Subgroups — C₅₃×#10624840 #50311725

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×#6880	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	6	no	—
7	C₅₃×#733	C₃²	1	9	yes	—
8	C₅₃×C₉⋊C₉	—	3	2	no	—
9	C₅₃×C₃²⋊C₉	—	3	1	no	—
10	C₉²×C₅₃	C₃²	1	3	yes	—
11	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
12	3₋¹⁺²×C₅₃	—	3	9	no	—
13	3₋¹⁺²×C₅₃	—	9	6	no	—
14	C₃×C₉×C₅₃	—	3	3	no	—
15	C₃×C₉×C₅₃	—	3	3	no	—
16	C₃×C₉×C₅₃	—	9	3	no	—
17	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
18	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
19	#10624840	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₅₃	—	9	1	no	—
21	C₃²×C₅₃	#732	1	3	yes	—
22	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
23	C₉×C₅₃	—	3	2	no	—
24	C₉×C₅₃	—	3	1	no	—
25	C₉×C₅₃	—	9	3	no	—
26	C₉×C₅₃	—	9	9	no	—
27	C₃×#733	C₃×C₅₃	1	3	yes	—
28	#6887	C₃×C₅₃	1	1	yes	characteristic
29	#6880	C₃×C₅₃	1	9	yes	complemented
30	C₃×C₅₃	—	3	1	no	—
31	C₃×C₅₃	—	3	3	no	—
32	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
33	C₃×3₋¹⁺²	—	3	3	no	—
34	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
35	#733	—	3	6	no	—
36	#733	C₃²×C₅₃	1	9	yes	—
37	C₉⋊C₉	—	3	2	no	—
38	C₃²⋊C₉	—	3	1	no	—
39	C₉²	C₃²×C₅₃	1	3	yes	—
40	C₅₃	#10624840	1	1	yes	characteristic, direct product factor, Sylow
41	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
42	3₋¹⁺²	—	3	9	no	—
43	3₋¹⁺²	—	9	6	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	3	3	no	—
46	C₃×C₉	—	9	3	no	—
47	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
48	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
49	C₃²	—	9	1	no	—
50	C₃²	C₅₃×#732	1	3	yes	—
51	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
52	C₉	—	3	2	no	—
53	C₉	—	3	1	no	—
54	C₉	—	9	3	no	—
55	C₉	—	9	9	no	—
56	C₃	—	3	1	no	—
57	C₃	—	3	3	no	—
58	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×#6880	C₃	semidirect
19. #10624840	C₅₃	direct (N × H)
29. #6880	C₃×C₅₃	semidirect
40. C₅₃	#10624840	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×#6880	3
#10624840	53

Subgroups — C53×#10624840 #50311725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624840 #50311725