Subgroups — C₃²×C₅²×D₄₉ #36347697

Order 22050.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₄₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×D₄₉	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×D₄₉	C₅	1	6	yes	maximal, direct product factor
4	C₃×C₅²×C₄₉	C₂×C₃	1	4	yes	complemented
5	C₃²×C₅²×D₇	—	7	1	no	maximal
6	C₅²×D₄₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₅×C₄₉	C₂×C₅	1	6	yes	complemented
8	C₃²×C₅²×C₇	D₇	1	1	yes	characteristic
9	C₃×C₅×D₄₉	C₃×C₅	1	24	yes	direct product factor
10	C₅²×C₄₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅²×D₇	—	7	4	no	—
12	C₃²×D₄₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×C₄₉	C₂×C₃×C₅	1	24	yes	complemented
14	C₃²×C₅×D₇	—	7	6	no	—
15	C₃×C₅²×C₇	C₃×D₇	1	4	yes	—
16	C₅×D₄₉	C₃²×C₅	1	6	yes	direct product factor
17	C₂×C₃²×C₅²	—	49	1	no	Hall
18	C₃²×C₄₉	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
19	C₅²×D₇	—	7	1	no	—
20	C₃²×C₅×C₇	C₅×D₇	1	6	yes	—
21	C₃×D₄₉	C₃×C₅²	1	4	yes	direct product factor
22	C₅×C₄₉	C₂×C₃²×C₅	1	6	yes	complemented
23	C₃²×C₅²	D₄₉	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₅×D₇	—	7	24	no	—
25	C₅²×C₇	C₃²×D₇	1	1	yes	characteristic
26	C₂×C₃×C₅²	—	49	4	no	—
27	C₃×C₄₉	C₂×C₃×C₅²	1	4	yes	complemented
28	C₃²×D₇	—	7	1	no	—
29	C₃×C₅×C₇	C₃×C₅×D₇	1	24	yes	—
30	D₄₉	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃²×C₅	—	49	6	no	—
32	C₃×C₅²	C₃×D₄₉	1	4	yes	direct product factor
33	C₅×D₇	—	7	6	no	—
34	C₃²×C₇	C₅²×D₇	1	1	yes	characteristic
35	C₂×C₅²	—	49	1	no	Hall
36	C₄₉	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
37	C₃²×C₅	C₅×D₄₉	1	6	yes	direct product factor
38	C₃×D₇	—	7	4	no	—
39	C₅×C₇	C₃²×C₅×D₇	1	6	yes	—
40	C₂×C₃×C₅	—	49	24	no	—
41	C₅²	C₃²×D₄₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₃×C₇	C₃×C₅²×D₇	1	4	yes	—
43	C₂×C₃²	—	49	1	no	Hall
44	C₃×C₅	C₃×C₅×D₄₉	1	24	yes	direct product factor
45	D₇	—	7	1	no	—
46	C₂×C₅	—	49	6	no	—
47	C₃²	C₅²×D₄₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₇	C₃²×C₅²×D₇	1	1	yes	characteristic
49	C₂×C₃	—	49	4	no	—
50	C₅	C₃²×C₅×D₄₉	1	6	yes	direct product factor
51	C₃	C₃×C₅²×D₄₉	1	4	yes	direct product factor
52	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅²	D₄₉
commutator, cosocle, hypocenter	C₄₉	C₂×C₃²×C₅²
Frattini	C₇	C₃²×C₅²×D₇
Fitting	C₃²×C₅²×C₄₉	C₂
socle	C₃²×C₅²×C₇	D₇
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₃²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₄₉	C₂	semidirect
2. C₃×C₅²×D₄₉	C₃	direct (N × H)
3. C₃²×C₅×D₄₉	C₅	direct (N × H)
4. C₃×C₅²×C₄₉	C₂×C₃	semidirect
6. C₅²×D₄₉	C₃²	direct (N × H)
7. C₃²×C₅×C₄₉	C₂×C₅	semidirect
9. C₃×C₅×D₄₉	C₃×C₅	direct (N × H)
10. C₅²×C₄₉	C₂×C₃²	semidirect
12. C₃²×D₄₉	C₅²	direct (N × H)
13. C₃×C₅×C₄₉	C₂×C₃×C₅	semidirect
16. C₅×D₄₉	C₃²×C₅	direct (N × H)
18. C₃²×C₄₉	C₂×C₅²	semidirect
21. C₃×D₄₉	C₃×C₅²	direct (N × H)
22. C₅×C₄₉	C₂×C₃²×C₅	semidirect
23. C₃²×C₅²	D₄₉	direct (N × H)
27. C₃×C₄₉	C₂×C₃×C₅²	semidirect
30. D₄₉	C₃²×C₅²	direct (N × H)
32. C₃×C₅²	C₃×D₄₉	direct (N × H)
36. C₄₉	C₂×C₃²×C₅²	semidirect
37. C₃²×C₅	C₅×D₄₉	direct (N × H)
41. C₅²	C₃²×D₄₉	direct (N × H)
44. C₃×C₅	C₃×C₅×D₄₉	direct (N × H)
47. C₃²	C₅²×D₄₉	direct (N × H)
50. C₅	C₃²×C₅×D₄₉	direct (N × H)
51. C₃	C₃×C₅²×D₄₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₅²×C₄₉	2
C₃×C₅²×D₄₉	3
C₃²×C₅×D₄₉	5
C₃²×C₅²×D₇	7

Subgroups — C32×C52×D49 #36347697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×D₄₉ #36347697