Subgroups — C₂²×C₅³×C₄₉ #72201766

Order 24500.

960 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅³×C₄₉	C₂	1	3	yes	maximal, direct product factor
2	C₅³×C₄₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₅³×C₇	C₇	1	1	yes	maximal, characteristic
4	C₂²×C₅²×C₄₉	C₅	1	31	yes	maximal, direct product factor
5	C₂×C₅²×C₄₉	C₂×C₅	1	93	yes	direct product factor
6	C₂×C₅³×C₇	C₂×C₇	1	3	yes	—
7	C₅²×C₄₉	C₂²×C₅	1	31	yes	direct product factor
8	C₂²×C₅×C₄₉	C₅²	1	31	yes	direct product factor
9	C₅³×C₇	C₂²×C₇	1	1	yes	characteristic
10	C₂²×C₅²×C₇	C₅×C₇	1	31	yes	—
11	C₂²×C₅³	C₄₉	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₅×C₄₉	C₂×C₅²	1	93	yes	direct product factor
13	C₂×C₅²×C₇	C₂×C₅×C₇	1	93	yes	—
14	C₂×C₅³	C₂×C₄₉	1	3	yes	direct product factor
15	C₅×C₄₉	C₂²×C₅²	1	31	yes	direct product factor
16	C₂²×C₄₉	C₅³	1	1	yes	characteristic, direct product factor, Hall
17	C₅²×C₇	C₂²×C₅×C₇	1	31	yes	—
18	C₂²×C₅×C₇	C₅²×C₇	1	31	yes	—
19	C₅³	C₂²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
20	C₂²×C₅²	C₅×C₄₉	1	31	yes	direct product factor
21	C₂×C₄₉	C₂×C₅³	1	3	yes	direct product factor
22	C₂×C₅×C₇	C₂×C₅²×C₇	1	93	yes	—
23	C₂×C₅²	C₂×C₅×C₄₉	1	93	yes	direct product factor
24	C₄₉	C₂²×C₅³	1	1	yes	characteristic, direct product factor, Sylow
25	C₅×C₇	C₂²×C₅²×C₇	1	31	yes	—
26	C₂²×C₇	C₅³×C₇	1	1	yes	characteristic
27	C₅²	C₂²×C₅×C₄₉	1	31	yes	direct product factor
28	C₂²×C₅	C₅²×C₄₉	1	31	yes	direct product factor
29	C₂×C₇	C₂×C₅³×C₇	1	3	yes	—
30	C₂×C₅	C₂×C₅²×C₄₉	1	93	yes	direct product factor
31	C₇	C₂²×C₅³×C₇	1	1	yes	characteristic
32	C₅	C₂²×C₅²×C₄₉	1	31	yes	direct product factor
33	C₂²	C₅³×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₂×C₅³×C₄₉	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₂²×C₅³×C₇
socle, prime radical	C₂²×C₅³×C₇	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×C₅³×C₄₉	C₂	direct (N × H)
2. C₅³×C₄₉	C₂²	direct (N × H)
4. C₂²×C₅²×C₄₉	C₅	direct (N × H)
5. C₂×C₅²×C₄₉	C₂×C₅	direct (N × H)
7. C₅²×C₄₉	C₂²×C₅	direct (N × H)
8. C₂²×C₅×C₄₉	C₅²	direct (N × H)
11. C₂²×C₅³	C₄₉	direct (N × H)
12. C₂×C₅×C₄₉	C₂×C₅²	direct (N × H)
14. C₂×C₅³	C₂×C₄₉	direct (N × H)
15. C₅×C₄₉	C₂²×C₅²	direct (N × H)
16. C₂²×C₄₉	C₅³	direct (N × H)
19. C₅³	C₂²×C₄₉	direct (N × H)
20. C₂²×C₅²	C₅×C₄₉	direct (N × H)
21. C₂×C₄₉	C₂×C₅³	direct (N × H)
23. C₂×C₅²	C₂×C₅×C₄₉	direct (N × H)
24. C₄₉	C₂²×C₅³	direct (N × H)
27. C₅²	C₂²×C₅×C₄₉	direct (N × H)
28. C₂²×C₅	C₅²×C₄₉	direct (N × H)
30. C₂×C₅	C₂×C₅²×C₄₉	direct (N × H)
32. C₅	C₂²×C₅²×C₄₉	direct (N × H)
33. C₂²	C₅³×C₄₉	direct (N × H)
34. C₂	C₂×C₅³×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×C₅³×C₄₉	2
C₂²×C₅³×C₇	7
C₂²×C₅²×C₄₉	5

Subgroups — C22×C53×C49 #72201766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅³×C₄₉ #72201766