Subgroups — C₅₃×#10624828 #50311713

Order 38637.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6875	C₃	1	9	yes	maximal, complemented
3	C₅₃×#6869	C₃	1	3	yes	maximal
4	C₃²×C₉×C₅₃	—	3	1	no	—
5	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
6	C₅₃×C₂₇⋊C₃	C₃²	1	9	yes	—
7	C₅₃×C₉⋊C₉	—	3	2	no	—
8	C₅₃×C₉⋊C₉	C₃²	1	3	yes	—
9	C₃³×C₅₃	3₋¹⁺²	1	1	yes	characteristic
10	C₃×C₉×C₅₃	—	3	3	no	—
11	C₃×C₉×C₅₃	—	3	3	no	—
12	C₃×C₉×C₅₃	—	9	3	no	—
13	C₃×C₉×C₅₃	3₋¹⁺²	1	2	yes	—
14	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
15	C₂₇×C₅₃	—	3	9	no	—
16	#10624828	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃²×C₅₃	—	3	3	no	—
18	C₃²×C₅₃	—	3	1	no	—
19	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
20	C₉×C₅₃	—	3	2	no	—
21	C₉×C₅₃	—	9	3	no	—
22	C₉×C₅₃	#737	1	3	yes	—
23	C₃×C₉⋊C₉	C₃×C₅₃	1	1	yes	characteristic
24	#6875	C₃×C₅₃	1	9	yes	complemented
25	#6869	C₃×C₅₃	1	3	yes	—
26	C₃×C₅₃	—	3	1	no	—
27	C₃×C₅₃	—	3	3	no	—
28	C₃×C₅₃	#6895	1	1	yes	characteristic
29	C₃²×C₉	—	3	1	no	—
30	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
31	C₂₇⋊C₃	C₃²×C₅₃	1	9	yes	—
32	C₉⋊C₉	—	3	2	no	—
33	C₉⋊C₉	C₃²×C₅₃	1	3	yes	—
34	C₅₃	#10624828	1	1	yes	characteristic, direct product factor, Sylow
35	C₃³	3₋¹⁺²×C₅₃	1	1	yes	characteristic
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	3	3	no	—
38	C₃×C₉	—	9	3	no	—
39	C₃×C₉	3₋¹⁺²×C₅₃	1	2	yes	—
40	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
41	C₂₇	—	3	9	no	—
42	C₃²	—	3	3	no	—
43	C₃²	—	3	1	no	—
44	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
45	C₉	—	3	2	no	—
46	C₉	—	9	3	no	—
47	C₉	C₅₃×#737	1	3	yes	—
48	C₃	—	3	1	no	—
49	C₃	—	3	3	no	—
50	C₃	C₅₃×#6895	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₅₃	#6895
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₅₃
prime radical	C₃³×C₅₃	3₋¹⁺²
LCS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ C₃×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#6875	C₃	semidirect
16. #10624828	C₅₃	direct (N × H)
24. #6875	C₃×C₅₃	semidirect
34. C₅₃	#10624828	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×C₉⋊C₉	3
C₅₃×#6875	3
C₅₃×#6869	3
#10624828	53

Subgroups — C53×#10624828 #50311713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624828 #50311713