Subgroups — C₅₃×#10624683 #50311568

Order 38637.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₃×C₅₃×C₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	C₅₃×C₃²⋊C₂₇	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₅₃	C₉	1	1	yes	characteristic
5	C₃²×C₉×C₅₃	—	3	3	no	—
6	C₃²×C₉×C₅₃	—	3	6	no	—
7	C₃²×C₉×C₅₃	C₉	1	2	yes	—
8	C₃²×C₉×C₅₃	C₉	1	6	yes	—
9	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
10	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
11	C₅₃×C₂₇⋊C₃	—	3	9	no	—
12	C₃×C₂₇×C₅₃	C₃²	1	9	yes	complemented
13	C₃³×C₅₃	—	3	3	no	—
14	C₃³×C₅₃	—	3	9	no	—
15	C₃³×C₅₃	C₃×C₉	1	1	yes	characteristic
16	C₃³×C₅₃	C₃×C₉	1	3	yes	—
17	C₃×C₉×C₅₃	—	3	6	no	—
18	C₃×C₉×C₅₃	—	3	18	no	—
19	C₃×C₉×C₅₃	—	3	3	no	—
20	C₃×C₉×C₅₃	—	3	6	no	—
21	C₃×C₉×C₅₃	C₃×C₉	1	6	yes	—
22	C₃×C₉×C₅₃	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
24	C₃×C₉×C₅₃	3₋¹⁺²	1	6	yes	—
25	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
26	C₂₇×C₅₃	—	3	9	no	—
27	#10624683	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₅₃	—	3	27	no	—
29	C₃²×C₅₃	—	3	3	no	—
30	C₃²×C₅₃	—	3	3	no	—
31	C₃²×C₅₃	—	3	6	no	—
32	C₃²×C₅₃	C₃²⋊C₉	1	3	yes	—
33	C₃²×C₅₃	C₂₇⋊C₃	1	9	yes	complemented
34	C₃²×C₅₃	C₃²×C₉	1	1	yes	characteristic
35	C₉×C₅₃	—	3	6	no	—
36	C₉×C₅₃	C₃²⋊C₉	1	6	yes	—
37	C₉×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₉×C₅₃	C₃×3₋¹⁺²	1	2	yes	—
39	C₃³×C₉	C₃×C₅₃	1	1	yes	characteristic
40	C₃×C₂₇⋊C₃	C₃×C₅₃	1	3	yes	complemented
41	C₃²⋊C₂₇	C₃×C₅₃	1	9	yes	complemented
42	C₃×C₅₃	—	3	3	no	—
43	C₃×C₅₃	—	3	9	no	—
44	C₃×C₅₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
45	C₃×C₅₃	C₃×C₂₇⋊C₃	1	3	yes	—
46	C₃⁴	C₉×C₅₃	1	1	yes	characteristic
47	C₃²×C₉	—	3	3	no	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	C₉×C₅₃	1	2	yes	—
50	C₃²×C₉	C₉×C₅₃	1	6	yes	—
51	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
52	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
53	C₂₇⋊C₃	—	3	9	no	—
54	C₃×C₂₇	C₃²×C₅₃	1	9	yes	complemented
55	C₅₃	#10624683	1	1	yes	characteristic, direct product factor, Sylow
56	C₃³	—	3	3	no	—
57	C₃³	—	3	9	no	—
58	C₃³	C₃×C₉×C₅₃	1	1	yes	characteristic
59	C₃³	C₃×C₉×C₅₃	1	3	yes	—
60	C₃×C₉	—	3	6	no	—
61	C₃×C₉	—	3	18	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	C₃×C₉×C₅₃	1	6	yes	—
65	C₃×C₉	C₃×C₉×C₅₃	1	2	yes	—
66	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
67	C₃×C₉	3₋¹⁺²×C₅₃	1	6	yes	—
68	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
69	C₂₇	—	3	9	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	6	no	—
74	C₃²	C₅₃×C₃²⋊C₉	1	3	yes	—
75	C₃²	C₅₃×C₂₇⋊C₃	1	9	yes	complemented
76	C₃²	C₃²×C₉×C₅₃	1	1	yes	characteristic
77	C₉	—	3	6	no	—
78	C₉	C₅₃×C₃²⋊C₉	1	6	yes	—
79	C₉	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
80	C₉	C₃×3₋¹⁺²×C₅₃	1	2	yes	—
81	C₃	—	3	3	no	—
82	C₃	—	3	9	no	—
83	C₃	C₃×C₅₃×C₃²⋊C₉	1	1	yes	characteristic
84	C₃	C₃×C₅₃×C₂₇⋊C₃	1	3	yes	—

	subgroup	G/H
center	C₃×C₉×C₅₃	C₃³
commutator	C₃²	C₃²×C₉×C₅₃
Frattini, cosocle	C₃×C₉	C₃³×C₅₃
socle	C₃²×C₅₃	C₃²×C₉
prime radical	C₃⁴×C₅₃	C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅₃×C₂₇⋊C₃	C₃	semidirect
3. C₅₃×C₃²⋊C₂₇	C₃	semidirect
12. C₃×C₂₇×C₅₃	C₃²	semidirect
27. #10624683	C₅₃	direct (N × H)
33. C₃²×C₅₃	C₂₇⋊C₃	semidirect
40. C₃×C₂₇⋊C₃	C₃×C₅₃	semidirect
41. C₃²⋊C₂₇	C₃×C₅₃	semidirect
54. C₃×C₂₇	C₃²×C₅₃	semidirect
55. C₅₃	#10624683	direct (N × H)
75. C₃²	C₅₃×C₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₅₃	3
C₃×C₅₃×C₂₇⋊C₃	3
C₅₃×C₃²⋊C₂₇	3
#10624683	53

Subgroups — C53×#10624683 #50311568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624683 #50311568