Subgroups — C₃×C₅₃×#6879 #50311720

Order 38637.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#732	C₃	1	3	yes	maximal, complemented
2	C₃×C₉²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#6879	C₃	1	9	yes	maximal, direct product factor
4	C₃×H(3,𝔽₃)×C₅₃	—	3	3	no	—
5	C₃²×C₉×C₅₃	—	3	1	no	—
6	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
7	C₅₃×#732	—	3	6	no	—
8	C₅₃×#732	C₃²	1	9	yes	complemented
9	C₉²×C₅₃	—	3	2	no	—
10	C₉²×C₅₃	C₃²	1	3	yes	complemented
11	C₃³×C₅₃	—	9	3	no	—
12	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
13	H(3,𝔽₃)×C₅₃	—	3	9	no	—
14	H(3,𝔽₃)×C₅₃	—	9	6	no	—
15	C₃×C₉×C₅₃	—	3	9	no	—
16	C₃×C₉×C₅₃	—	3	2	no	—
17	C₃×C₉×C₅₃	—	3	1	no	—
18	C₃×C₉×C₅₃	—	3	3	no	—
19	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
20	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
21	C₃×#6879	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₅₃	—	3	3	no	—
23	C₃²×C₅₃	—	9	9	no	—
24	C₃²×C₅₃	—	27	9	no	—
25	C₃²×C₅₃	#732	1	2	yes	—
26	C₃²×C₅₃	#732	1	1	yes	characteristic
27	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₉×C₅₃	—	3	9	no	—
29	C₉×C₅₃	—	3	2	no	—
30	C₉×C₅₃	—	3	1	no	—
31	C₃×#732	C₃×C₅₃	1	3	yes	complemented
32	C₃×C₉²	C₃×C₅₃	1	1	yes	characteristic, complemented
33	#6879	C₃×C₅₃	1	9	yes	direct product factor
34	C₃×C₅₃	—	3	1	no	—
35	C₃×C₅₃	—	3	2	no	—
36	C₃×C₅₃	—	27	9	no	—
37	C₃×C₅₃	#6879	1	3	yes	direct product factor
38	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
39	C₃×H(3,𝔽₃)	—	3	3	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
42	#732	—	3	6	no	—
43	#732	C₃²×C₅₃	1	9	yes	complemented
44	C₉²	—	3	2	no	—
45	C₉²	C₃²×C₅₃	1	3	yes	complemented
46	C₅₃	C₃×#6879	1	1	yes	characteristic, direct product factor, Sylow
47	C₃³	—	9	3	no	—
48	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
49	H(3,𝔽₃)	—	3	9	no	—
50	H(3,𝔽₃)	—	9	6	no	—
51	C₃×C₉	—	3	9	no	—
52	C₃×C₉	—	3	2	no	—
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
56	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
57	C₃²	—	3	3	no	—
58	C₃²	—	9	9	no	—
59	C₃²	—	27	9	no	—
60	C₃²	C₅₃×#732	1	2	yes	—
61	C₃²	C₅₃×#732	1	1	yes	characteristic
62	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
63	C₉	—	3	9	no	—
64	C₉	—	3	2	no	—
65	C₉	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	—	27	9	no	—
69	C₃	C₅₃×#6879	1	3	yes	direct product factor
70	C₃	C₃×C₅₃×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₅₃	#732
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₅₃
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×#732	C₃	semidirect
2. C₃×C₉²×C₅₃	C₃	semidirect
3. C₅₃×#6879	C₃	direct (N × H)
8. C₅₃×#732	C₃²	semidirect
10. C₉²×C₅₃	C₃²	semidirect
21. C₃×#6879	C₅₃	direct (N × H)
31. C₃×#732	C₃×C₅₃	semidirect
32. C₃×C₉²	C₃×C₅₃	semidirect
33. #6879	C₃×C₅₃	direct (N × H)
37. C₃×C₅₃	#6879	direct (N × H)
43. #732	C₃²×C₅₃	semidirect
45. C₉²	C₃²×C₅₃	semidirect
46. C₅₃	C₃×#6879	direct (N × H)
69. C₃	C₅₃×#6879	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×#732	3
C₃×C₉²×C₅₃	3
C₅₃×#6879	3
C₃×#6879	53

Subgroups — C3×C53×#6879 #50311720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×#6879 #50311720