Subgroups — C₂×C₃×C₇²×C₄₉ #72192585

Order 14406.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇³	C₇	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₄₉	C₇	1	56	yes	maximal, direct product factor
3	C₃×C₇²×C₄₉	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₇²×C₄₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇²×C₄₉	C₂×C₃	1	1	yes	characteristic, direct product factor, Sylow
6	C₃×C₇³	C₂×C₇	1	1	yes	characteristic
7	C₃×C₇×C₄₉	C₂×C₇	1	56	yes	direct product factor
8	C₂×C₇³	C₃×C₇	1	1	yes	characteristic
9	C₂×C₇×C₄₉	C₃×C₇	1	56	yes	direct product factor
10	C₇³	C₂×C₃×C₇	1	1	yes	characteristic
11	C₇×C₄₉	C₂×C₃×C₇	1	56	yes	direct product factor
12	C₂×C₃×C₇²	C₄₉	1	49	yes	direct product factor
13	C₂×C₃×C₇²	C₇²	1	8	yes	—
14	C₂×C₃×C₄₉	C₇²	1	49	yes	direct product factor
15	C₃×C₇²	C₂×C₄₉	1	49	yes	direct product factor
16	C₃×C₇²	C₂×C₇²	1	8	yes	—
17	C₃×C₄₉	C₂×C₇²	1	49	yes	direct product factor
18	C₂×C₇²	C₃×C₄₉	1	49	yes	direct product factor
19	C₂×C₇²	C₃×C₇²	1	8	yes	—
20	C₂×C₄₉	C₃×C₇²	1	49	yes	direct product factor
21	C₇²	C₂×C₃×C₄₉	1	49	yes	direct product factor
22	C₇²	C₂×C₃×C₇²	1	8	yes	—
23	C₄₉	C₂×C₃×C₇²	1	49	yes	direct product factor
24	C₂×C₃×C₇	C₇×C₄₉	1	56	yes	direct product factor
25	C₂×C₃×C₇	C₇³	1	1	yes	characteristic
26	C₃×C₇	C₂×C₇×C₄₉	1	56	yes	direct product factor
27	C₃×C₇	C₂×C₇³	1	1	yes	characteristic
28	C₂×C₇	C₃×C₇×C₄₉	1	56	yes	direct product factor
29	C₂×C₇	C₃×C₇³	1	1	yes	characteristic
30	C₇	C₂×C₃×C₇×C₄₉	1	56	yes	direct product factor
31	C₇	C₂×C₃×C₇³	1	1	yes	characteristic
32	C₂×C₃	C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Hall
33	C₃	C₂×C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₃×C₇²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₂×C₃×C₇³
socle, prime radical	C₂×C₃×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×C₃×C₇×C₄₉	C₇	direct (N × H)
3. C₃×C₇²×C₄₉	C₂	direct (N × H)
4. C₂×C₇²×C₄₉	C₃	direct (N × H)
5. C₇²×C₄₉	C₂×C₃	direct (N × H)
7. C₃×C₇×C₄₉	C₂×C₇	direct (N × H)
9. C₂×C₇×C₄₉	C₃×C₇	direct (N × H)
11. C₇×C₄₉	C₂×C₃×C₇	direct (N × H)
12. C₂×C₃×C₇²	C₄₉	direct (N × H)
14. C₂×C₃×C₄₉	C₇²	direct (N × H)
15. C₃×C₇²	C₂×C₄₉	direct (N × H)
17. C₃×C₄₉	C₂×C₇²	direct (N × H)
18. C₂×C₇²	C₃×C₄₉	direct (N × H)
20. C₂×C₄₉	C₃×C₇²	direct (N × H)
21. C₇²	C₂×C₃×C₄₉	direct (N × H)
23. C₄₉	C₂×C₃×C₇²	direct (N × H)
24. C₂×C₃×C₇	C₇×C₄₉	direct (N × H)
26. C₃×C₇	C₂×C₇×C₄₉	direct (N × H)
28. C₂×C₇	C₃×C₇×C₄₉	direct (N × H)
30. C₇	C₂×C₃×C₇×C₄₉	direct (N × H)
32. C₂×C₃	C₇²×C₄₉	direct (N × H)
33. C₃	C₂×C₇²×C₄₉	direct (N × H)
34. C₂	C₃×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×C₃×C₇³	7
C₂×C₃×C₇×C₄₉	7
C₃×C₇²×C₄₉	2
C₂×C₇²×C₄₉	3

Subgroups — C2×C3×C72×C49 #72192585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇²×C₄₉ #72192585