Subgroups — C₂²×C₅×C₂₇×C₄₇ #72202608

Order 25380.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂₇×C₄₇	C₂	1	3	yes	maximal, direct product factor
2	C₅×C₂₇×C₄₇	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₂₇×C₄₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₂₇×C₄₇	C₂×C₅	1	3	yes	direct product factor
5	C₂²×C₅×C₉×C₄₇	C₃	1	1	yes	maximal, characteristic
6	C₂×C₅×C₉×C₄₇	C₂×C₃	1	3	yes	—
7	C₂²×C₃×C₅×C₄₇	C₉	1	1	yes	characteristic
8	C₅×C₉×C₄₇	C₂²×C₃	1	1	yes	characteristic
9	C₂²×C₉×C₄₇	C₃×C₅	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄₇	C₂×C₉	1	3	yes	—
11	C₂₇×C₄₇	C₂²×C₅	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₅×C₄₇	C₂₇	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₉×C₄₇	C₂×C₃×C₅	1	3	yes	—
14	C₃×C₅×C₄₇	C₂²×C₉	1	1	yes	characteristic
15	C₂²×C₃×C₄₇	C₅×C₉	1	1	yes	characteristic
16	C₂²×C₅×C₂₇	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×C₄₇	C₂×C₂₇	1	3	yes	direct product factor
18	C₉×C₄₇	C₂²×C₃×C₅	1	1	yes	characteristic
19	C₂×C₃×C₄₇	C₂×C₅×C₉	1	3	yes	—
20	C₂×C₅×C₂₇	C₂×C₄₇	1	3	yes	direct product factor
21	C₅×C₄₇	C₂²×C₂₇	1	1	yes	characteristic, direct product factor, Hall
22	C₂²×C₄₇	C₅×C₂₇	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₅×C₉	C₃×C₄₇	1	1	yes	characteristic
24	C₃×C₄₇	C₂²×C₅×C₉	1	1	yes	characteristic
25	C₅×C₂₇	C₂²×C₄₇	1	1	yes	characteristic, direct product factor, Hall
26	C₂²×C₂₇	C₅×C₄₇	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₄₇	C₂×C₅×C₂₇	1	3	yes	direct product factor
28	C₂×C₅×C₉	C₂×C₃×C₄₇	1	3	yes	—
29	C₂²×C₃×C₅	C₉×C₄₇	1	1	yes	characteristic
30	C₂×C₂₇	C₂×C₅×C₄₇	1	3	yes	direct product factor
31	C₄₇	C₂²×C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
32	C₅×C₉	C₂²×C₃×C₄₇	1	1	yes	characteristic
33	C₂²×C₉	C₃×C₅×C₄₇	1	1	yes	characteristic
34	C₂×C₃×C₅	C₂×C₉×C₄₇	1	3	yes	—
35	C₂₇	C₂²×C₅×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₅	C₂₇×C₄₇	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₉	C₂×C₃×C₅×C₄₇	1	3	yes	—
38	C₃×C₅	C₂²×C₉×C₄₇	1	1	yes	characteristic
39	C₂²×C₃	C₅×C₉×C₄₇	1	1	yes	characteristic
40	C₂×C₅	C₂×C₂₇×C₄₇	1	3	yes	direct product factor
41	C₉	C₂²×C₃×C₅×C₄₇	1	1	yes	characteristic
42	C₂×C₃	C₂×C₅×C₉×C₄₇	1	3	yes	—
43	C₅	C₂²×C₂₇×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²	C₅×C₂₇×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	C₂²×C₅×C₉×C₄₇	1	1	yes	characteristic
46	C₂	C₂×C₅×C₂₇×C₄₇	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂²×C₃×C₅×C₄₇
socle, prime radical	C₂²×C₃×C₅×C₄₇	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₂₇×C₄₇	C₂	direct (N × H)
2. C₅×C₂₇×C₄₇	C₂²	direct (N × H)
3. C₂²×C₂₇×C₄₇	C₅	direct (N × H)
4. C₂×C₂₇×C₄₇	C₂×C₅	direct (N × H)
11. C₂₇×C₄₇	C₂²×C₅	direct (N × H)
12. C₂²×C₅×C₄₇	C₂₇	direct (N × H)
16. C₂²×C₅×C₂₇	C₄₇	direct (N × H)
17. C₂×C₅×C₄₇	C₂×C₂₇	direct (N × H)
20. C₂×C₅×C₂₇	C₂×C₄₇	direct (N × H)
21. C₅×C₄₇	C₂²×C₂₇	direct (N × H)
22. C₂²×C₄₇	C₅×C₂₇	direct (N × H)
25. C₅×C₂₇	C₂²×C₄₇	direct (N × H)
26. C₂²×C₂₇	C₅×C₄₇	direct (N × H)
27. C₂×C₄₇	C₂×C₅×C₂₇	direct (N × H)
30. C₂×C₂₇	C₂×C₅×C₄₇	direct (N × H)
31. C₄₇	C₂²×C₅×C₂₇	direct (N × H)
35. C₂₇	C₂²×C₅×C₄₇	direct (N × H)
36. C₂²×C₅	C₂₇×C₄₇	direct (N × H)
40. C₂×C₅	C₂×C₂₇×C₄₇	direct (N × H)
43. C₅	C₂²×C₂₇×C₄₇	direct (N × H)
44. C₂²	C₅×C₂₇×C₄₇	direct (N × H)
46. C₂	C₂×C₅×C₂₇×C₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₂₇×C₄₇	2
C₂²×C₂₇×C₄₇	5
C₂²×C₅×C₉×C₄₇	3
C₂²×C₅×C₂₇	47

Subgroups — C22×C5×C27×C47 #72202608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂₇×C₄₇ #72202608