Subgroups — C₅₃×#10624773 #50311658

Order 38637.

464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6891	C₃	1	3	yes	maximal, complemented
3	C₅₃×#6866	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₅₃	C₃²	1	3	yes	—
5	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×H(3,𝔽₃)×C₅₃	—	3	2	no	—
8	C₃×H(3,𝔽₃)×C₅₃	—	3	1	no	—
9	C₃×H(3,𝔽₃)×C₅₃	—	3	9	no	—
10	C₅₃×C₃²⋊C₉	—	3	6	no	—
11	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃³×C₅₃	—	3	3	no	—
13	C₃³×C₅₃	—	3	6	no	—
14	C₃³×C₅₃	—	3	3	no	—
15	C₃³×C₅₃	—	9	3	no	—
16	C₃³×C₅₃	—	9	9	no	—
17	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
18	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃³×C₅₃	3₋¹⁺²	1	9	yes	complemented
20	C₃³×C₅₃	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₅₃	—	9	9	no	—
22	H(3,𝔽₃)×C₅₃	—	3	18	no	—
23	H(3,𝔽₃)×C₅₃	—	3	9	no	—
24	C₃×C₉×C₅₃	—	3	9	no	—
25	#10624773	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₅₃	—	3	3	no	—
27	C₃²×C₅₃	—	3	2	no	—
28	C₃²×C₅₃	—	3	1	no	—
29	C₃²×C₅₃	—	3	9	no	—
30	C₃²×C₅₃	—	3	3	no	—
31	C₃²×C₅₃	—	3	6	no	—
32	C₃²×C₅₃	—	9	9	no	—
33	C₃²×C₅₃	—	9	9	no	—
34	C₃²×C₅₃	—	9	18	no	—
35	C₃²×C₅₃	—	9	9	no	—
36	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
38	C₉×C₅₃	—	9	9	no	—
39	C₃²×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic
40	#6891	C₃×C₅₃	1	3	yes	complemented
41	#6866	C₃×C₅₃	1	9	yes	complemented
42	C₃×C₅₃	—	3	3	no	—
43	C₃×C₅₃	—	9	1	no	—
44	C₃×C₅₃	—	9	2	no	—
45	C₃×C₅₃	—	9	9	no	—
46	C₃×C₅₃	#6891	1	1	yes	characteristic
47	C₃×C₅₃	#6909	1	3	yes	—
48	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
49	C₃⁴	C₃²×C₅₃	1	3	yes	—
50	C₃×3₋¹⁺²	—	3	3	no	—
51	C₃×H(3,𝔽₃)	—	3	2	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	—	3	9	no	—
54	C₃²⋊C₉	—	3	6	no	—
55	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
56	C₅₃	#10624773	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	3	3	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	6	no	—
60	C₃³	—	9	3	no	—
61	C₃³	—	9	9	no	—
62	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
63	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
64	C₃³	3₋¹⁺²×C₅₃	1	9	yes	complemented
65	C₃³	C₃³×C₅₃	1	1	yes	characteristic
66	3₋¹⁺²	—	9	9	no	—
67	H(3,𝔽₃)	—	3	18	no	—
68	H(3,𝔽₃)	—	3	9	no	—
69	C₃×C₉	—	3	9	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	9	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	9	9	no	—
77	C₃²	—	9	9	no	—
78	C₃²	—	9	18	no	—
79	C₃²	—	9	9	no	—
80	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
81	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
82	C₉	—	9	9	no	—
83	C₃	—	3	3	no	—
84	C₃	—	9	1	no	—
85	C₃	—	9	2	no	—
86	C₃	—	9	9	no	—
87	C₃	C₅₃×#6891	1	1	yes	characteristic
88	C₃	C₅₃×#6909	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃²×H(3,𝔽₃)×C₅₃	C₃
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
2. C₅₃×#6891	C₃	semidirect
3. C₅₃×#6866	C₃	semidirect
19. C₃³×C₅₃	3₋¹⁺²	semidirect
25. #10624773	C₅₃	direct (N × H)
40. #6891	C₃×C₅₃	semidirect
41. #6866	C₃×C₅₃	semidirect
56. C₅₃	#10624773	direct (N × H)
64. C₃³	3₋¹⁺²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×H(3,𝔽₃)×C₅₃	3
C₅₃×#6891	3
C₅₃×#6866	3
#10624773	53

Subgroups — C53×#10624773 #50311658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624773 #50311658