Subgroups — C₅₃×#10624483 #50311368

Order 38637.

132 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
2	C₅₃×#6868	C₃	1	1	yes	maximal, characteristic
3	C₅₃×#6855	C₃	1	1	yes	maximal, characteristic
4	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
5	C₃²×C₉×C₅₃	—	3	1	no	—
6	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
7	C₅₃×#733	—	3	6	no	—
8	C₅₃×C₃²⋊C₉	—	3	1	no	—
9	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
10	3₋¹⁺²×C₅₃	—	3	6	no	—
11	3₋¹⁺²×C₅₃	—	9	2	no	—
12	3₋¹⁺²×C₅₃	—	9	2	no	—
13	C₃×C₉×C₅₃	—	3	3	no	—
14	C₃×C₉×C₅₃	—	3	2	no	—
15	C₃×C₉×C₅₃	—	3	1	no	—
16	C₃×C₉×C₅₃	—	3	1	no	—
17	C₃×C₉×C₅₃	—	9	2	no	—
18	C₃×C₉×C₅₃	—	9	1	no	—
19	C₃×C₉×C₅₃	—	9	2	no	—
20	C₃×C₉×C₅₃	—	9	1	no	—
21	#10624483	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₅₃	—	3	1	no	—
23	C₃²×C₅₃	—	3	2	no	—
24	C₃²×C₅₃	AΣL(1,𝔽₂₇)	1	2	yes	—
25	C₃²×C₅₃	#732	1	1	yes	characteristic
26	C₃²×C₅₃	#732	1	1	yes	characteristic
27	C₉×C₅₃	—	9	3	no	—
28	C₉×C₅₃	—	9	1	no	—
29	C₉×C₅₃	—	9	6	no	—
30	C₉×C₅₃	—	27	1	no	—
31	C₃×#733	C₃×C₅₃	1	2	yes	—
32	#6868	C₃×C₅₃	1	1	yes	characteristic
33	#6855	C₃×C₅₃	1	1	yes	characteristic
34	C₃×C₅₃	—	3	1	no	—
35	C₃×C₅₃	—	3	2	no	—
36	C₃×C₅₃	#6856	1	1	yes	characteristic
37	C₃×C₅₃	#6878	1	1	yes	characteristic
38	C₃×C₅₃	#6882	1	2	yes	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
42	#733	—	3	6	no	—
43	C₃²⋊C₉	—	3	1	no	—
44	C₅₃	#10624483	1	1	yes	characteristic, direct product factor, Sylow
45	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
46	3₋¹⁺²	—	3	6	no	—
47	3₋¹⁺²	—	9	2	no	—
48	3₋¹⁺²	—	9	2	no	—
49	C₃×C₉	—	3	3	no	—
50	C₃×C₉	—	3	2	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	3	1	no	—
53	C₃×C₉	—	9	2	no	—
54	C₃×C₉	—	9	1	no	—
55	C₃×C₉	—	9	2	no	—
56	C₃×C₉	—	9	1	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	2	no	—
59	C₃²	C₅₃×AΣL(1,𝔽₂₇)	1	2	yes	—
60	C₃²	C₅₃×#732	1	1	yes	characteristic
61	C₃²	C₅₃×#732	1	1	yes	characteristic
62	C₉	—	9	3	no	—
63	C₉	—	9	1	no	—
64	C₉	—	9	6	no	—
65	C₉	—	27	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	C₅₃×#6856	1	1	yes	characteristic
69	C₃	C₅₃×#6878	1	1	yes	characteristic
70	C₃	C₅₃×#6882	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
21. #10624483	C₅₃	direct (N × H)
44. C₅₃	#10624483	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×#6868	3
C₅₃×#6855	3
#10624483	53

Subgroups — C53×#10624483 #50311368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624483 #50311368