Subgroups — C₃×C₅₃×C₈₁⋊C₃ #50311717

Order 38637.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₅₃×C₈₁⋊C₃	C₃	1	9	yes	maximal, direct product factor
3	C₃×C₅₃×C₈₁	C₃	1	3	yes	maximal, complemented
4	C₃²×C₉×C₅₃	C₉	1	1	yes	characteristic
5	C₃×C₂₇×C₅₃	C₉	1	2	yes	—
6	C₃×C₂₇×C₅₃	C₉	1	6	yes	—
7	C₃×C₂₇×C₅₃	C₃²	1	3	yes	—
8	C₃×C₂₇×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×C₈₁	C₃²	1	9	yes	complemented
10	C₃³×C₅₃	C₂₇	1	1	yes	characteristic
11	C₃×C₉×C₅₃	C₂₇	1	6	yes	—
12	C₃×C₉×C₅₃	C₂₇	1	2	yes	—
13	C₃×C₉×C₅₃	C₃×C₉	1	3	yes	—
14	C₃×C₉×C₅₃	C₃×C₉	1	1	yes	characteristic
15	C₂₇×C₅₃	C₃×C₉	1	6	yes	—
16	C₂₇×C₅₃	C₃×C₉	1	2	yes	—
17	C₂₇×C₅₃	C₃³	1	1	yes	characteristic
18	C₃×C₈₁⋊C₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃²×C₅₃	—	3	3	no	—
20	C₃²×C₅₃	C₃×C₂₇	1	3	yes	—
21	C₃²×C₅₃	C₃×C₂₇	1	1	yes	characteristic
22	C₉×C₅₃	C₃×C₂₇	1	6	yes	—
23	C₉×C₅₃	C₃×C₂₇	1	2	yes	—
24	C₉×C₅₃	C₃²×C₉	1	1	yes	characteristic
25	C₃²×C₂₇	C₃×C₅₃	1	1	yes	characteristic
26	C₈₁⋊C₃	C₃×C₅₃	1	9	yes	direct product factor
27	C₃×C₈₁	C₃×C₅₃	1	3	yes	complemented
28	C₃×C₅₃	—	3	3	no	—
29	C₃×C₅₃	C₈₁⋊C₃	1	3	yes	direct product factor
30	C₃×C₅₃	C₃²×C₂₇	1	1	yes	characteristic
31	C₃²×C₉	C₉×C₅₃	1	1	yes	characteristic
32	C₃×C₂₇	C₉×C₅₃	1	2	yes	—
33	C₃×C₂₇	C₉×C₅₃	1	6	yes	—
34	C₃×C₂₇	C₃²×C₅₃	1	3	yes	—
35	C₃×C₂₇	C₃²×C₅₃	1	1	yes	characteristic
36	C₈₁	C₃²×C₅₃	1	9	yes	complemented
37	C₅₃	C₃×C₈₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
38	C₃³	C₂₇×C₅₃	1	1	yes	characteristic
39	C₃×C₉	C₂₇×C₅₃	1	6	yes	—
40	C₃×C₉	C₂₇×C₅₃	1	2	yes	—
41	C₃×C₉	C₃×C₉×C₅₃	1	3	yes	—
42	C₃×C₉	C₃×C₉×C₅₃	1	1	yes	characteristic
43	C₂₇	C₃×C₉×C₅₃	1	6	yes	—
44	C₂₇	C₃×C₉×C₅₃	1	2	yes	—
45	C₂₇	C₃³×C₅₃	1	1	yes	characteristic
46	C₃²	—	3	3	no	—
47	C₃²	C₃×C₂₇×C₅₃	1	3	yes	—
48	C₃²	C₃×C₂₇×C₅₃	1	1	yes	characteristic
49	C₉	C₃×C₂₇×C₅₃	1	6	yes	—
50	C₉	C₃×C₂₇×C₅₃	1	2	yes	—
51	C₉	C₃²×C₉×C₅₃	1	1	yes	characteristic
52	C₃	—	3	3	no	—
53	C₃	C₅₃×C₈₁⋊C₃	1	3	yes	direct product factor
54	C₃	C₃²×C₂₇×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×C₈₁⋊C₃	C₃	direct (N × H)
3. C₃×C₅₃×C₈₁	C₃	semidirect
9. C₅₃×C₈₁	C₃²	semidirect
18. C₃×C₈₁⋊C₃	C₅₃	direct (N × H)
26. C₈₁⋊C₃	C₃×C₅₃	direct (N × H)
27. C₃×C₈₁	C₃×C₅₃	semidirect
29. C₃×C₅₃	C₈₁⋊C₃	direct (N × H)
36. C₈₁	C₃²×C₅₃	semidirect
37. C₅₃	C₃×C₈₁⋊C₃	direct (N × H)
53. C₃	C₅₃×C₈₁⋊C₃	direct (N × H)

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
C₅₃×C₈₁⋊C₃	3
C₃×C₅₃×C₈₁	3
C₃×C₈₁⋊C₃	53

Subgroups — C3×C53×C81⋊C3 #50311717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×C₈₁⋊C₃ #50311717