Subgroups — C₂³×C₂₅×C₁₂₁ #72201506

Order 24200.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₂₁	C₅	1	1	yes	maximal, characteristic
2	C₂³×C₁₁×C₂₅	C₁₁	1	1	yes	maximal, characteristic
3	C₂²×C₂₅×C₁₂₁	C₂	1	7	yes	maximal, direct product factor
4	C₂×C₂₅×C₁₂₁	C₂²	1	7	yes	direct product factor
5	C₂₅×C₁₂₁	C₂³	1	1	yes	characteristic, direct product factor, Hall
6	C₂²×C₅×C₁₂₁	C₂×C₅	1	7	yes	—
7	C₂×C₅×C₁₂₁	C₂²×C₅	1	7	yes	—
8	C₂²×C₁₁×C₂₅	C₂×C₁₁	1	7	yes	—
9	C₂³×C₁₂₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
10	C₅×C₁₂₁	C₂³×C₅	1	1	yes	characteristic
11	C₂×C₁₁×C₂₅	C₂²×C₁₁	1	7	yes	—
12	C₂²×C₁₂₁	C₂×C₂₅	1	7	yes	direct product factor
13	C₂³×C₅×C₁₁	C₅×C₁₁	1	1	yes	characteristic
14	C₁₁×C₂₅	C₂³×C₁₁	1	1	yes	characteristic
15	C₂×C₁₂₁	C₂²×C₂₅	1	7	yes	direct product factor
16	C₂²×C₅×C₁₁	C₂×C₅×C₁₁	1	7	yes	—
17	C₂³×C₂₅	C₁₂₁	1	1	yes	characteristic, direct product factor, Hall
18	C₁₂₁	C₂³×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₅×C₁₁	C₂²×C₅×C₁₁	1	7	yes	—
20	C₂²×C₂₅	C₂×C₁₂₁	1	7	yes	direct product factor
21	C₂³×C₁₁	C₁₁×C₂₅	1	1	yes	characteristic
22	C₅×C₁₁	C₂³×C₅×C₁₁	1	1	yes	characteristic
23	C₂×C₂₅	C₂²×C₁₂₁	1	7	yes	direct product factor
24	C₂²×C₁₁	C₂×C₁₁×C₂₅	1	7	yes	—
25	C₂³×C₅	C₅×C₁₂₁	1	1	yes	characteristic
26	C₂₅	C₂³×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₁₁	C₂²×C₁₁×C₂₅	1	7	yes	—
28	C₂²×C₅	C₂×C₅×C₁₂₁	1	7	yes	—
29	C₁₁	C₂³×C₁₁×C₂₅	1	1	yes	characteristic
30	C₂×C₅	C₂²×C₅×C₁₂₁	1	7	yes	—
31	C₂³	C₂₅×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
32	C₅	C₂³×C₅×C₁₂₁	1	1	yes	characteristic
33	C₂²	C₂×C₂₅×C₁₂₁	1	7	yes	direct product factor
34	C₂	C₂²×C₂₅×C₁₂₁	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅×C₁₁	C₂³×C₅×C₁₁
socle, prime radical	C₂³×C₅×C₁₁	C₅×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₂₅×C₁₂₁	C₂	direct (N × H)
4. C₂×C₂₅×C₁₂₁	C₂²	direct (N × H)
5. C₂₅×C₁₂₁	C₂³	direct (N × H)
9. C₂³×C₁₂₁	C₂₅	direct (N × H)
12. C₂²×C₁₂₁	C₂×C₂₅	direct (N × H)
15. C₂×C₁₂₁	C₂²×C₂₅	direct (N × H)
17. C₂³×C₂₅	C₁₂₁	direct (N × H)
18. C₁₂₁	C₂³×C₂₅	direct (N × H)
20. C₂²×C₂₅	C₂×C₁₂₁	direct (N × H)
23. C₂×C₂₅	C₂²×C₁₂₁	direct (N × H)
26. C₂₅	C₂³×C₁₂₁	direct (N × H)
31. C₂³	C₂₅×C₁₂₁	direct (N × H)
33. C₂²	C₂×C₂₅×C₁₂₁	direct (N × H)
34. C₂	C₂²×C₂₅×C₁₂₁	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×C₅×C₁₂₁	5
C₂³×C₁₁×C₂₅	11
C₂²×C₂₅×C₁₂₁	2

Subgroups — C23×C25×C121 #72201506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₅×C₁₂₁ #72201506