Subgroups — C₂²×C₃×C₄×C₅₆₉ #72204367

Order 27312.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅₆₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₅₆₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅₆₉	C₄	1	4	yes	direct product factor
5	C₂²×C₃×C₅₆₉	C₂²	1	3	yes	—
6	C₃×C₄×C₅₆₉	C₂²	1	4	yes	direct product factor
7	C₂³×C₅₆₉	C₂×C₃	1	1	yes	characteristic
8	C₂×C₄×C₅₆₉	C₂×C₃	1	6	yes	direct product factor
9	C₂×C₃×C₅₆₉	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₃×C₅₆₉	C₂³	1	1	yes	characteristic
11	C₂²×C₅₆₉	C₃×C₄	1	4	yes	direct product factor
12	C₂²×C₅₆₉	C₂²×C₃	1	3	yes	—
13	C₄×C₅₆₉	C₂²×C₃	1	4	yes	direct product factor
14	C₃×C₅₆₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₅₆₉	C₂×C₃×C₄	1	6	yes	direct product factor
16	C₂×C₅₆₉	C₂³×C₃	1	1	yes	characteristic
17	C₅₆₉	C₂²×C₃×C₄	1	1	yes	characteristic, direct product factor, Sylow
18	C₂²×C₃×C₄	C₅₆₉	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₃	C₂×C₅₆₉	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₅₆₉	1	6	yes	direct product factor
21	C₂²×C₄	C₃×C₅₆₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃	C₄×C₅₆₉	1	4	yes	direct product factor
23	C₂²×C₃	C₂²×C₅₆₉	1	3	yes	—
24	C₃×C₄	C₂²×C₅₆₉	1	4	yes	direct product factor
25	C₂³	C₂×C₃×C₅₆₉	1	1	yes	characteristic
26	C₂×C₄	C₂×C₃×C₅₆₉	1	6	yes	direct product factor
27	C₂×C₃	C₂×C₄×C₅₆₉	1	6	yes	direct product factor
28	C₂×C₃	C₂³×C₅₆₉	1	1	yes	characteristic
29	C₂²	C₃×C₄×C₅₆₉	1	4	yes	direct product factor
30	C₂²	C₂²×C₃×C₅₆₉	1	3	yes	—
31	C₄	C₂²×C₃×C₅₆₉	1	4	yes	direct product factor
32	C₃	C₂²×C₄×C₅₆₉	1	1	yes	characteristic, direct product factor, Sylow
33	C₂	C₂×C₃×C₄×C₅₆₉	1	6	yes	direct product factor
34	C₂	C₂³×C₃×C₅₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₃×C₅₆₉
socle, prime radical	C₂³×C₃×C₅₆₉	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₅₆₉	C₂	direct (N × H)
3. C₂²×C₄×C₅₆₉	C₃	direct (N × H)
4. C₂²×C₃×C₅₆₉	C₄	direct (N × H)
6. C₃×C₄×C₅₆₉	C₂²	direct (N × H)
8. C₂×C₄×C₅₆₉	C₂×C₃	direct (N × H)
9. C₂×C₃×C₅₆₉	C₂×C₄	direct (N × H)
11. C₂²×C₅₆₉	C₃×C₄	direct (N × H)
13. C₄×C₅₆₉	C₂²×C₃	direct (N × H)
14. C₃×C₅₆₉	C₂²×C₄	direct (N × H)
15. C₂×C₅₆₉	C₂×C₃×C₄	direct (N × H)
17. C₅₆₉	C₂²×C₃×C₄	direct (N × H)
18. C₂²×C₃×C₄	C₅₆₉	direct (N × H)
20. C₂×C₃×C₄	C₂×C₅₆₉	direct (N × H)
21. C₂²×C₄	C₃×C₅₆₉	direct (N × H)
22. C₂²×C₃	C₄×C₅₆₉	direct (N × H)
24. C₃×C₄	C₂²×C₅₆₉	direct (N × H)
26. C₂×C₄	C₂×C₃×C₅₆₉	direct (N × H)
27. C₂×C₃	C₂×C₄×C₅₆₉	direct (N × H)
29. C₂²	C₃×C₄×C₅₆₉	direct (N × H)
31. C₄	C₂²×C₃×C₅₆₉	direct (N × H)
32. C₃	C₂²×C₄×C₅₆₉	direct (N × H)
33. C₂	C₂×C₃×C₄×C₅₆₉	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×C₃×C₅₆₉	2
C₂×C₃×C₄×C₅₆₉	2
C₂²×C₄×C₅₆₉	3
C₂²×C₃×C₄	569

Subgroups — C22×C3×C4×C569 #72204367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×C₅₆₉ #72204367