Subgroups — C₂³×C₂₇×C₅₉ #72191069

Order 12744.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₇×C₅₉	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₉×C₅₉	C₃	1	1	yes	maximal, characteristic
3	C₂×C₂₇×C₅₉	C₂²	1	7	yes	direct product factor
4	C₂²×C₉×C₅₉	C₂×C₃	1	7	yes	—
5	C₂₇×C₅₉	C₂³	1	1	yes	characteristic, direct product factor, Hall
6	C₂³×C₃×C₅₉	C₉	1	1	yes	characteristic
7	C₂×C₉×C₅₉	C₂²×C₃	1	7	yes	—
8	C₂²×C₃×C₅₉	C₂×C₉	1	7	yes	—
9	C₉×C₅₉	C₂³×C₃	1	1	yes	characteristic
10	C₂³×C₅₉	C₂₇	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₅₉	C₂²×C₉	1	7	yes	—
12	C₂²×C₅₉	C₂×C₂₇	1	7	yes	direct product factor
13	C₂³×C₂₇	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×C₅₉	C₂³×C₉	1	1	yes	characteristic
15	C₂×C₅₉	C₂²×C₂₇	1	7	yes	direct product factor
16	C₂²×C₂₇	C₂×C₅₉	1	7	yes	direct product factor
17	C₂³×C₉	C₃×C₅₉	1	1	yes	characteristic
18	C₅₉	C₂³×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₂₇	C₂²×C₅₉	1	7	yes	direct product factor
20	C₂²×C₉	C₂×C₃×C₅₉	1	7	yes	—
21	C₂₇	C₂³×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	C₉×C₅₉	1	1	yes	characteristic
23	C₂×C₉	C₂²×C₃×C₅₉	1	7	yes	—
24	C₂²×C₃	C₂×C₉×C₅₉	1	7	yes	—
25	C₉	C₂³×C₃×C₅₉	1	1	yes	characteristic
26	C₂³	C₂₇×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃	C₂²×C₉×C₅₉	1	7	yes	—
28	C₂²	C₂×C₂₇×C₅₉	1	7	yes	direct product factor
29	C₃	C₂³×C₉×C₅₉	1	1	yes	characteristic
30	C₂	C₂²×C₂₇×C₅₉	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂³×C₃×C₅₉
socle, prime radical	C₂³×C₃×C₅₉	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₇×C₅₉	C₂	direct (N × H)
3. C₂×C₂₇×C₅₉	C₂²	direct (N × H)
5. C₂₇×C₅₉	C₂³	direct (N × H)
10. C₂³×C₅₉	C₂₇	direct (N × H)
12. C₂²×C₅₉	C₂×C₂₇	direct (N × H)
13. C₂³×C₂₇	C₅₉	direct (N × H)
15. C₂×C₅₉	C₂²×C₂₇	direct (N × H)
16. C₂²×C₂₇	C₂×C₅₉	direct (N × H)
18. C₅₉	C₂³×C₂₇	direct (N × H)
19. C₂×C₂₇	C₂²×C₅₉	direct (N × H)
21. C₂₇	C₂³×C₅₉	direct (N × H)
26. C₂³	C₂₇×C₅₉	direct (N × H)
28. C₂²	C₂×C₂₇×C₅₉	direct (N × H)
30. C₂	C₂²×C₂₇×C₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₇×C₅₉	2
C₂³×C₉×C₅₉	3
C₂³×C₂₇	59

Subgroups — C23×C27×C59 #72191069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₇×C₅₉ #72191069