Subgroups — C₅₃×#10624787 #50311672

Order 38637.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#733	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6894	C₃	1	3	yes	maximal
3	C₃×C₉²×C₅₃	C₃	1	1	yes	maximal, characteristic
4	C₅₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal
5	C₅₃×#6868	C₃	1	2	yes	maximal
6	C₃×3₋¹⁺²×C₅₃	C₃²	1	3	yes	—
7	C₃²×C₉×C₅₃	C₃²	1	2	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
10	C₅₃×#733	—	3	3	no	—
11	C₅₃×C₉⋊C₉	—	3	6	no	—
12	C₅₃×C₉⋊C₉	—	3	3	no	—
13	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₉²×C₅₃	—	3	2	no	—
15	C₉²×C₅₃	—	3	1	no	—
16	C₃³×C₅₃	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₅₃	—	3	3	no	—
18	3₋¹⁺²×C₅₃	—	3	6	no	—
19	C₃×C₉×C₅₃	—	3	1	no	—
20	C₃×C₉×C₅₃	—	3	2	no	—
21	C₃×C₉×C₅₃	—	3	2	no	—
22	C₃×C₉×C₅₃	—	3	3	no	—
23	C₃×C₉×C₅₃	—	3	6	no	—
24	C₃×C₉×C₅₃	—	3	6	no	—
25	C₃×C₉×C₅₃	—	3	3	no	—
26	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
27	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₉×C₅₃	3₋¹⁺²	1	3	yes	—
29	#10624787	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃²×C₅₃	—	3	1	no	—
31	C₃²×C₅₃	—	3	2	no	—
32	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₃²×C₅₃	#737	1	2	yes	—
35	C₉×C₅₃	—	3	3	no	—
36	C₉×C₅₃	—	3	2	no	—
37	C₉×C₅₃	—	3	6	no	—
38	C₉×C₅₃	—	9	3	no	—
39	C₉×C₅₃	—	9	6	no	—
40	C₉×C₅₃	#733	1	3	yes	complemented
41	C₃×#733	C₃×C₅₃	1	1	yes	characteristic
42	#6894	C₃×C₅₃	1	3	yes	—
43	C₃×C₉²	C₃×C₅₃	1	1	yes	characteristic
44	3₋¹⁺²⋊C₉	C₃×C₅₃	1	6	yes	—
45	#6868	C₃×C₅₃	1	2	yes	—
46	C₃×C₅₃	—	3	1	no	—
47	C₃×C₅₃	—	3	2	no	—
48	C₃×C₅₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₅₃	C₃×#733	1	1	yes	characteristic
50	C₃×C₅₃	#6908	1	2	yes	—
51	C₃×3₋¹⁺²	C₃²×C₅₃	1	3	yes	—
52	C₃²×C₉	C₃²×C₅₃	1	2	yes	—
53	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
54	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
55	#733	—	3	3	no	—
56	C₉⋊C₉	—	3	6	no	—
57	C₉⋊C₉	—	3	3	no	—
58	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
59	C₉²	—	3	2	no	—
60	C₉²	—	3	1	no	—
61	C₅₃	#10624787	1	1	yes	characteristic, direct product factor, Sylow
62	C₃³	C₃³×C₅₃	1	1	yes	characteristic
63	3₋¹⁺²	—	3	3	no	—
64	3₋¹⁺²	—	3	6	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	3	6	no	—
70	C₃×C₉	—	3	6	no	—
71	C₃×C₉	—	3	3	no	—
72	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
73	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
74	C₃×C₉	3₋¹⁺²×C₅₃	1	3	yes	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
78	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
79	C₃²	C₅₃×#737	1	2	yes	—
80	C₉	—	3	3	no	—
81	C₉	—	3	2	no	—
82	C₉	—	3	6	no	—
83	C₉	—	9	3	no	—
84	C₉	—	9	6	no	—
85	C₉	C₅₃×#733	1	3	yes	complemented
86	C₃	—	3	1	no	—
87	C₃	—	3	2	no	—
88	C₃	C₅₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
89	C₃	C₃×C₅₃×#733	1	1	yes	characteristic
90	C₃	C₅₃×#6908	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃³×C₅₃	C₃³
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
29. #10624787	C₅₃	direct (N × H)
40. C₉×C₅₃	#733	semidirect
61. C₅₃	#10624787	direct (N × H)
85. C₉	C₅₃×#733	semidirect

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
C₅₃×#6894	3
C₃×C₉²×C₅₃	3
C₅₃×3₋¹⁺²⋊C₉	3
C₅₃×#6868	3
#10624787	53

Subgroups — C53×#10624787 #50311672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624787 #50311672