Subgroups — C₅₃×#10624790 #50311675

Order 38637.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#732	C₃	1	1	yes	maximal, characteristic
2	C₅₃×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₃×C₅₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×H(3,𝔽₃)⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₅₃×#6868	C₃	1	2	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₅₃	C₃²	1	2	yes	—
10	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
11	C₅₃×#732	—	3	3	no	—
12	C₅₃×C₉⋊C₉	—	3	3	no	—
13	C₅₃×C₃²⋊C₉	—	3	6	no	—
14	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃³×C₅₃	—	3	3	no	—
16	C₃³×C₅₃	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₅₃	—	9	9	no	—
18	H(3,𝔽₃)×C₅₃	—	3	3	no	—
19	H(3,𝔽₃)×C₅₃	—	3	6	no	—
20	C₃×C₉×C₅₃	—	3	1	no	—
21	C₃×C₉×C₅₃	—	3	2	no	—
22	C₃×C₉×C₅₃	—	3	6	no	—
23	C₃×C₉×C₅₃	—	3	2	no	—
24	C₃×C₉×C₅₃	—	9	1	no	—
25	C₃×C₉×C₅₃	—	9	2	no	—
26	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
27	C₃×C₉×C₅₃	3₋¹⁺²	1	3	yes	complemented
28	#10624790	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	3	2	no	—
30	C₃²×C₅₃	—	3	1	no	—
31	C₃²×C₅₃	—	9	6	no	—
32	C₃²×C₅₃	—	9	3	no	—
33	C₃²×C₅₃	—	9	3	no	—
34	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
36	C₃²×C₅₃	#737	1	2	yes	—
37	C₉×C₅₃	—	3	3	no	—
38	C₉×C₅₃	—	3	3	no	—
39	C₉×C₅₃	—	9	2	no	—
40	C₉×C₅₃	—	9	6	no	—
41	C₃×#732	C₃×C₅₃	1	1	yes	characteristic
42	C₉⋊H(3,𝔽₃)	C₃×C₅₃	1	3	yes	complemented
43	C₃×C₉⋊C₉	C₃×C₅₃	1	1	yes	characteristic, complemented
44	H(3,𝔽₃)⋊C₉	C₃×C₅₃	1	6	yes	complemented
45	#6868	C₃×C₅₃	1	2	yes	complemented
46	C₃×C₅₃	—	3	1	no	—
47	C₃×C₅₃	—	3	2	no	—
48	C₃×C₅₃	—	27	3	no	—
49	C₃×C₅₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
50	C₃×C₅₃	#6911	1	1	yes	characteristic
51	C₃×C₅₃	#6913	1	2	yes	—
52	C₃×3₋¹⁺²	—	3	3	no	—
53	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
54	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
55	C₃²×C₉	C₃²×C₅₃	1	2	yes	—
56	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
57	#732	—	3	3	no	—
58	C₉⋊C₉	—	3	3	no	—
59	C₃²⋊C₉	—	3	6	no	—
60	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
61	C₅₃	#10624790	1	1	yes	characteristic, direct product factor, Sylow
62	C₃³	—	3	3	no	—
63	C₃³	C₃³×C₅₃	1	1	yes	characteristic
64	3₋¹⁺²	—	9	9	no	—
65	H(3,𝔽₃)	—	3	3	no	—
66	H(3,𝔽₃)	—	3	6	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	6	no	—
70	C₃×C₉	—	3	2	no	—
71	C₃×C₉	—	9	1	no	—
72	C₃×C₉	—	9	2	no	—
73	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
74	C₃×C₉	3₋¹⁺²×C₅₃	1	3	yes	complemented
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	9	3	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	9	6	no	—
80	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
81	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
82	C₃²	C₅₃×#737	1	2	yes	—
83	C₉	—	3	3	no	—
84	C₉	—	3	3	no	—
85	C₉	—	9	6	no	—
86	C₉	—	9	2	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	2	no	—
89	C₃	—	27	3	no	—
90	C₃	C₅₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
91	C₃	C₅₃×#6911	1	1	yes	characteristic
92	C₃	C₅₃×#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃×C₅₃×#732	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₃×C₅₃×C₉⋊C₉	C₃	semidirect
4. C₅₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
5. C₅₃×#6868	C₃	semidirect
27. C₃×C₉×C₅₃	3₋¹⁺²	semidirect
28. #10624790	C₅₃	direct (N × H)
42. C₉⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
43. C₃×C₉⋊C₉	C₃×C₅₃	semidirect
44. H(3,𝔽₃)⋊C₉	C₃×C₅₃	semidirect
45. #6868	C₃×C₅₃	semidirect
61. C₅₃	#10624790	direct (N × H)
74. 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
C₃×C₅₃×#732	3
C₅₃×C₉⋊H(3,𝔽₃)	3
C₃×C₅₃×C₉⋊C₉	3
C₅₃×H(3,𝔽₃)⋊C₉	3
C₅₃×#6868	3
#10624790	53

Subgroups — C53×#10624790 #50311675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624790 #50311675