Subgroups — C₅₃×#10624769 #50311654

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	3	no	—
8	C₃×H(3,𝔽₃)×C₅₃	—	3	9	no	—
9	C₅₃×C₃²⋊C₉	—	3	6	no	—
10	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
11	C₃³×C₅₃	—	3	3	no	—
12	C₃³×C₅₃	—	3	6	no	—
13	C₃³×C₅₃	—	3	3	no	—
14	C₃³×C₅₃	—	9	3	no	—
15	C₃³×C₅₃	—	9	9	no	—
16	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
17	C₃³×C₅₃	3₋¹⁺²	1	9	yes	complemented
18	C₃³×C₅₃	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₅₃	—	9	9	no	—
20	H(3,𝔽₃)×C₅₃	—	3	27	no	—
21	C₃×C₉×C₅₃	—	3	9	no	—
22	#10624769	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₅₃	—	3	3	no	—
24	C₃²×C₅₃	—	3	3	no	—
25	C₃²×C₅₃	—	3	9	no	—
26	C₃²×C₅₃	—	3	3	no	—
27	C₃²×C₅₃	—	3	6	no	—
28	C₃²×C₅₃	—	9	9	no	—
29	C₃²×C₅₃	—	9	9	no	—
30	C₃²×C₅₃	—	9	18	no	—
31	C₃²×C₅₃	—	9	9	no	—
32	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
34	C₉×C₅₃	—	9	9	no	—
35	C₃²×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic
36	#6891	C₃×C₅₃	1	3	yes	complemented
37	#6866	C₃×C₅₃	1	9	yes	complemented
38	C₃×C₅₃	—	3	3	no	—
39	C₃×C₅₃	—	9	3	no	—
40	C₃×C₅₃	—	9	9	no	—
41	C₃×C₅₃	#6891	1	1	yes	characteristic
42	C₃×C₅₃	#6909	1	3	yes	—
43	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
44	C₃⁴	C₃²×C₅₃	1	3	yes	—
45	C₃×3₋¹⁺²	—	3	3	no	—
46	C₃×H(3,𝔽₃)	—	3	3	no	—
47	C₃×H(3,𝔽₃)	—	3	9	no	—
48	C₃²⋊C₉	—	3	6	no	—
49	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
50	C₅₃	#10624769	1	1	yes	characteristic, direct product factor, Sylow
51	C₃³	—	3	3	no	—
52	C₃³	—	3	3	no	—
53	C₃³	—	3	6	no	—
54	C₃³	—	9	3	no	—
55	C₃³	—	9	9	no	—
56	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
57	C₃³	3₋¹⁺²×C₅₃	1	9	yes	complemented
58	C₃³	C₃³×C₅₃	1	1	yes	characteristic
59	3₋¹⁺²	—	9	9	no	—
60	H(3,𝔽₃)	—	3	27	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	9	no	—
65	C₃²	—	3	3	no	—
66	C₃²	—	3	6	no	—
67	C₃²	—	9	9	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	18	no	—
70	C₃²	—	9	9	no	—
71	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
72	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
73	C₉	—	9	9	no	—
74	C₃	—	3	3	no	—
75	C₃	—	9	3	no	—
76	C₃	—	9	9	no	—
77	C₃	C₅₃×#6891	1	1	yes	characteristic
78	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
17. C₃³×C₅₃	3₋¹⁺²	semidirect
22. #10624769	C₅₃	direct (N × H)
36. #6891	C₃×C₅₃	semidirect
37. #6866	C₃×C₅₃	semidirect
50. C₅₃	#10624769	direct (N × H)
57. 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
#10624769	53

Subgroups — C53×#10624769 #50311654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624769 #50311654