Subgroups — C₂²×C₅×C₂₅×C₄₉ #72201764

Order 24500.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂₅×C₄₉	C₂	1	3	yes	maximal, direct product factor
2	C₅×C₂₅×C₄₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₅×C₇×C₂₅	C₇	1	1	yes	maximal, characteristic
4	C₂²×C₅²×C₄₉	C₅	1	1	yes	maximal, characteristic
5	C₂²×C₂₅×C₄₉	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₅²×C₄₉	C₂×C₅	1	3	yes	—
7	C₂×C₂₅×C₄₉	C₂×C₅	1	15	yes	direct product factor
8	C₂×C₅×C₇×C₂₅	C₂×C₇	1	3	yes	—
9	C₅²×C₄₉	C₂²×C₅	1	1	yes	characteristic
10	C₂₅×C₄₉	C₂²×C₅	1	5	yes	direct product factor
11	C₂²×C₅×C₄₉	C₂₅	1	5	yes	direct product factor
12	C₂²×C₅×C₄₉	C₅²	1	1	yes	characteristic
13	C₅×C₇×C₂₅	C₂²×C₇	1	1	yes	characteristic
14	C₂²×C₅²×C₇	C₅×C₇	1	1	yes	characteristic
15	C₂²×C₇×C₂₅	C₅×C₇	1	5	yes	—
16	C₂²×C₅×C₂₅	C₄₉	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₅×C₄₉	C₂×C₂₅	1	15	yes	direct product factor
18	C₂×C₅×C₄₉	C₂×C₅²	1	3	yes	—
19	C₂×C₅²×C₇	C₂×C₅×C₇	1	3	yes	—
20	C₂×C₇×C₂₅	C₂×C₅×C₇	1	15	yes	—
21	C₂×C₅×C₂₅	C₂×C₄₉	1	3	yes	direct product factor
22	C₅×C₄₉	C₂²×C₂₅	1	5	yes	direct product factor
23	C₅×C₄₉	C₂²×C₅²	1	1	yes	characteristic
24	C₂²×C₄₉	C₅×C₂₅	1	1	yes	characteristic, direct product factor, Hall
25	C₅²×C₇	C₂²×C₅×C₇	1	1	yes	characteristic
26	C₇×C₂₅	C₂²×C₅×C₇	1	5	yes	—
27	C₂²×C₅×C₇	C₇×C₂₅	1	5	yes	—
28	C₂²×C₅×C₇	C₅²×C₇	1	1	yes	characteristic
29	C₅×C₂₅	C₂²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₅²	C₅×C₄₉	1	1	yes	characteristic
31	C₂²×C₂₅	C₅×C₄₉	1	5	yes	direct product factor
32	C₂×C₄₉	C₂×C₅×C₂₅	1	3	yes	direct product factor
33	C₂×C₅×C₇	C₂×C₇×C₂₅	1	15	yes	—
34	C₂×C₅×C₇	C₂×C₅²×C₇	1	3	yes	—
35	C₂×C₅²	C₂×C₅×C₄₉	1	3	yes	—
36	C₂×C₂₅	C₂×C₅×C₄₉	1	15	yes	direct product factor
37	C₄₉	C₂²×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
38	C₅×C₇	C₂²×C₇×C₂₅	1	5	yes	—
39	C₅×C₇	C₂²×C₅²×C₇	1	1	yes	characteristic
40	C₂²×C₇	C₅×C₇×C₂₅	1	1	yes	characteristic
41	C₅²	C₂²×C₅×C₄₉	1	1	yes	characteristic
42	C₂₅	C₂²×C₅×C₄₉	1	5	yes	direct product factor
43	C₂²×C₅	C₂₅×C₄₉	1	5	yes	direct product factor
44	C₂²×C₅	C₅²×C₄₉	1	1	yes	characteristic
45	C₂×C₇	C₂×C₅×C₇×C₂₅	1	3	yes	—
46	C₂×C₅	C₂×C₂₅×C₄₉	1	15	yes	direct product factor
47	C₂×C₅	C₂×C₅²×C₄₉	1	3	yes	—
48	C₇	C₂²×C₅×C₇×C₂₅	1	1	yes	characteristic
49	C₅	C₂²×C₂₅×C₄₉	1	5	yes	direct product factor
50	C₅	C₂²×C₅²×C₄₉	1	1	yes	characteristic
51	C₂²	C₅×C₂₅×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	C₂×C₅×C₂₅×C₄₉	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅×C₇	C₂²×C₅²×C₇
socle, prime radical	C₂²×C₅²×C₇	C₅×C₇
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×C₂₅×C₄₉	C₂	direct (N × H)
2. C₅×C₂₅×C₄₉	C₂²	direct (N × H)
5. C₂²×C₂₅×C₄₉	C₅	direct (N × H)
7. C₂×C₂₅×C₄₉	C₂×C₅	direct (N × H)
10. C₂₅×C₄₉	C₂²×C₅	direct (N × H)
11. C₂²×C₅×C₄₉	C₂₅	direct (N × H)
16. C₂²×C₅×C₂₅	C₄₉	direct (N × H)
17. C₂×C₅×C₄₉	C₂×C₂₅	direct (N × H)
21. C₂×C₅×C₂₅	C₂×C₄₉	direct (N × H)
22. C₅×C₄₉	C₂²×C₂₅	direct (N × H)
24. C₂²×C₄₉	C₅×C₂₅	direct (N × H)
29. C₅×C₂₅	C₂²×C₄₉	direct (N × H)
31. C₂²×C₂₅	C₅×C₄₉	direct (N × H)
32. C₂×C₄₉	C₂×C₅×C₂₅	direct (N × H)
36. C₂×C₂₅	C₂×C₅×C₄₉	direct (N × H)
37. C₄₉	C₂²×C₅×C₂₅	direct (N × H)
42. C₂₅	C₂²×C₅×C₄₉	direct (N × H)
43. C₂²×C₅	C₂₅×C₄₉	direct (N × H)
46. C₂×C₅	C₂×C₂₅×C₄₉	direct (N × H)
49. C₅	C₂²×C₂₅×C₄₉	direct (N × H)
51. C₂²	C₅×C₂₅×C₄₉	direct (N × H)
52. C₂	C₂×C₅×C₂₅×C₄₉	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×C₂₅×C₄₉	2
C₂²×C₅×C₇×C₂₅	7
C₂²×C₅²×C₄₉	5
C₂²×C₂₅×C₄₉	5

Subgroups — C22×C5×C25×C49 #72201764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂₅×C₄₉ #72201764