Subgroups — C₂⁴×C₉×C₂₂₇ #72209356

Order 32688.

402 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉×C₂₂₇	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₃×C₂₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₃×C₂₂₇	C₂×C₃	1	15	yes	—
4	C₂²×C₉×C₂₂₇	C₂²	1	35	yes	direct product factor
5	C₂×C₉×C₂₂₇	C₂³	1	15	yes	direct product factor
6	C₂⁴×C₂₂₇	C₉	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₃×C₂₂₇	C₂²×C₃	1	35	yes	—
8	C₉×C₂₂₇	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
9	C₂³×C₂₂₇	C₂×C₉	1	15	yes	direct product factor
10	C₂×C₃×C₂₂₇	C₂³×C₃	1	15	yes	—
11	C₂²×C₂₂₇	C₂²×C₉	1	35	yes	direct product factor
12	C₃×C₂₂₇	C₂⁴×C₃	1	1	yes	characteristic
13	C₂×C₂₂₇	C₂³×C₉	1	15	yes	direct product factor
14	C₂₂₇	C₂⁴×C₉	1	1	yes	characteristic, direct product factor, Sylow
15	C₂⁴×C₉	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₉	C₂×C₂₂₇	1	15	yes	direct product factor
17	C₂⁴×C₃	C₃×C₂₂₇	1	1	yes	characteristic
18	C₂²×C₉	C₂²×C₂₂₇	1	35	yes	direct product factor
19	C₂³×C₃	C₂×C₃×C₂₂₇	1	15	yes	—
20	C₂×C₉	C₂³×C₂₂₇	1	15	yes	direct product factor
21	C₂⁴	C₉×C₂₂₇	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃	C₂²×C₃×C₂₂₇	1	35	yes	—
23	C₉	C₂⁴×C₂₂₇	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³	C₂×C₉×C₂₂₇	1	15	yes	direct product factor
25	C₂×C₃	C₂³×C₃×C₂₂₇	1	15	yes	—
26	C₂²	C₂²×C₉×C₂₂₇	1	35	yes	direct product factor
27	C₃	C₂⁴×C₃×C₂₂₇	1	1	yes	characteristic
28	C₂	C₂³×C₉×C₂₂₇	1	15	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂⁴×C₃×C₂₂₇
socle, prime radical	C₂⁴×C₃×C₂₂₇	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₉×C₂₂₇	C₂	direct (N × H)
4. C₂²×C₉×C₂₂₇	C₂²	direct (N × H)
5. C₂×C₉×C₂₂₇	C₂³	direct (N × H)
6. C₂⁴×C₂₂₇	C₉	direct (N × H)
8. C₉×C₂₂₇	C₂⁴	direct (N × H)
9. C₂³×C₂₂₇	C₂×C₉	direct (N × H)
11. C₂²×C₂₂₇	C₂²×C₉	direct (N × H)
13. C₂×C₂₂₇	C₂³×C₉	direct (N × H)
14. C₂₂₇	C₂⁴×C₉	direct (N × H)
15. C₂⁴×C₉	C₂₂₇	direct (N × H)
16. C₂³×C₉	C₂×C₂₂₇	direct (N × H)
18. C₂²×C₉	C₂²×C₂₂₇	direct (N × H)
20. C₂×C₉	C₂³×C₂₂₇	direct (N × H)
21. C₂⁴	C₉×C₂₂₇	direct (N × H)
23. C₉	C₂⁴×C₂₂₇	direct (N × H)
24. C₂³	C₂×C₉×C₂₂₇	direct (N × H)
26. C₂²	C₂²×C₉×C₂₂₇	direct (N × H)
28. C₂	C₂³×C₉×C₂₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₉×C₂₂₇	2
C₂⁴×C₃×C₂₂₇	3
C₂⁴×C₉	227

Subgroups — C24×C9×C227 #72209356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₉×C₂₂₇ #72209356