Subgroups — C₅₃×#10624643 #50311528

Order 38637.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6891	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6891	C₃	1	2	yes	maximal
3	C₃×C₅₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₅₃×#6861	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
7	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
8	C₃²×C₉×C₅₃	—	3	1	no	—
9	C₅₃×C₃²⋊C₉	—	3	2	no	—
10	C₅₃×C₃²⋊C₉	—	3	2	no	—
11	C₅₃×C₃²⋊C₉	—	3	2	no	—
12	C₅₃×C₃²⋊C₉	—	3	2	no	—
13	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃³×C₅₃	—	3	3	no	—
17	C₃³×C₅₃	—	9	2	no	—
18	C₃³×C₅₃	—	9	1	no	—
19	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
20	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₅₃	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₅₃	—	9	6	no	—
23	3₋¹⁺²×C₅₃	—	9	3	no	—
24	C₃×C₉×C₅₃	—	3	3	no	—
25	C₃×C₉×C₅₃	—	3	3	no	—
26	C₃×C₉×C₅₃	—	3	6	no	—
27	C₃×C₉×C₅₃	—	9	3	no	—
28	#10624643	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	3	2	no	—
30	C₃²×C₅₃	—	3	1	no	—
31	C₃²×C₅₃	—	3	4	no	—
32	C₃²×C₅₃	—	3	2	no	—
33	C₃²×C₅₃	—	3	2	no	—
34	C₃²×C₅₃	—	3	1	no	—
35	C₃²×C₅₃	—	3	2	no	—
36	C₃²×C₅₃	—	3	1	no	—
37	C₃²×C₅₃	—	9	9	no	—
38	C₃²×C₅₃	#731	1	3	yes	—
39	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₉×C₅₃	—	9	3	no	—
41	C₉×C₅₃	—	9	3	no	—
42	C₉×C₅₃	—	9	6	no	—
43	#6891	C₃×C₅₃	1	1	yes	characteristic
44	#6891	C₃×C₅₃	1	2	yes	—
45	C₃×C₃²⋊C₉	C₃×C₅₃	1	1	yes	characteristic
46	#6861	C₃×C₅₃	1	9	yes	complemented
47	C₃×C₅₃	—	3	3	no	—
48	C₃×C₅₃	—	9	2	no	—
49	C₃×C₅₃	—	9	1	no	—
50	C₃×C₅₃	C₃×#731	1	1	yes	characteristic
51	C₃×C₅₃	#6911	1	1	yes	characteristic
52	C₃×C₅₃	#6912	1	2	yes	—
53	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	3	2	no	—
55	C₃×3₋¹⁺²	—	3	1	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²⋊C₉	—	3	2	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
62	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
63	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
64	C₅₃	#10624643	1	1	yes	characteristic, direct product factor, Sylow
65	C₃³	—	3	3	no	—
66	C₃³	—	9	1	no	—
67	C₃³	—	9	2	no	—
68	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
69	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
70	C₃³	C₃³×C₅₃	1	1	yes	characteristic
71	3₋¹⁺²	—	9	6	no	—
72	3₋¹⁺²	—	9	3	no	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	—	3	6	no	—
76	C₃×C₉	—	9	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	2	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	4	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	9	9	no	—
86	C₃²	C₅₃×#731	1	3	yes	—
87	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
88	C₉	—	9	3	no	—
89	C₉	—	9	3	no	—
90	C₉	—	9	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	2	no	—
93	C₃	—	9	1	no	—
94	C₃	C₃×C₅₃×#731	1	1	yes	characteristic
95	C₃	C₅₃×#6911	1	1	yes	characteristic
96	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 ▷ C₃² ▷ 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×#6861	C₃	semidirect
28. #10624643	C₅₃	direct (N × H)
46. #6861	C₃×C₅₃	semidirect
64. C₅₃	#10624643	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×#6861	3
#10624643	53

Subgroups — C53×#10624643 #50311528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624643 #50311528