Subgroups — C₅₃×#10624463 #50311348

Order 38637.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇×C₅₃	C₃	1	3	yes	maximal
2	C₃×C₉²×C₅₃	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉×C₅₃	C₉	1	3	yes	—
4	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
5	C₃×C₂₇×C₅₃	—	3	3	no	—
6	C₃×C₂₇×C₅₃	—	3	6	no	—
7	C₃×C₂₇×C₅₃	C₉	1	9	yes	complemented
8	C₉²×C₅₃	—	3	1	no	—
9	C₉²×C₅₃	—	3	2	no	—
10	C₃³×C₅₃	C₃×C₉	1	1	yes	characteristic
11	C₃×C₉×C₅₃	—	3	3	no	—
12	C₃×C₉×C₅₃	—	3	6	no	—
13	C₃×C₉×C₅₃	C₂₇	1	9	yes	complemented
14	C₃×C₉×C₅₃	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₅₃	3₋¹⁺²	1	6	yes	—
17	C₃×C₉×C₅₃	3₋¹⁺²	1	2	yes	—
18	C₂₇×C₅₃	—	3	9	no	—
19	#10624463	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₅₃	C₉²	1	1	yes	characteristic
21	C₃²×C₅₃	C₃²⋊C₉	1	1	yes	characteristic
22	C₃²×C₅₃	C₉⋊C₉	1	2	yes	—
23	C₃²×C₅₃	C₃×C₂₇	1	3	yes	—
24	C₃²×C₅₃	C₂₇⋊C₃	1	6	yes	—
25	C₉×C₅₃	—	3	9	no	—
26	C₉×C₅₃	C₃²⋊C₉	1	3	yes	—
27	C₉×C₅₃	C₉⋊C₉	1	6	yes	—
28	C₃²×C₂₇	C₃×C₅₃	1	3	yes	—
29	C₃×C₉²	C₃×C₅₃	1	1	yes	characteristic
30	C₃×C₅₃	#6855	1	1	yes	characteristic
31	C₃×C₅₃	C₉×C₂₇	1	1	yes	characteristic
32	C₃×C₅₃	#6864	1	2	yes	—
33	C₃×C₅₃	C₃²⋊C₂₇	1	3	yes	—
34	C₃×C₅₃	C₉⋊C₂₇	1	6	yes	—
35	C₃²×C₉	C₉×C₅₃	1	3	yes	—
36	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
37	C₃×C₂₇	—	3	3	no	—
38	C₃×C₂₇	—	3	6	no	—
39	C₃×C₂₇	C₉×C₅₃	1	9	yes	complemented
40	C₉²	—	3	1	no	—
41	C₉²	—	3	2	no	—
42	C₅₃	#10624463	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	C₃×C₉×C₅₃	1	1	yes	characteristic
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	3	6	no	—
46	C₃×C₉	C₂₇×C₅₃	1	9	yes	complemented
47	C₃×C₉	C₃×C₉×C₅₃	1	3	yes	—
48	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
49	C₃×C₉	3₋¹⁺²×C₅₃	1	6	yes	—
50	C₃×C₉	3₋¹⁺²×C₅₃	1	2	yes	—
51	C₂₇	—	3	9	no	—
52	C₃²	C₉²×C₅₃	1	1	yes	characteristic
53	C₃²	C₅₃×C₃²⋊C₉	1	1	yes	characteristic
54	C₃²	C₅₃×C₉⋊C₉	1	2	yes	—
55	C₃²	C₃×C₂₇×C₅₃	1	3	yes	—
56	C₃²	C₅₃×C₂₇⋊C₃	1	6	yes	—
57	C₉	—	3	9	no	—
58	C₉	C₅₃×C₃²⋊C₉	1	3	yes	—
59	C₉	C₅₃×C₉⋊C₉	1	6	yes	—
60	C₃	C₅₃×#6855	1	1	yes	characteristic
61	C₃	C₉×C₂₇×C₅₃	1	1	yes	characteristic
62	C₃	C₅₃×#6864	1	2	yes	—
63	C₃	C₅₃×C₃²⋊C₂₇	1	3	yes	—
64	C₃	C₅₃×C₉⋊C₂₇	1	6	yes	—

	subgroup	G/H
center	C₃²×C₉×C₅₃	C₃²
commutator	C₃	C₉×C₂₇×C₅₃
Frattini, cosocle	C₃²×C₉	C₃²×C₅₃
socle, prime radical	C₃³×C₅₃	C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃×C₂₇×C₅₃	C₉	semidirect
13. C₃×C₉×C₅₃	C₂₇	semidirect
19. #10624463	C₅₃	direct (N × H)
39. C₃×C₂₇	C₉×C₅₃	semidirect
42. C₅₃	#10624463	direct (N × H)
46. C₃×C₉	C₂₇×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇×C₅₃	3
C₃×C₉²×C₅₃	3
#10624463	53

Subgroups — C53×#10624463 #50311348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624463 #50311348