Subgroups — C₅₃×#10624791 #50311676

Order 38637.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#733	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6894	C₃	1	3	yes	maximal
3	C₃×C₅₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₅₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal
5	C₅₃×#6868	C₃	1	2	yes	maximal
6	C₃×3₋¹⁺²×C₅₃	C₃²	1	3	yes	—
7	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₅₃	C₃²	1	2	yes	—
9	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
10	C₅₃×#733	—	3	3	no	—
11	C₅₃×C₉⋊C₉	—	3	3	no	—
12	C₅₃×C₉⋊C₉	—	3	9	no	—
13	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₃³×C₅₃	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₅₃	—	3	3	no	—
16	3₋¹⁺²×C₅₃	—	3	6	no	—
17	C₃×C₉×C₅₃	—	3	1	no	—
18	C₃×C₉×C₅₃	—	3	2	no	—
19	C₃×C₉×C₅₃	—	3	6	no	—
20	C₃×C₉×C₅₃	—	3	2	no	—
21	C₃×C₉×C₅₃	—	3	3	no	—
22	C₃×C₉×C₅₃	—	9	1	no	—
23	C₃×C₉×C₅₃	—	9	2	no	—
24	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
25	C₃×C₉×C₅₃	3₋¹⁺²	1	3	yes	—
26	#10624791	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₅₃	—	3	1	no	—
28	C₃²×C₅₃	—	3	2	no	—
29	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₃²×C₅₃	#737	1	2	yes	—
32	C₉×C₅₃	—	3	3	no	—
33	C₉×C₅₃	—	3	3	no	—
34	C₉×C₅₃	—	9	6	no	—
35	C₉×C₅₃	—	9	2	no	—
36	C₉×C₅₃	—	9	3	no	—
37	C₃×#733	C₃×C₅₃	1	1	yes	characteristic
38	#6894	C₃×C₅₃	1	3	yes	—
39	C₃×C₉⋊C₉	C₃×C₅₃	1	1	yes	characteristic
40	3₋¹⁺²⋊C₉	C₃×C₅₃	1	6	yes	—
41	#6868	C₃×C₅₃	1	2	yes	—
42	C₃×C₅₃	—	3	1	no	—
43	C₃×C₅₃	—	3	2	no	—
44	C₃×C₅₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₅₃	#6912	1	1	yes	characteristic
46	C₃×C₅₃	#6913	1	2	yes	—
47	C₃×3₋¹⁺²	C₃²×C₅₃	1	3	yes	—
48	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
49	C₃²×C₉	C₃²×C₅₃	1	2	yes	—
50	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
51	#733	—	3	3	no	—
52	C₉⋊C₉	—	3	3	no	—
53	C₉⋊C₉	—	3	9	no	—
54	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
55	C₅₃	#10624791	1	1	yes	characteristic, direct product factor, Sylow
56	C₃³	C₃³×C₅₃	1	1	yes	characteristic
57	3₋¹⁺²	—	3	3	no	—
58	3₋¹⁺²	—	3	6	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	9	1	no	—
65	C₃×C₉	—	9	2	no	—
66	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
67	C₃×C₉	3₋¹⁺²×C₅₃	1	3	yes	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	2	no	—
70	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
71	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
72	C₃²	C₅₃×#737	1	2	yes	—
73	C₉	—	3	3	no	—
74	C₉	—	3	3	no	—
75	C₉	—	9	3	no	—
76	C₉	—	9	6	no	—
77	C₉	—	9	2	no	—
78	C₃	—	3	1	no	—
79	C₃	—	3	2	no	—
80	C₃	C₅₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
81	C₃	C₅₃×#6912	1	1	yes	characteristic
82	C₃	C₅₃×#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃³×C₅₃	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ 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
26. #10624791	C₅₃	direct (N × H)
55. C₅₃	#10624791	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅₃×#733	3
C₅₃×#6894	3
C₃×C₅₃×C₉⋊C₉	3
C₅₃×3₋¹⁺²⋊C₉	3
C₅₃×#6868	3
#10624791	53

Subgroups — C53×#10624791 #50311676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624791 #50311676