Subgroups — C₂×C₅²×C₈×C₆₁ #72201688

Order 24400.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅²×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₈×C₆₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×C₈×C₆₁	C₅	1	6	yes	maximal, direct product factor
4	C₂×C₄×C₅×C₆₁	C₂×C₅	1	6	yes	—
5	C₅×C₈×C₆₁	C₂×C₅	1	12	yes	direct product factor
6	C₂²×C₅²×C₆₁	C₄	1	1	yes	characteristic
7	C₄×C₅²×C₆₁	C₄	1	1	yes	characteristic
8	C₄×C₅²×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₆₁	C₈	1	2	yes	direct product factor
10	C₂×C₅²×C₆₁	C₂×C₄	1	1	yes	characteristic
11	C₅²×C₆₁	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₅×C₆₁	C₄×C₅	1	6	yes	—
13	C₄×C₅×C₆₁	C₄×C₅	1	6	yes	—
14	C₄×C₅×C₆₁	C₂²×C₅	1	6	yes	—
15	C₂×C₈×C₆₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₅×C₆₁	C₅×C₈	1	12	yes	direct product factor
17	C₂×C₅×C₆₁	C₂×C₄×C₅	1	6	yes	—
18	C₂×C₄×C₆₁	C₂×C₅²	1	1	yes	characteristic
19	C₈×C₆₁	C₂×C₅²	1	2	yes	direct product factor
20	C₂×C₅²×C₈	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×C₆₁	C₂×C₅×C₈	1	6	yes	direct product factor
22	C₂²×C₆₁	C₄×C₅²	1	1	yes	characteristic
23	C₄×C₆₁	C₄×C₅²	1	1	yes	characteristic
24	C₄×C₆₁	C₂²×C₅²	1	1	yes	characteristic
25	C₂×C₄×C₅²	C₂×C₆₁	1	1	yes	characteristic
26	C₅²×C₈	C₂×C₆₁	1	2	yes	direct product factor
27	C₂×C₆₁	C₅²×C₈	1	2	yes	direct product factor
28	C₂×C₆₁	C₂×C₄×C₅²	1	1	yes	characteristic
29	C₂²×C₅²	C₄×C₆₁	1	1	yes	characteristic
30	C₄×C₅²	C₄×C₆₁	1	1	yes	characteristic
31	C₄×C₅²	C₂²×C₆₁	1	1	yes	characteristic
32	C₂×C₅×C₈	C₅×C₆₁	1	6	yes	direct product factor
33	C₆₁	C₂×C₅²×C₈	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₅²	C₈×C₆₁	1	2	yes	direct product factor
35	C₂×C₅²	C₂×C₄×C₆₁	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂×C₅×C₆₁	1	6	yes	—
37	C₅×C₈	C₂×C₅×C₆₁	1	12	yes	direct product factor
38	C₅²	C₂×C₈×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₅	C₄×C₅×C₆₁	1	6	yes	—
40	C₄×C₅	C₄×C₅×C₆₁	1	6	yes	—
41	C₄×C₅	C₂²×C₅×C₆₁	1	6	yes	—
42	C₂×C₈	C₅²×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₅	C₅×C₈×C₆₁	1	12	yes	direct product factor
44	C₂×C₅	C₂×C₄×C₅×C₆₁	1	6	yes	—
45	C₂×C₄	C₂×C₅²×C₆₁	1	1	yes	characteristic
46	C₈	C₂×C₅²×C₆₁	1	2	yes	direct product factor
47	C₅	C₂×C₅×C₈×C₆₁	1	6	yes	direct product factor
48	C₂²	C₄×C₅²×C₆₁	1	1	yes	characteristic
49	C₄	C₄×C₅²×C₆₁	1	1	yes	characteristic
50	C₄	C₂²×C₅²×C₆₁	1	1	yes	characteristic
51	C₂	C₅²×C₈×C₆₁	1	2	yes	direct product factor
52	C₂	C₂×C₄×C₅²×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂²×C₅²×C₆₁
socle, prime radical	C₂²×C₅²×C₆₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅²×C₈×C₆₁	C₂	direct (N × H)
3. C₂×C₅×C₈×C₆₁	C₅	direct (N × H)
5. C₅×C₈×C₆₁	C₂×C₅	direct (N × H)
9. C₂×C₅²×C₆₁	C₈	direct (N × H)
11. C₅²×C₆₁	C₂×C₈	direct (N × H)
15. C₂×C₈×C₆₁	C₅²	direct (N × H)
16. C₂×C₅×C₆₁	C₅×C₈	direct (N × H)
19. C₈×C₆₁	C₂×C₅²	direct (N × H)
20. C₂×C₅²×C₈	C₆₁	direct (N × H)
21. C₅×C₆₁	C₂×C₅×C₈	direct (N × H)
26. C₅²×C₈	C₂×C₆₁	direct (N × H)
27. C₂×C₆₁	C₅²×C₈	direct (N × H)
32. C₂×C₅×C₈	C₅×C₆₁	direct (N × H)
33. C₆₁	C₂×C₅²×C₈	direct (N × H)
34. C₂×C₅²	C₈×C₆₁	direct (N × H)
37. C₅×C₈	C₂×C₅×C₆₁	direct (N × H)
38. C₅²	C₂×C₈×C₆₁	direct (N × H)
42. C₂×C₈	C₅²×C₆₁	direct (N × H)
43. C₂×C₅	C₅×C₈×C₆₁	direct (N × H)
46. C₈	C₂×C₅²×C₆₁	direct (N × H)
47. C₅	C₂×C₅×C₈×C₆₁	direct (N × H)
51. C₂	C₅²×C₈×C₆₁	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×C₅²×C₆₁	2
C₅²×C₈×C₆₁	2
C₂×C₅×C₈×C₆₁	5
C₂×C₅²×C₈	61

Subgroups — C2×C52×C8×C61 #72201688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₈×C₆₁ #72201688