Subgroups — C₂×C₄×C₅²×C₁₂₁ #72201504

Order 24200.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₂₁	C₅	1	6	yes	maximal, direct product factor
2	C₂×C₄×C₅²×C₁₁	C₁₁	1	1	yes	maximal, characteristic
3	C₂²×C₅²×C₁₂₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅²×C₁₂₁	C₂	1	2	yes	maximal, direct product factor
5	C₂×C₅²×C₁₂₁	C₄	1	2	yes	direct product factor
6	C₂×C₅²×C₁₂₁	C₂²	1	1	yes	characteristic
7	C₅²×C₁₂₁	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
8	C₂²×C₅×C₁₂₁	C₂×C₅	1	6	yes	—
9	C₄×C₅×C₁₂₁	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₅×C₁₂₁	C₄×C₅	1	12	yes	direct product factor
11	C₂×C₅×C₁₂₁	C₂²×C₅	1	6	yes	—
12	C₂²×C₅²×C₁₁	C₂×C₁₁	1	1	yes	characteristic
13	C₄×C₅²×C₁₁	C₂×C₁₁	1	2	yes	—
14	C₂×C₄×C₁₂₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₅×C₁₂₁	C₂×C₄×C₅	1	6	yes	direct product factor
16	C₂×C₅²×C₁₁	C₄×C₁₁	1	2	yes	—
17	C₂×C₅²×C₁₁	C₂²×C₁₁	1	1	yes	characteristic
18	C₂²×C₁₂₁	C₂×C₅²	1	1	yes	characteristic
19	C₄×C₁₂₁	C₂×C₅²	1	2	yes	direct product factor
20	C₂×C₄×C₅×C₁₁	C₅×C₁₁	1	6	yes	—
21	C₅²×C₁₁	C₂×C₄×C₁₁	1	1	yes	characteristic
22	C₂×C₁₂₁	C₄×C₅²	1	2	yes	direct product factor
23	C₂×C₁₂₁	C₂²×C₅²	1	1	yes	characteristic
24	C₂²×C₅×C₁₁	C₂×C₅×C₁₁	1	6	yes	—
25	C₄×C₅×C₁₁	C₂×C₅×C₁₁	1	12	yes	—
26	C₂×C₄×C₅²	C₁₂₁	1	1	yes	characteristic, direct product factor, Hall
27	C₁₂₁	C₂×C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₅×C₁₁	C₄×C₅×C₁₁	1	12	yes	—
29	C₂×C₅×C₁₁	C₂²×C₅×C₁₁	1	6	yes	—
30	C₂²×C₅²	C₂×C₁₂₁	1	1	yes	characteristic
31	C₄×C₅²	C₂×C₁₂₁	1	2	yes	direct product factor
32	C₂×C₄×C₁₁	C₅²×C₁₁	1	1	yes	characteristic
33	C₅×C₁₁	C₂×C₄×C₅×C₁₁	1	6	yes	—
34	C₂×C₅²	C₄×C₁₂₁	1	2	yes	direct product factor
35	C₂×C₅²	C₂²×C₁₂₁	1	1	yes	characteristic
36	C₂²×C₁₁	C₂×C₅²×C₁₁	1	1	yes	characteristic
37	C₄×C₁₁	C₂×C₅²×C₁₁	1	2	yes	—
38	C₂×C₄×C₅	C₅×C₁₂₁	1	6	yes	direct product factor
39	C₅²	C₂×C₄×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₁₁	C₄×C₅²×C₁₁	1	2	yes	—
41	C₂×C₁₁	C₂²×C₅²×C₁₁	1	1	yes	characteristic
42	C₂²×C₅	C₂×C₅×C₁₂₁	1	6	yes	—
43	C₄×C₅	C₂×C₅×C₁₂₁	1	12	yes	direct product factor
44	C₁₁	C₂×C₄×C₅²×C₁₁	1	1	yes	characteristic
45	C₂×C₅	C₄×C₅×C₁₂₁	1	12	yes	direct product factor
46	C₂×C₅	C₂²×C₅×C₁₂₁	1	6	yes	—
47	C₂×C₄	C₅²×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
48	C₅	C₂×C₄×C₅×C₁₂₁	1	6	yes	direct product factor
49	C₂²	C₂×C₅²×C₁₂₁	1	1	yes	characteristic
50	C₄	C₂×C₅²×C₁₂₁	1	2	yes	direct product factor
51	C₂	C₄×C₅²×C₁₂₁	1	2	yes	direct product factor
52	C₂	C₂²×C₅²×C₁₂₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₁	C₂²×C₅²×C₁₁
socle, prime radical	C₂²×C₅²×C₁₁	C₂×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅×C₁₂₁	C₅	direct (N × H)
4. C₄×C₅²×C₁₂₁	C₂	direct (N × H)
5. C₂×C₅²×C₁₂₁	C₄	direct (N × H)
7. C₅²×C₁₂₁	C₂×C₄	direct (N × H)
9. C₄×C₅×C₁₂₁	C₂×C₅	direct (N × H)
10. C₂×C₅×C₁₂₁	C₄×C₅	direct (N × H)
14. C₂×C₄×C₁₂₁	C₅²	direct (N × H)
15. C₅×C₁₂₁	C₂×C₄×C₅	direct (N × H)
19. C₄×C₁₂₁	C₂×C₅²	direct (N × H)
22. C₂×C₁₂₁	C₄×C₅²	direct (N × H)
26. C₂×C₄×C₅²	C₁₂₁	direct (N × H)
27. C₁₂₁	C₂×C₄×C₅²	direct (N × H)
31. C₄×C₅²	C₂×C₁₂₁	direct (N × H)
34. C₂×C₅²	C₄×C₁₂₁	direct (N × H)
38. C₂×C₄×C₅	C₅×C₁₂₁	direct (N × H)
39. C₅²	C₂×C₄×C₁₂₁	direct (N × H)
43. C₄×C₅	C₂×C₅×C₁₂₁	direct (N × H)
45. C₂×C₅	C₄×C₅×C₁₂₁	direct (N × H)
47. C₂×C₄	C₅²×C₁₂₁	direct (N × H)
48. C₅	C₂×C₄×C₅×C₁₂₁	direct (N × H)
50. C₄	C₂×C₅²×C₁₂₁	direct (N × H)
51. C₂	C₄×C₅²×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
C₂×C₄×C₅×C₁₂₁	5
C₂×C₄×C₅²×C₁₁	11
C₂²×C₅²×C₁₂₁	2
C₄×C₅²×C₁₂₁	2

Subgroups — C2×C4×C52×C121 #72201504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅²×C₁₂₁ #72201504