Subgroups — C₅₃×#10624830 #50311715

Order 38637.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6887	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	C₃²	1	1	yes	characteristic
5	C₅₃×C₂₇⋊C₃	C₃²	1	9	yes	—
6	C₅₃×C₉⋊C₉	—	3	1	no	—
7	C₅₃×C₉⋊C₉	C₃²	1	3	yes	—
8	C₅₃×C₃²⋊C₉	—	3	1	no	—
9	C₉²×C₅₃	—	3	1	no	—
10	C₃³×C₅₃	3₋¹⁺²	1	1	yes	characteristic
11	C₃×C₉×C₅₃	—	3	3	no	—
12	C₃×C₉×C₅₃	—	3	3	no	—
13	C₃×C₉×C₅₃	3₋¹⁺²	1	1	yes	characteristic
14	C₃×C₉×C₅₃	3₋¹⁺²	1	1	yes	characteristic
15	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
16	C₂₇×C₅₃	—	3	9	no	—
17	#10624830	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₃²×C₅₃	—	3	1	no	—
19	C₃²×C₅₃	—	9	1	no	—
20	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
21	C₉×C₅₃	—	3	1	no	—
22	C₉×C₅₃	—	3	1	no	—
23	C₉×C₅₃	—	9	3	no	—
24	C₉×C₅₃	#737	1	3	yes	—
25	#6887	C₃×C₅₃	1	1	yes	characteristic
26	#6875	C₃×C₅₃	1	9	yes	complemented
27	#6869	C₃×C₅₃	1	3	yes	—
28	C₃×C₅₃	—	3	1	no	—
29	C₃×C₅₃	—	9	1	no	—
30	C₃×C₅₃	#6895	1	1	yes	characteristic
31	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
32	C₂₇⋊C₃	C₃²×C₅₃	1	9	yes	—
33	C₉⋊C₉	—	3	1	no	—
34	C₉⋊C₉	C₃²×C₅₃	1	3	yes	—
35	C₃²⋊C₉	—	3	1	no	—
36	C₉²	—	3	1	no	—
37	C₅₃	#10624830	1	1	yes	characteristic, direct product factor, Sylow
38	C₃³	3₋¹⁺²×C₅₃	1	1	yes	characteristic
39	C₃×C₉	—	3	3	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	3₋¹⁺²×C₅₃	1	1	yes	characteristic
42	C₃×C₉	3₋¹⁺²×C₅₃	1	1	yes	characteristic
43	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
44	C₂₇	—	3	9	no	—
45	C₃²	—	3	1	no	—
46	C₃²	—	9	1	no	—
47	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	3	1	no	—
50	C₉	—	9	3	no	—
51	C₉	C₅₃×#737	1	3	yes	—
52	C₃	—	3	1	no	—
53	C₃	—	9	1	no	—
54	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
17. #10624830	C₅₃	direct (N × H)
26. #6875	C₃×C₅₃	semidirect
37. C₅₃	#10624830	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#6887	3
C₅₃×#6875	3
C₅₃×#6869	3
#10624830	53

Subgroups — C53×#10624830 #50311715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624830 #50311715