3	no	—
33	C₃×C₅₃	#6891	1	1	yes	characteristic
34	C₃×C₅₃	#6893	1	3	yes	—
35	C₃×C₅₃	#6899	1	9	yes	—
36	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃²×C₉	C₃²×C₅₃	1	9	yes	—
40	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
41	C₃²⋊C₉	—	3	9	no	—
42	C₅₃	#10624570	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	—	3	3	no	—
44	C₃³	—	3	9	no	—
45	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
46	C₃³	C₃³×C₅₃	1	1	yes	characteristic
47	3₋¹⁺²	—	3	27	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	3	6	no	—
50	C₃×C₉	—	3	9	no	—
51	C₃×C₉	—	3	27	no	—
52	C₃×C₉	3₋¹⁺²×C₅₃	1	9	yes	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	9	no	—
55	C₃²	—	9	9	no	—
56	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
57	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
58	C₃²	C₅₃×#737	1	9	yes	—
59	C₉	—	3	9	no	—
60	C₉	—	9	9	no	—
61	C₃	—	9	3	no	—
62	C₃	C₅₃×#6891	1	1	yes	characteristic
63	C₃	C₅₃×#6893	1	3	yes	—
64	C₃	C₅₃×#6899	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#6855	C₃	semidirect
20. #10624570	C₅₃	direct (N × H)
31. #6855	C₃×C₅₃	semidirect
42. C₅₃	#10624570	direct (N × H)

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×C₃²⋊C₉	3
C₅₃×#6855	3
#10624570	53

Subgroups — C53×#10624570 #50311455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624570 #50311455