Subgroups — C₃×C₅₃×#6867 #50311571

Order 38637.

476 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
2	C₃×C₅₃×C₃²⋊C₉	C₃	1	3	yes	maximal
3	C₅₃×#6867	C₃	1	9	yes	maximal, direct product factor
4	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
5	C₃²×C₉×C₅₃	—	3	2	no	—
6	C₃²×C₉×C₅₃	—	3	1	no	—
7	C₃²×C₉×C₅₃	—	3	6	no	—
8	C₃²×C₉×C₅₃	—	3	3	no	—
9	C₃²×C₉×C₅₃	C₉	1	9	yes	complemented
10	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
11	C₅₃×C₃²⋊C₉	—	3	6	no	—
12	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
13	C₃³×C₅₃	—	3	9	no	—
14	C₃³×C₅₃	C₃×C₉	1	3	yes	—
15	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
16	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₅₃	3₋¹⁺²	1	6	yes	—
18	C₃³×C₅₃	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₅₃	—	3	18	no	—
20	C₃×C₉×C₅₃	—	3	9	no	—
21	C₃×C₉×C₅₃	—	3	3	no	—
22	C₃×C₉×C₅₃	—	3	6	no	—
23	C₃×C₉×C₅₃	—	3	9	no	—
24	C₃×C₉×C₅₃	—	9	9	no	—
25	C₃×C₉×C₅₃	C₃×C₉	1	9	yes	complemented
26	C₃×#6867	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₅₃	—	3	9	no	—
28	C₃²×C₅₃	—	3	18	no	—
29	C₃²×C₅₃	—	3	6	no	—
30	C₃²×C₅₃	—	3	3	no	—
31	C₃²×C₅₃	C₃²⋊C₉	1	6	yes	—
32	C₃²×C₅₃	C₃²⋊C₉	1	3	yes	—
33	C₃²×C₅₃	#731	1	9	yes	—
34	C₃²×C₅₃	C₃²×C₉	1	1	yes	characteristic
35	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₃²×C₅₃	C₃×3₋¹⁺²	1	2	yes	—
37	C₉×C₅₃	—	3	9	no	—
38	C₉×C₅₃	—	9	9	no	—
39	C₃³×C₉	C₃×C₅₃	1	1	yes	characteristic
40	C₃×C₃²⋊C₉	C₃×C₅₃	1	3	yes	—
41	#6867	C₃×C₅₃	1	9	yes	direct product factor
42	C₃×C₅₃	—	3	6	no	—
43	C₃×C₅₃	—	3	1	no	—
44	C₃×C₅₃	—	3	2	no	—
45	C₃×C₅₃	#6867	1	9	yes	direct product factor
46	C₃×C₅₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
47	C₃×C₅₃	C₃×#731	1	3	yes	—
48	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	3	1	no	—
51	C₃²×C₉	—	3	6	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	C₉×C₅₃	1	9	yes	complemented
54	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
55	C₃²⋊C₉	—	3	6	no	—
56	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
57	C₅₃	C₃×#6867	1	1	yes	characteristic, direct product factor, Sylow
58	C₃³	—	3	9	no	—
59	C₃³	C₃×C₉×C₅₃	1	3	yes	—
60	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
61	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
62	C₃³	3₋¹⁺²×C₅₃	1	6	yes	—
63	C₃³	C₃³×C₅₃	1	1	yes	characteristic
64	C₃×C₉	—	3	18	no	—
65	C₃×C₉	—	3	9	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	3	9	no	—
69	C₃×C₉	—	9	9	no	—
70	C₃×C₉	C₃×C₉×C₅₃	1	9	yes	complemented
71	C₃²	—	3	6	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	9	no	—
74	C₃²	—	3	18	no	—
75	C₃²	C₅₃×C₃²⋊C₉	1	6	yes	—
76	C₃²	C₅₃×C₃²⋊C₉	1	3	yes	—
77	C₃²	C₅₃×#731	1	9	yes	—
78	C₃²	C₃²×C₉×C₅₃	1	1	yes	characteristic
79	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
80	C₃²	C₃×3₋¹⁺²×C₅₃	1	2	yes	—
81	C₉	—	3	9	no	—
82	C₉	—	9	9	no	—
83	C₃	—	3	6	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	2	no	—
86	C₃	C₅₃×#6867	1	9	yes	direct product factor
87	C₃	C₃×C₅₃×C₃²⋊C₉	1	1	yes	characteristic
88	C₃	C₃×C₅₃×#731	1	3	yes	—

	subgroup	G/H
center, socle	C₃³×C₅₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₅₃
Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₅₃×#6867	C₃	direct (N × H)
9. C₃²×C₉×C₅₃	C₉	semidirect
25. C₃×C₉×C₅₃	C₃×C₉	semidirect
26. C₃×#6867	C₅₃	direct (N × H)
41. #6867	C₃×C₅₃	direct (N × H)
45. C₃×C₅₃	#6867	direct (N × H)
53. C₃²×C₉	C₉×C₅₃	semidirect
57. C₅₃	C₃×#6867	direct (N × H)
70. C₃×C₉	C₃×C₉×C₅₃	semidirect
86. C₃	C₅₃×#6867	direct (N × H)

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×C₃²⋊C₉	3
C₅₃×#6867	3
C₃×#6867	53

Subgroups — C3×C53×#6867 #50311571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×#6867 #50311571