Subgroups — C₅₃×#10624735 #50311620

Order 38637.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅₃×AΣL(1,𝔽₂₇)	C₃	1	9	yes	maximal, complemented
3	C₅₃×#6891	C₃	1	2	yes	maximal, complemented
4	C₅₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃⁴×C₅₃	C₃²	1	3	yes	—
6	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₅₃	C₃²	1	6	yes	—
8	C₃×H(3,𝔽₃)×C₅₃	—	3	3	no	—
9	C₃×H(3,𝔽₃)×C₅₃	—	3	9	no	—
10	C₃×H(3,𝔽₃)×C₅₃	—	3	3	no	—
11	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
12	C₅₃×AΣL(1,𝔽₂₇)	—	3	27	no	—
13	C₅₃×C₃²⋊C₉	—	3	6	no	—
14	C₃³×C₅₃	—	3	6	no	—
15	C₃³×C₅₃	—	3	3	no	—
16	C₃³×C₅₃	—	3	3	no	—
17	C₃³×C₅₃	—	3	3	no	—
18	C₃³×C₅₃	—	9	3	no	—
19	C₃³×C₅₃	—	9	9	no	—
20	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
21	C₃³×C₅₃	H(3,𝔽₃)	1	9	yes	complemented
22	C₃³×C₅₃	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₅₃	—	3	18	no	—
24	H(3,𝔽₃)×C₅₃	—	3	27	no	—
25	H(3,𝔽₃)×C₅₃	—	3	9	no	—
26	H(3,𝔽₃)×C₅₃	—	9	9	no	—
27	C₃×C₉×C₅₃	—	3	6	no	—
28	#10624735	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	3	3	no	—
30	C₃²×C₅₃	—	3	3	no	—
31	C₃²×C₅₃	—	3	9	no	—
32	C₃²×C₅₃	—	3	3	no	—
33	C₃²×C₅₃	—	3	6	no	—
34	C₃²×C₅₃	—	9	9	no	—
35	C₃²×C₅₃	—	9	9	no	—
36	C₃²×C₅₃	—	9	18	no	—
37	C₃²×C₅₃	—	9	9	no	—
38	C₃²×C₅₃	—	9	3	no	—
39	C₃²×C₅₃	—	9	9	no	—
40	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	3	yes	—
41	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₉×C₅₃	—	9	6	no	—
43	C₃²×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic, complemented
44	C₃×AΣL(1,𝔽₂₇)	C₃×C₅₃	1	9	yes	complemented
45	#6891	C₃×C₅₃	1	2	yes	complemented
46	C₃²⋊H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic, complemented
47	C₃×C₅₃	—	3	3	no	—
48	C₃×C₅₃	—	9	3	no	—
49	C₃×C₅₃	—	9	9	no	—
50	C₃×C₅₃	—	27	3	no	—
51	C₃×C₅₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₅₃	#6909	1	3	yes	—
53	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
54	C₃⁴	C₃²×C₅₃	1	3	yes	—
55	C₃×3₋¹⁺²	C₃²×C₅₃	1	6	yes	—
56	C₃×H(3,𝔽₃)	—	3	3	no	—
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	—	3	3	no	—
59	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
60	AΣL(1,𝔽₂₇)	—	3	27	no	—
61	C₃²⋊C₉	—	3	6	no	—
62	C₅₃	#10624735	1	1	yes	characteristic, direct product factor, Sylow
63	C₃³	—	3	3	no	—
64	C₃³	—	3	6	no	—
65	C₃³	—	3	3	no	—
66	C₃³	—	3	3	no	—
67	C₃³	—	9	3	no	—
68	C₃³	—	9	9	no	—
69	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
70	C₃³	H(3,𝔽₃)×C₅₃	1	9	yes	complemented
71	C₃³	C₃³×C₅₃	1	1	yes	characteristic
72	3₋¹⁺²	—	3	18	no	—
73	H(3,𝔽₃)	—	3	27	no	—
74	H(3,𝔽₃)	—	3	9	no	—
75	H(3,𝔽₃)	—	9	9	no	—
76	C₃×C₉	—	3	6	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	9	no	—
80	C₃²	—	3	3	no	—
81	C₃²	—	3	6	no	—
82	C₃²	—	9	9	no	—
83	C₃²	—	9	9	no	—
84	C₃²	—	9	18	no	—
85	C₃²	—	9	9	no	—
86	C₃²	—	9	3	no	—
87	C₃²	—	9	9	no	—
88	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	3	yes	—
89	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
90	C₉	—	9	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	3	no	—
93	C₃	—	9	9	no	—
94	C₃	—	27	3	no	—
95	C₃	C₅₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₃	C₅₃×#6909	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×H(3,𝔽₃)×C₅₃	C₃	semidirect
2. C₃×C₅₃×AΣL(1,𝔽₂₇)	C₃	semidirect
3. C₅₃×#6891	C₃	semidirect
4. C₅₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
21. C₃³×C₅₃	H(3,𝔽₃)	semidirect
28. #10624735	C₅₃	direct (N × H)
43. C₃²×H(3,𝔽₃)	C₃×C₅₃	semidirect
44. C₃×AΣL(1,𝔽₂₇)	C₃×C₅₃	semidirect
45. #6891	C₃×C₅₃	semidirect
46. C₃²⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
62. C₅₃	#10624735	direct (N × H)
70. C₃³	H(3,𝔽₃)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×H(3,𝔽₃)×C₅₃	3
C₃×C₅₃×AΣL(1,𝔽₂₇)	3
C₅₃×#6891	3
C₅₃×C₃²⋊H(3,𝔽₃)	3
#10624735	53

Subgroups — C53×#10624735 #50311620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624735 #50311620