Subgroups — C₅₃×#10624736 #50311621

Order 38637.

488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅₃×#731	C₃	1	9	yes	maximal, complemented
3	C₅₃×#6891	C₃	1	2	yes	maximal, complemented
4	C₅₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×3₋¹⁺²×C₅₃	—	3	9	no	—
8	C₃×3₋¹⁺²×C₅₃	C₃²	1	6	yes	—
9	C₃×H(3,𝔽₃)×C₅₃	—	3	3	no	—
10	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
11	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
12	C₅₃×#731	—	3	27	no	—
13	C₅₃×C₃²⋊C₉	—	3	6	no	—
14	C₃³×C₅₃	—	3	3	no	—
15	C₃³×C₅₃	—	3	3	no	—
16	C₃³×C₅₃	—	9	3	no	—
17	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₅₃	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₅₃	—	3	27	no	—
20	3₋¹⁺²×C₅₃	—	3	18	no	—
21	H(3,𝔽₃)×C₅₃	—	3	9	no	—
22	H(3,𝔽₃)×C₅₃	—	9	9	no	—
23	C₃×C₉×C₅₃	—	3	6	no	—
24	C₃×C₉×C₅₃	—	3	3	no	—
25	C₃×C₉×C₅₃	—	3	6	no	—
26	C₃×C₉×C₅₃	H(3,𝔽₃)	1	9	yes	complemented
27	#10624736	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₅₃	—	3	3	no	—
29	C₃²×C₅₃	—	3	3	no	—
30	C₃²×C₅₃	—	3	3	no	—
31	C₃²×C₅₃	—	3	6	no	—
32	C₃²×C₅₃	—	9	9	no	—
33	C₃²×C₅₃	—	9	3	no	—
34	C₃²×C₅₃	—	9	9	no	—
35	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	3	yes	—
36	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₉×C₅₃	—	3	9	no	—
38	C₉×C₅₃	—	9	6	no	—
39	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic, complemented
40	C₃×#731	C₃×C₅₃	1	9	yes	complemented
41	#6891	C₃×C₅₃	1	2	yes	complemented
42	C₃²⋊H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic
43	C₃×C₅₃	—	3	3	no	—
44	C₃×C₅₃	—	9	3	no	—
45	C₃×C₅₃	—	27	3	no	—
46	C₃×C₅₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₅₃	#6910	1	3	yes	—
48	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	3	3	no	—
50	C₃×3₋¹⁺²	—	3	9	no	—
51	C₃×3₋¹⁺²	C₃²×C₅₃	1	6	yes	—
52	C₃×H(3,𝔽₃)	—	3	3	no	—
53	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
54	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
55	#731	—	3	27	no	—
56	C₃²⋊C₉	—	3	6	no	—
57	C₅₃	#10624736	1	1	yes	characteristic, direct product factor, Sylow
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	9	3	no	—
61	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
62	C₃³	C₃³×C₅₃	1	1	yes	characteristic
63	3₋¹⁺²	—	3	27	no	—
64	3₋¹⁺²	—	3	18	no	—
65	H(3,𝔽₃)	—	3	9	no	—
66	H(3,𝔽₃)	—	9	9	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	3	6	no	—
70	C₃×C₉	H(3,𝔽₃)×C₅₃	1	9	yes	complemented
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	9	no	—
78	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	3	yes	—
79	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
80	C₉	—	3	9	no	—
81	C₉	—	9	6	no	—
82	C₃	—	3	3	no	—
83	C₃	—	9	3	no	—
84	C₃	—	27	3	no	—
85	C₃	C₅₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
86	C₃	C₅₃×#6910	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₅₃×C₃²⋊H(3,𝔽₃)	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
1. C₃²×3₋¹⁺²×C₅₃	C₃	semidirect
2. C₃×C₅₃×#731	C₃	semidirect
3. C₅₃×#6891	C₃	semidirect
26. C₃×C₉×C₅₃	H(3,𝔽₃)	semidirect
27. #10624736	C₅₃	direct (N × H)
39. C₃²×3₋¹⁺²	C₃×C₅₃	semidirect
40. C₃×#731	C₃×C₅₃	semidirect
41. #6891	C₃×C₅₃	semidirect
57. C₅₃	#10624736	direct (N × H)
70. C₃×C₉	H(3,𝔽₃)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₅₃	3
C₃×C₅₃×#731	3
C₅₃×#6891	3
C₅₃×C₃²⋊H(3,𝔽₃)	3
#10624736	53

Subgroups — C53×#10624736 #50311621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624736 #50311621