Subgroups — C₅₃×#10624838 #50311723

Order 38637.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#733	C₃	1	2	yes	maximal, complemented
2	C₃×C₅₃×#732	C₃	1	1	yes	maximal, characteristic
3	C₅₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#6878	C₃	1	9	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
6	C₃×H(3,𝔽₃)×C₅₃	—	3	1	no	—
7	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
8	C₅₃×#733	—	3	4	no	—
9	C₅₃×#733	C₃²	1	6	yes	complemented
10	C₅₃×#732	—	3	2	no	—
11	C₅₃×#732	C₃²	1	3	yes	—
12	C₅₃×C₉⋊C₉	—	3	2	no	—
13	C₅₃×C₃²⋊C₉	—	3	1	no	—
14	C₉²×C₅₃	C₃²	1	3	yes	complemented
15	C₃³×C₅₃	—	9	1	no	—
16	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
17	3₋¹⁺²×C₅₃	—	3	6	no	—
18	3₋¹⁺²×C₅₃	—	9	2	no	—
19	3₋¹⁺²×C₅₃	—	9	2	no	—
20	H(3,𝔽₃)×C₅₃	—	3	3	no	—
21	H(3,𝔽₃)×C₅₃	—	9	2	no	—
22	C₃×C₉×C₅₃	—	3	2	no	—
23	C₃×C₉×C₅₃	—	3	1	no	—
24	C₃×C₉×C₅₃	—	3	3	no	—
25	C₃×C₉×C₅₃	—	9	2	no	—
26	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
27	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
28	#10624838	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	9	1	no	—
30	C₃²×C₅₃	—	9	3	no	—
31	C₃²×C₅₃	—	27	3	no	—
32	C₃²×C₅₃	#732	1	2	yes	—
33	C₃²×C₅₃	#732	1	1	yes	characteristic
34	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₉×C₅₃	—	3	2	no	—
36	C₉×C₅₃	—	3	1	no	—
37	C₉×C₅₃	—	9	3	no	—
38	C₉×C₅₃	—	9	6	no	—
39	C₃×#733	C₃×C₅₃	1	2	yes	complemented
40	C₃×#732	C₃×C₅₃	1	1	yes	characteristic
41	#6887	C₃×C₅₃	1	1	yes	characteristic, complemented
42	#6878	C₃×C₅₃	1	9	yes	complemented
43	C₃×C₅₃	—	3	1	no	—
44	C₃×C₅₃	—	3	1	no	—
45	C₃×C₅₃	—	3	2	no	—
46	C₃×C₅₃	—	27	3	no	—
47	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
51	#733	—	3	4	no	—
52	#733	C₃²×C₅₃	1	6	yes	complemented
53	#732	—	3	2	no	—
54	#732	C₃²×C₅₃	1	3	yes	—
55	C₉⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	1	no	—
57	C₉²	C₃²×C₅₃	1	3	yes	complemented
58	C₅₃	#10624838	1	1	yes	characteristic, direct product factor, Sylow
59	C₃³	—	9	1	no	—
60	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
61	3₋¹⁺²	—	3	6	no	—
62	3₋¹⁺²	—	9	2	no	—
63	3₋¹⁺²	—	9	2	no	—
64	H(3,𝔽₃)	—	3	3	no	—
65	H(3,𝔽₃)	—	9	2	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	9	2	no	—
70	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
71	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
72	C₃²	—	9	1	no	—
73	C₃²	—	9	3	no	—
74	C₃²	—	27	3	no	—
75	C₃²	C₅₃×#732	1	2	yes	—
76	C₃²	C₅₃×#732	1	1	yes	characteristic
77	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
78	C₉	—	3	2	no	—
79	C₉	—	3	1	no	—
80	C₉	—	9	3	no	—
81	C₉	—	9	6	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	2	no	—
85	C₃	—	27	3	no	—
86	C₃	C₃×C₅₃×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₅₃	C₃×#732
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₅₃
prime radical	C₃×C₅₃×#732	C₃
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ C₃³×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×#733	C₃	semidirect
3. C₅₃×#6887	C₃	semidirect
4. C₅₃×#6878	C₃	semidirect
9. C₅₃×#733	C₃²	semidirect
14. C₉²×C₅₃	C₃²	semidirect
28. #10624838	C₅₃	direct (N × H)
39. C₃×#733	C₃×C₅₃	semidirect
41. #6887	C₃×C₅₃	semidirect
42. #6878	C₃×C₅₃	semidirect
52. #733	C₃²×C₅₃	semidirect
57. C₉²	C₃²×C₅₃	semidirect
58. C₅₃	#10624838	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅₃×#733	3
C₃×C₅₃×#732	3
C₅₃×#6887	3
C₅₃×#6878	3
#10624838	53

Subgroups — C53×#10624838 #50311723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624838 #50311723