Subgroups — C₅₃×#10624768 #50311653

Order 38637.

428 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	C₅₃×#6891	C₃	1	3	yes	maximal
3	C₅₃×#6868	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
6	C₃²×C₉×C₅₃	—	3	1	no	—
7	C₃²×C₉×C₅₃	—	3	2	no	—
8	C₃²×C₉×C₅₃	—	3	9	no	—
9	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
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	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₅₃	H(3,𝔽₃)	1	2	yes	—
17	C₃³×C₅₃	C₃³	1	1	yes	characteristic
18	3₋¹⁺²×C₅₃	—	9	9	no	—
19	C₃×C₉×C₅₃	—	3	9	no	—
20	C₃×C₉×C₅₃	—	3	18	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	3₋¹⁺²	1	9	yes	complemented
25	#10624768	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₅₃	—	3	27	no	—
27	C₃²×C₅₃	—	3	1	no	—
28	C₃²×C₅₃	—	3	2	no	—
29	C₃²×C₅₃	—	3	2	no	—
30	C₃²×C₅₃	—	3	1	no	—
31	C₃²×C₅₃	—	3	2	no	—
32	C₃²×C₅₃	—	3	2	no	—
33	C₃²×C₅₃	—	3	2	no	—
34	C₃²×C₅₃	#732	1	3	yes	—
35	C₃²×C₅₃	#733	1	6	yes	—
36	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₅₃	C₃×3₋¹⁺²	1	3	yes	—
38	C₉×C₅₃	—	3	9	no	—
39	C₉×C₅₃	—	9	9	no	—
40	C₃³×C₉	C₃×C₅₃	1	1	yes	characteristic
41	#6891	C₃×C₅₃	1	3	yes	—
42	#6868	C₃×C₅₃	1	9	yes	complemented
43	C₃×C₅₃	—	3	3	no	—
44	C₃×C₅₃	—	3	6	no	—
45	C₃×C₅₃	—	3	3	no	—
46	C₃×C₅₃	#6891	1	1	yes	characteristic
47	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
48	C₃×C₅₃	C₃×#733	1	2	yes	—
49	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	3	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	9	no	—
54	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
55	C₃²⋊C₉	—	3	6	no	—
56	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
57	C₅₃	#10624768	1	1	yes	characteristic, direct product factor, Sylow
58	C₃³	—	3	3	no	—
59	C₃³	—	3	6	no	—
60	C₃³	—	3	3	no	—
61	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
62	C₃³	H(3,𝔽₃)×C₅₃	1	2	yes	—
63	C₃³	C₃³×C₅₃	1	1	yes	characteristic
64	3₋¹⁺²	—	9	9	no	—
65	C₃×C₉	—	3	9	no	—
66	C₃×C₉	—	3	18	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	—	3	9	no	—
70	C₃×C₉	3₋¹⁺²×C₅₃	1	9	yes	complemented
71	C₃²	—	3	27	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	C₅₃×#732	1	3	yes	—
80	C₃²	C₅₃×#733	1	6	yes	—
81	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
82	C₃²	C₃×3₋¹⁺²×C₅₃	1	3	yes	—
83	C₉	—	3	9	no	—
84	C₉	—	9	9	no	—
85	C₃	—	3	3	no	—
86	C₃	—	3	6	no	—
87	C₃	—	3	3	no	—
88	C₃	C₅₃×#6891	1	1	yes	characteristic
89	C₃	C₃×C₅₃×#732	1	1	yes	characteristic
90	C₃	C₃×C₅₃×#733	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
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
3. C₅₃×#6868	C₃	semidirect
24. C₃×C₉×C₅₃	3₋¹⁺²	semidirect
25. #10624768	C₅₃	direct (N × H)
42. #6868	C₃×C₅₃	semidirect
57. C₅₃	#10624768	direct (N × H)
70. C₃×C₉	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
C₃³×C₉×C₅₃	3
C₅₃×#6891	3
C₅₃×#6868	3
#10624768	53

Subgroups — C53×#10624768 #50311653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624768 #50311653