Subgroups — C₂×C₅×C₇²×C₄₉ #72201301

Order 24010.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇³	C₇	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₄₉	C₇	1	56	yes	maximal, direct product factor
3	C₅×C₇²×C₄₉	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₇²×C₄₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇²×C₄₉	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
6	C₅×C₇³	C₂×C₇	1	1	yes	characteristic
7	C₅×C₇×C₄₉	C₂×C₇	1	56	yes	direct product factor
8	C₂×C₇³	C₅×C₇	1	1	yes	characteristic
9	C₂×C₇×C₄₉	C₅×C₇	1	56	yes	direct product factor
10	C₂×C₅×C₇²	C₄₉	1	49	yes	direct product factor
11	C₂×C₅×C₇²	C₇²	1	8	yes	—
12	C₂×C₅×C₄₉	C₇²	1	49	yes	direct product factor
13	C₇³	C₂×C₅×C₇	1	1	yes	characteristic
14	C₇×C₄₉	C₂×C₅×C₇	1	56	yes	direct product factor
15	C₅×C₇²	C₂×C₄₉	1	49	yes	direct product factor
16	C₅×C₇²	C₂×C₇²	1	8	yes	—
17	C₅×C₄₉	C₂×C₇²	1	49	yes	direct product factor
18	C₂×C₇²	C₅×C₄₉	1	49	yes	direct product factor
19	C₂×C₇²	C₅×C₇²	1	8	yes	—
20	C₂×C₄₉	C₅×C₇²	1	49	yes	direct product factor
21	C₂×C₅×C₇	C₇×C₄₉	1	56	yes	direct product factor
22	C₂×C₅×C₇	C₇³	1	1	yes	characteristic
23	C₇²	C₂×C₅×C₄₉	1	49	yes	direct product factor
24	C₇²	C₂×C₅×C₇²	1	8	yes	—
25	C₄₉	C₂×C₅×C₇²	1	49	yes	direct product factor
26	C₅×C₇	C₂×C₇×C₄₉	1	56	yes	direct product factor
27	C₅×C₇	C₂×C₇³	1	1	yes	characteristic
28	C₂×C₇	C₅×C₇×C₄₉	1	56	yes	direct product factor
29	C₂×C₇	C₅×C₇³	1	1	yes	characteristic
30	C₂×C₅	C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Hall
31	C₇	C₂×C₅×C₇×C₄₉	1	56	yes	direct product factor
32	C₇	C₂×C₅×C₇³	1	1	yes	characteristic
33	C₅	C₂×C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₅×C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₂×C₅×C₇³
socle, prime radical	C₂×C₅×C₇³	C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₇×C₄₉	C₇	direct (N × H)
3. C₅×C₇²×C₄₉	C₂	direct (N × H)
4. C₂×C₇²×C₄₉	C₅	direct (N × H)
5. C₇²×C₄₉	C₂×C₅	direct (N × H)
7. C₅×C₇×C₄₉	C₂×C₇	direct (N × H)
9. C₂×C₇×C₄₉	C₅×C₇	direct (N × H)
10. C₂×C₅×C₇²	C₄₉	direct (N × H)
12. C₂×C₅×C₄₉	C₇²	direct (N × H)
14. C₇×C₄₉	C₂×C₅×C₇	direct (N × H)
15. C₅×C₇²	C₂×C₄₉	direct (N × H)
17. C₅×C₄₉	C₂×C₇²	direct (N × H)
18. C₂×C₇²	C₅×C₄₉	direct (N × H)
20. C₂×C₄₉	C₅×C₇²	direct (N × H)
21. C₂×C₅×C₇	C₇×C₄₉	direct (N × H)
23. C₇²	C₂×C₅×C₄₉	direct (N × H)
25. C₄₉	C₂×C₅×C₇²	direct (N × H)
26. C₅×C₇	C₂×C₇×C₄₉	direct (N × H)
28. C₂×C₇	C₅×C₇×C₄₉	direct (N × H)
30. C₂×C₅	C₇²×C₄₉	direct (N × H)
31. C₇	C₂×C₅×C₇×C₄₉	direct (N × H)
33. C₅	C₂×C₇²×C₄₉	direct (N × H)
34. C₂	C₅×C₇²×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×C₇³	7
C₂×C₅×C₇×C₄₉	7
C₅×C₇²×C₄₉	2
C₂×C₇²×C₄₉	5

Subgroups — C2×C5×C72×C49 #72201301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇²×C₄₉ #72201301