Subgroups — C₅₃×#10624847 #50311732

Order 38637.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#733	C₃	1	3	yes	maximal
2	C₅₃×#6887	C₃	1	1	yes	maximal, characteristic
3	C₅₃×#6882	C₃	1	9	yes	maximal, complemented
4	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
5	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
6	C₅₃×#733	—	3	3	no	—
7	C₅₃×#733	—	3	3	no	—
8	C₅₃×#733	C₃²	1	9	yes	—
9	C₅₃×C₉⋊C₉	—	3	1	no	—
10	C₅₃×C₉⋊C₉	C₃²	1	3	yes	—
11	C₅₃×C₃²⋊C₉	—	3	1	no	—
12	C₉²×C₅₃	—	3	1	no	—
13	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
14	3₋¹⁺²×C₅₃	—	3	9	no	—
15	3₋¹⁺²×C₅₃	—	9	6	no	—
16	C₃×C₉×C₅₃	—	3	3	no	—
17	C₃×C₉×C₅₃	—	3	3	no	—
18	C₃×C₉×C₅₃	—	9	3	no	—
19	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
22	#10624847	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₅₃	—	9	1	no	—
24	C₃²×C₅₃	#732	1	3	yes	—
25	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₉×C₅₃	—	3	1	no	—
27	C₉×C₅₃	—	3	1	no	—
28	C₉×C₅₃	—	3	1	no	—
29	C₉×C₅₃	—	9	3	no	—
30	C₉×C₅₃	—	9	9	no	—
31	C₃×#733	C₃×C₅₃	1	3	yes	—
32	#6887	C₃×C₅₃	1	1	yes	characteristic
33	#6882	C₃×C₅₃	1	9	yes	complemented
34	C₃×C₅₃	—	3	1	no	—
35	C₃×C₅₃	—	3	3	no	—
36	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
39	#733	—	3	3	no	—
40	#733	—	3	3	no	—
41	#733	C₃²×C₅₃	1	9	yes	—
42	C₉⋊C₉	—	3	1	no	—
43	C₉⋊C₉	C₃²×C₅₃	1	3	yes	—
44	C₃²⋊C₉	—	3	1	no	—
45	C₉²	—	3	1	no	—
46	C₅₃	#10624847	1	1	yes	characteristic, direct product factor, Sylow
47	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
48	3₋¹⁺²	—	3	9	no	—
49	3₋¹⁺²	—	9	6	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	3	3	no	—
52	C₃×C₉	—	9	3	no	—
53	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
54	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
55	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
56	C₃²	—	9	1	no	—
57	C₃²	C₅₃×#732	1	3	yes	—
58	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
59	C₉	—	3	1	no	—
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	9	3	no	—
63	C₉	—	9	9	no	—
64	C₃	—	3	1	no	—
65	C₃	—	3	3	no	—
66	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	C₃³×C₅₃	H(3,𝔽₃)
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
3. C₅₃×#6882	C₃	semidirect
22. #10624847	C₅₃	direct (N × H)
33. #6882	C₃×C₅₃	semidirect
46. C₅₃	#10624847	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
C₅₃×#6887	3
C₅₃×#6882	3
#10624847	53

Subgroups — C53×#10624847 #50311732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624847 #50311732