Subgroups — C₃²×C₅₃×C₈₁ #50311716

Order 38637.

188 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	C₃×C₅₃×C₈₁	C₃	1	12	yes	maximal, direct product factor
3	C₃²×C₉×C₅₃	C₉	1	1	yes	characteristic
4	C₃×C₂₇×C₅₃	C₉	1	8	yes	—
5	C₃×C₂₇×C₅₃	C₃²	1	4	yes	—
6	C₅₃×C₈₁	C₃²	1	9	yes	direct product factor
7	C₃³×C₅₃	C₂₇	1	1	yes	characteristic
8	C₃×C₉×C₅₃	C₂₇	1	8	yes	—
9	C₃×C₉×C₅₃	C₃×C₉	1	4	yes	—
10	C₂₇×C₅₃	C₃×C₉	1	8	yes	—
11	C₂₇×C₅₃	C₃³	1	1	yes	characteristic
12	C₃²×C₈₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃²×C₅₃	C₈₁	1	9	yes	direct product factor
14	C₃²×C₅₃	C₃×C₂₇	1	4	yes	—
15	C₉×C₅₃	C₃×C₂₇	1	8	yes	—
16	C₉×C₅₃	C₃²×C₉	1	1	yes	characteristic
17	C₃²×C₂₇	C₃×C₅₃	1	1	yes	characteristic
18	C₃×C₈₁	C₃×C₅₃	1	12	yes	direct product factor
19	C₃×C₅₃	C₃×C₈₁	1	12	yes	direct product factor
20	C₃×C₅₃	C₃²×C₂₇	1	1	yes	characteristic
21	C₃²×C₉	C₉×C₅₃	1	1	yes	characteristic
22	C₃×C₂₇	C₉×C₅₃	1	8	yes	—
23	C₃×C₂₇	C₃²×C₅₃	1	4	yes	—
24	C₈₁	C₃²×C₅₃	1	9	yes	direct product factor
25	C₅₃	C₃²×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
26	C₃³	C₂₇×C₅₃	1	1	yes	characteristic
27	C₃×C₉	C₂₇×C₅₃	1	8	yes	—
28	C₃×C₉	C₃×C₉×C₅₃	1	4	yes	—
29	C₂₇	C₃×C₉×C₅₃	1	8	yes	—
30	C₂₇	C₃³×C₅₃	1	1	yes	characteristic
31	C₃²	C₅₃×C₈₁	1	9	yes	direct product factor
32	C₃²	C₃×C₂₇×C₅₃	1	4	yes	—
33	C₉	C₃×C₂₇×C₅₃	1	8	yes	—
34	C₉	C₃²×C₉×C₅₃	1	1	yes	characteristic
35	C₃	C₃×C₅₃×C₈₁	1	12	yes	direct product factor
36	C₃	C₃²×C₂₇×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅₃×C₈₁	C₃	direct (N × H)
6. C₅₃×C₈₁	C₃²	direct (N × H)
12. C₃²×C₈₁	C₅₃	direct (N × H)
13. C₃²×C₅₃	C₈₁	direct (N × H)
18. C₃×C₈₁	C₃×C₅₃	direct (N × H)
19. C₃×C₅₃	C₃×C₈₁	direct (N × H)
24. C₈₁	C₃²×C₅₃	direct (N × H)
25. C₅₃	C₃²×C₈₁	direct (N × H)
31. C₃²	C₅₃×C₈₁	direct (N × H)
35. C₃	C₃×C₅₃×C₈₁	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
C₃×C₅₃×C₈₁	3
C₃²×C₈₁	53

Subgroups — C32×C53×C81 #50311716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅₃×C₈₁ #50311716