Subgroups — C₃×C₂₅×C₂₅₆ #72196871

Order 19200.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₅×C₁₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂₅×C₂₅₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₂₅×C₆₄	C₄	1	1	yes	characteristic
4	C₃×C₅×C₂₅₆	C₅	1	1	yes	maximal, characteristic
5	C₂₅×C₁₂₈	C₂×C₃	1	1	yes	characteristic
6	C₃×C₂₅×C₃₂	C₈	1	1	yes	characteristic
7	C₃×C₅×C₁₂₈	C₂×C₅	1	1	yes	characteristic
8	C₂₅×C₆₄	C₃×C₄	1	1	yes	characteristic
9	C₅×C₂₅₆	C₃×C₅	1	1	yes	characteristic
10	C₃×C₁₆×C₂₅	C₁₆	1	1	yes	characteristic
11	C₃×C₅×C₆₄	C₄×C₅	1	1	yes	characteristic
12	C₂₅×C₃₂	C₃×C₈	1	1	yes	characteristic
13	C₃×C₂₅₆	C₂₅	1	1	yes	characteristic, direct product factor, Hall
14	C₅×C₁₂₈	C₂×C₃×C₅	1	1	yes	characteristic
15	C₃×C₈×C₂₅	C₃₂	1	1	yes	characteristic
16	C₃×C₅×C₃₂	C₅×C₈	1	1	yes	characteristic
17	C₁₆×C₂₅	C₃×C₁₆	1	1	yes	characteristic
18	C₃×C₁₂₈	C₂×C₂₅	1	1	yes	characteristic
19	C₅×C₆₄	C₃×C₄×C₅	1	1	yes	characteristic
20	C₃×C₄×C₂₅	C₆₄	1	1	yes	characteristic
21	C₂₅₆	C₃×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×C₅×C₁₆	C₅×C₁₆	1	1	yes	characteristic
23	C₈×C₂₅	C₃×C₃₂	1	1	yes	characteristic
24	C₃×C₆₄	C₄×C₂₅	1	1	yes	characteristic
25	C₅×C₃₂	C₃×C₅×C₈	1	1	yes	characteristic
26	C₂×C₃×C₂₅	C₁₂₈	1	1	yes	characteristic
27	C₁₂₈	C₂×C₃×C₂₅	1	1	yes	characteristic
28	C₃×C₅×C₈	C₅×C₃₂	1	1	yes	characteristic
29	C₄×C₂₅	C₃×C₆₄	1	1	yes	characteristic
30	C₃×C₃₂	C₈×C₂₅	1	1	yes	characteristic
31	C₅×C₁₆	C₃×C₅×C₁₆	1	1	yes	characteristic
32	C₃×C₂₅	C₂₅₆	1	1	yes	characteristic, direct product factor, Hall
33	C₆₄	C₃×C₄×C₂₅	1	1	yes	characteristic
34	C₃×C₄×C₅	C₅×C₆₄	1	1	yes	characteristic
35	C₂×C₂₅	C₃×C₁₂₈	1	1	yes	characteristic
36	C₃×C₁₆	C₁₆×C₂₅	1	1	yes	characteristic
37	C₅×C₈	C₃×C₅×C₃₂	1	1	yes	characteristic
38	C₃₂	C₃×C₈×C₂₅	1	1	yes	characteristic
39	C₂×C₃×C₅	C₅×C₁₂₈	1	1	yes	characteristic
40	C₂₅	C₃×C₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₈	C₂₅×C₃₂	1	1	yes	characteristic
42	C₄×C₅	C₃×C₅×C₆₄	1	1	yes	characteristic
43	C₁₆	C₃×C₁₆×C₂₅	1	1	yes	characteristic
44	C₃×C₅	C₅×C₂₅₆	1	1	yes	characteristic
45	C₃×C₄	C₂₅×C₆₄	1	1	yes	characteristic
46	C₂×C₅	C₃×C₅×C₁₂₈	1	1	yes	characteristic
47	C₈	C₃×C₂₅×C₃₂	1	1	yes	characteristic
48	C₂×C₃	C₂₅×C₁₂₈	1	1	yes	characteristic
49	C₅	C₃×C₅×C₂₅₆	1	1	yes	characteristic
50	C₄	C₃×C₂₅×C₆₄	1	1	yes	characteristic
51	C₃	C₂₅×C₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	C₃×C₂₅×C₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅×C₁₂₈	C₂×C₃×C₅
socle, prime radical	C₂×C₃×C₅	C₅×C₁₂₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₅×C₂₅₆	C₃	direct (N × H)
13. C₃×C₂₅₆	C₂₅	direct (N × H)
21. C₂₅₆	C₃×C₂₅	direct (N × H)
32. C₃×C₂₅	C₂₅₆	direct (N × H)
40. C₂₅	C₃×C₂₅₆	direct (N × H)
51. C₃	C₂₅×C₂₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₂₅×C₁₂₈	2
C₂₅×C₂₅₆	3
C₃×C₅×C₂₅₆	5

Subgroups — C3×C25×C256 #72196871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂₅×C₂₅₆ #72196871