Subgroups — C₅×C₇×C₈×C₄₉ #72191920

Order 13720.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₇×C₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₄₉	C₄	1	1	yes	characteristic
3	C₇×C₈×C₄₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₇²×C₈	C₇	1	1	yes	maximal, characteristic
5	C₅×C₈×C₄₉	C₇	1	7	yes	maximal, direct product factor
6	C₅×C₇×C₄₉	C₈	1	1	yes	characteristic, direct product factor, Hall
7	C₄×C₇×C₄₉	C₂×C₅	1	1	yes	characteristic
8	C₄×C₅×C₇²	C₂×C₇	1	1	yes	characteristic
9	C₄×C₅×C₄₉	C₂×C₇	1	7	yes	—
10	C₂×C₇×C₄₉	C₄×C₅	1	1	yes	characteristic
11	C₂×C₅×C₇²	C₄×C₇	1	1	yes	characteristic
12	C₂×C₅×C₄₉	C₄×C₇	1	7	yes	—
13	C₇²×C₈	C₅×C₇	1	1	yes	characteristic
14	C₈×C₄₉	C₅×C₇	1	7	yes	direct product factor
15	C₇×C₄₉	C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
16	C₅×C₇×C₈	C₄₉	1	7	yes	direct product factor
17	C₅×C₇×C₈	C₇²	1	1	yes	characteristic
18	C₅×C₇²	C₇×C₈	1	1	yes	characteristic
19	C₅×C₄₉	C₇×C₈	1	7	yes	direct product factor
20	C₄×C₇²	C₂×C₅×C₇	1	1	yes	characteristic
21	C₄×C₄₉	C₂×C₅×C₇	1	7	yes	—
22	C₄×C₅×C₇	C₂×C₄₉	1	7	yes	—
23	C₄×C₅×C₇	C₂×C₇²	1	1	yes	characteristic
24	C₂×C₇²	C₄×C₅×C₇	1	1	yes	characteristic
25	C₂×C₄₉	C₄×C₅×C₇	1	7	yes	—
26	C₂×C₅×C₇	C₄×C₄₉	1	7	yes	—
27	C₂×C₅×C₇	C₄×C₇²	1	1	yes	characteristic
28	C₇×C₈	C₅×C₄₉	1	7	yes	direct product factor
29	C₇×C₈	C₅×C₇²	1	1	yes	characteristic
30	C₇²	C₅×C₇×C₈	1	1	yes	characteristic
31	C₄₉	C₅×C₇×C₈	1	7	yes	direct product factor
32	C₅×C₈	C₇×C₄₉	1	1	yes	characteristic, direct product factor, Hall
33	C₅×C₇	C₈×C₄₉	1	7	yes	direct product factor
34	C₅×C₇	C₇²×C₈	1	1	yes	characteristic
35	C₄×C₇	C₂×C₅×C₄₉	1	7	yes	—
36	C₄×C₇	C₂×C₅×C₇²	1	1	yes	characteristic
37	C₄×C₅	C₂×C₇×C₄₉	1	1	yes	characteristic
38	C₂×C₇	C₄×C₅×C₄₉	1	7	yes	—
39	C₂×C₇	C₄×C₅×C₇²	1	1	yes	characteristic
40	C₂×C₅	C₄×C₇×C₄₉	1	1	yes	characteristic
41	C₈	C₅×C₇×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₇	C₅×C₈×C₄₉	1	7	yes	direct product factor
43	C₇	C₅×C₇²×C₈	1	1	yes	characteristic
44	C₅	C₇×C₈×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₄	C₂×C₅×C₇×C₄₉	1	1	yes	characteristic
46	C₂	C₄×C₅×C₇×C₄₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₇	C₂×C₅×C₇²
socle, prime radical	C₂×C₅×C₇²	C₄×C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×C₈×C₄₉	C₅	direct (N × H)
5. C₅×C₈×C₄₉	C₇	direct (N × H)
6. C₅×C₇×C₄₉	C₈	direct (N × H)
14. C₈×C₄₉	C₅×C₇	direct (N × H)
15. C₇×C₄₉	C₅×C₈	direct (N × H)
16. C₅×C₇×C₈	C₄₉	direct (N × H)
19. C₅×C₄₉	C₇×C₈	direct (N × H)
28. C₇×C₈	C₅×C₄₉	direct (N × H)
31. C₄₉	C₅×C₇×C₈	direct (N × H)
32. C₅×C₈	C₇×C₄₉	direct (N × H)
33. C₅×C₇	C₈×C₄₉	direct (N × H)
41. C₈	C₅×C₇×C₄₉	direct (N × H)
42. C₇	C₅×C₈×C₄₉	direct (N × H)
44. C₅	C₇×C₈×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
C₄×C₅×C₇×C₄₉	2
C₇×C₈×C₄₉	5
C₅×C₇²×C₈	7
C₅×C₈×C₄₉	7

Subgroups — C5×C7×C8×C49 #72191920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×C₈×C₄₉ #72191920