Subgroups — C₅₃×#10624837 #50311722

Order 38637.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6908	C₃	1	3	yes	maximal, complemented
2	C₃×C₉²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#6880	C₃	1	2	yes	maximal, complemented
4	C₅₃×#6879	C₃	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#6878	C₃	1	6	yes	maximal, complemented
6	C₅₃×#737	—	3	3	no	—
7	C₃²×C₉×C₅₃	—	3	1	no	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×#733	C₃²	1	6	yes	—
10	C₅₃×#732	C₃²	1	3	yes	—
11	C₅₃×#731	—	3	3	no	—
12	C₅₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
13	C₉²×C₅₃	—	3	1	no	—
14	C₉²×C₅₃	—	3	1	no	—
15	C₉²×C₅₃	C₃²	1	2	yes	—
16	C₉²×C₅₃	C₃²	1	1	yes	characteristic
17	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
18	3₋¹⁺²×C₅₃	—	3	6	no	—
19	3₋¹⁺²×C₅₃	—	9	6	no	—
20	H(3,𝔽₃)×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	—	3	9	no	—
22	C₃×C₉×C₅₃	—	3	2	no	—
23	C₃×C₉×C₅₃	—	3	1	no	—
24	C₃×C₉×C₅₃	—	3	2	no	—
25	C₃×C₉×C₅₃	—	3	1	no	—
26	C₃×C₉×C₅₃	—	9	3	no	—
27	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
30	#10624837	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₃²×C₅₃	—	3	3	no	—
32	C₃²×C₅₃	—	3	1	no	—
33	C₃²×C₅₃	—	9	3	no	—
34	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₉×C₅₃	—	3	6	no	—
36	C₉×C₅₃	—	3	3	no	—
37	C₉×C₅₃	—	3	1	no	—
38	C₉×C₅₃	—	3	1	no	—
39	C₉×C₅₃	—	9	6	no	—
40	C₉×C₅₃	#732	1	2	yes	—
41	C₉×C₅₃	#732	1	1	yes	characteristic
42	#6908	C₃×C₅₃	1	3	yes	complemented
43	C₃×C₉²	C₃×C₅₃	1	1	yes	characteristic, complemented
44	#6880	C₃×C₅₃	1	2	yes	complemented
45	#6879	C₃×C₅₃	1	1	yes	characteristic, complemented
46	#6878	C₃×C₅₃	1	6	yes	complemented
47	C₃×C₅₃	—	3	1	no	—
48	C₃×C₅₃	—	3	3	no	—
49	C₃×C₅₃	—	27	3	no	—
50	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
51	#737	—	3	3	no	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
54	#733	C₃²×C₅₃	1	6	yes	—
55	#732	C₃²×C₅₃	1	3	yes	—
56	#731	—	3	3	no	—
57	AΣL(1,𝔽₂₇)	—	3	3	no	—
58	C₉²	—	3	1	no	—
59	C₉²	—	3	1	no	—
60	C₉²	C₃²×C₅₃	1	2	yes	—
61	C₉²	C₃²×C₅₃	1	1	yes	characteristic
62	C₅₃	#10624837	1	1	yes	characteristic, direct product factor, Sylow
63	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
64	3₋¹⁺²	—	3	6	no	—
65	3₋¹⁺²	—	9	6	no	—
66	H(3,𝔽₃)	—	3	3	no	—
67	C₃×C₉	—	3	9	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	2	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	9	3	no	—
73	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
74	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
75	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
76	C₃²	—	3	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	9	3	no	—
79	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
80	C₉	—	3	6	no	—
81	C₉	—	3	3	no	—
82	C₉	—	3	1	no	—
83	C₉	—	3	1	no	—
84	C₉	—	9	6	no	—
85	C₉	C₅₃×#732	1	2	yes	—
86	C₉	C₅₃×#732	1	1	yes	characteristic
87	C₃	—	3	1	no	—
88	C₃	—	3	3	no	—
89	C₃	—	27	3	no	—
90	C₃	C₃×C₅₃×#732	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#6908	C₃	semidirect
2. C₃×C₉²×C₅₃	C₃	semidirect
3. C₅₃×#6880	C₃	semidirect
4. C₅₃×#6879	C₃	semidirect
5. C₅₃×#6878	C₃	semidirect
30. #10624837	C₅₃	direct (N × H)
42. #6908	C₃×C₅₃	semidirect
43. C₃×C₉²	C₃×C₅₃	semidirect
44. #6880	C₃×C₅₃	semidirect
45. #6879	C₃×C₅₃	semidirect
46. #6878	C₃×C₅₃	semidirect
62. C₅₃	#10624837	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#6908	3
C₃×C₉²×C₅₃	3
C₅₃×#6880	3
C₅₃×#6879	3
C₅₃×#6878	3
#10624837	53

Subgroups — C53×#10624837 #50311722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624837 #50311722