Subgroups — C₅₃×#10624806 #50311691

Order 38637.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6891	C₃	1	2	yes	maximal
2	C₃×C₅₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃×C₅₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₅₃×#6868	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
7	C₃²×C₉×C₅₃	—	3	1	no	—
8	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
9	C₅₃×C₃²⋊C₉	—	3	2	no	—
10	C₅₃×C₃²⋊C₉	—	3	1	no	—
11	C₅₃×C₃²⋊C₉	—	3	2	no	—
12	C₅₃×C₃²⋊C₉	—	3	4	no	—
13	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃³×C₅₃	—	3	3	no	—
16	C₃³×C₅₃	—	3	3	no	—
17	C₃³×C₅₃	—	9	2	no	—
18	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
19	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₅₃	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₅₃	—	9	6	no	—
22	C₃×C₉×C₅₃	—	3	3	no	—
23	C₃×C₉×C₅₃	—	3	3	no	—
24	C₃×C₉×C₅₃	—	3	6	no	—
25	C₃×C₉×C₅₃	—	3	9	no	—
26	C₃×C₉×C₅₃	—	9	3	no	—
27	#10624806	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₅₃	—	3	2	no	—
29	C₃²×C₅₃	—	3	2	no	—
30	C₃²×C₅₃	—	3	1	no	—
31	C₃²×C₅₃	—	3	1	no	—
32	C₃²×C₅₃	—	3	4	no	—
33	C₃²×C₅₃	—	3	2	no	—
34	C₃²×C₅₃	—	3	2	no	—
35	C₃²×C₅₃	—	9	9	no	—
36	C₃²×C₅₃	#732	1	3	yes	—
37	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₅₃	#737	1	3	yes	—
39	C₉×C₅₃	—	9	3	no	—
40	C₉×C₅₃	—	9	3	no	—
41	C₉×C₅₃	—	9	6	no	—
42	#6891	C₃×C₅₃	1	2	yes	—
43	C₃×C₃²⋊C₉	C₃×C₅₃	1	1	yes	characteristic
44	C₃×C₃²⋊C₉	C₃×C₅₃	1	1	yes	characteristic
45	#6868	C₃×C₅₃	1	9	yes	complemented
46	C₃×C₅₃	—	3	3	no	—
47	C₃×C₅₃	—	3	3	no	—
48	C₃×C₅₃	—	9	2	no	—
49	C₃×C₅₃	#6893	1	1	yes	characteristic
50	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
51	C₃×C₅₃	#6912	1	2	yes	—
52	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	3	2	no	—
54	C₃²×C₉	—	3	1	no	—
55	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	1	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	4	no	—
60	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
61	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
62	C₅₃	#10624806	1	1	yes	characteristic, direct product factor, Sylow
63	C₃³	—	3	3	no	—
64	C₃³	—	3	3	no	—
65	C₃³	—	9	2	no	—
66	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
67	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
68	C₃³	C₃³×C₅₃	1	1	yes	characteristic
69	3₋¹⁺²	—	9	6	no	—
70	C₃×C₉	—	3	3	no	—
71	C₃×C₉	—	3	3	no	—
72	C₃×C₉	—	3	6	no	—
73	C₃×C₉	—	3	9	no	—
74	C₃×C₉	—	9	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	4	no	—
82	C₃²	—	9	9	no	—
83	C₃²	C₅₃×#732	1	3	yes	—
84	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
85	C₃²	C₅₃×#737	1	3	yes	—
86	C₉	—	9	3	no	—
87	C₉	—	9	3	no	—
88	C₉	—	9	6	no	—
89	C₃	—	3	3	no	—
90	C₃	—	3	3	no	—
91	C₃	—	9	2	no	—
92	C₃	C₅₃×#6893	1	1	yes	characteristic
93	C₃	C₃×C₅₃×#732	1	1	yes	characteristic
94	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
4. C₅₃×#6868	C₃	semidirect
27. #10624806	C₅₃	direct (N × H)
45. #6868	C₃×C₅₃	semidirect
62. C₅₃	#10624806	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×#6891	3
C₃×C₅₃×C₃²⋊C₉	3
C₅₃×#6868	3
#10624806	53

Subgroups — C53×#10624806 #50311691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624806 #50311691