Subgroups — C₅₃×#10624732 #50311617

Order 38637.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅₃×#732	C₃	1	9	yes	maximal, complemented
3	C₅₃×C₃²⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
4	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
5	C₃×H(3,𝔽₃)×C₅₃	—	3	9	no	—
6	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	9	yes	—
7	C₃²×C₉×C₅₃	—	3	3	no	—
8	C₃²×C₉×C₅₃	—	3	9	no	—
9	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
10	C₅₃×#732	—	3	27	no	—
11	C₃³×C₅₃	—	3	9	no	—
12	C₃³×C₅₃	—	3	3	no	—
13	C₃³×C₅₃	—	3	9	no	—
14	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
15	C₃³×C₅₃	C₃³	1	1	yes	characteristic
16	H(3,𝔽₃)×C₅₃	—	3	27	no	—
17	H(3,𝔽₃)×C₅₃	—	9	27	no	—
18	C₃×C₉×C₅₃	—	3	27	no	—
19	C₃×C₉×C₅₃	—	3	6	no	—
20	C₃×C₉×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	H(3,𝔽₃)	1	9	yes	complemented
22	#10624732	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₅₃	—	3	27	no	—
24	C₃²×C₅₃	—	3	3	no	—
25	C₃²×C₅₃	—	3	6	no	—
26	C₃²×C₅₃	—	3	3	no	—
27	C₃²×C₅₃	—	9	9	no	—
28	C₃²×C₅₃	—	9	27	no	—
29	C₃²×C₅₃	#732	1	9	yes	complemented
30	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	3	yes	—
31	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₉×C₅₃	—	3	9	no	—
33	C₃³×C₉	C₃×C₅₃	1	1	yes	characteristic, complemented
34	C₃×#732	C₃×C₅₃	1	9	yes	complemented
35	C₃²⋊H(3,𝔽₃)	C₃×C₅₃	1	3	yes	complemented
36	C₃×C₅₃	—	3	9	no	—
37	C₃×C₅₃	—	3	3	no	—
38	C₃×C₅₃	—	27	9	no	—
39	C₃×C₅₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₅₃	C₃×#732	1	3	yes	—
41	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
42	C₃×H(3,𝔽₃)	—	3	9	no	—
43	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	9	yes	—
44	C₃²×C₉	—	3	3	no	—
45	C₃²×C₉	—	3	9	no	—
46	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
47	#732	—	3	27	no	—
48	C₅₃	#10624732	1	1	yes	characteristic, direct product factor, Sylow
49	C₃³	—	3	9	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	9	no	—
52	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
53	C₃³	C₃³×C₅₃	1	1	yes	characteristic
54	H(3,𝔽₃)	—	3	27	no	—
55	H(3,𝔽₃)	—	9	27	no	—
56	C₃×C₉	—	3	27	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	H(3,𝔽₃)×C₅₃	1	9	yes	complemented
60	C₃²	—	3	27	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	6	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	9	9	no	—
65	C₃²	—	9	27	no	—
66	C₃²	C₅₃×#732	1	9	yes	complemented
67	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	3	yes	—
68	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
69	C₉	—	3	9	no	—
70	C₃	—	3	9	no	—
71	C₃	—	3	3	no	—
72	C₃	—	27	9	no	—
73	C₃	C₅₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
74	C₃	C₃×C₅₃×#732	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
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
1. C₃³×C₉×C₅₃	C₃	semidirect
2. C₃×C₅₃×#732	C₃	semidirect
3. C₅₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
21. C₃×C₉×C₅₃	H(3,𝔽₃)	semidirect
22. #10624732	C₅₃	direct (N × H)
29. C₃²×C₅₃	#732	semidirect
33. C₃³×C₉	C₃×C₅₃	semidirect
34. C₃×#732	C₃×C₅₃	semidirect
35. C₃²⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
48. C₅₃	#10624732	direct (N × H)
59. C₃×C₉	H(3,𝔽₃)×C₅₃	semidirect
66. C₃²	C₅₃×#732	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₅₃	3
C₃×C₅₃×#732	3
C₅₃×C₃²⋊H(3,𝔽₃)	3
#10624732	53

Subgroups — C53×#10624732 #50311617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624732 #50311617