Subgroups — C₅×C₉×C₂²⋊C₄ #10623613

Order 720.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₉	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₉	—	2	2	no	—
5	C₂²×C₅×C₉	C₄	1	2	yes	complemented
6	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₄×C₅×C₉	—	2	2	no	—
8	C₉×C₂²⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₅	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃×C₄×C₅	C₂×C₃	1	2	yes	—
11	C₂×C₅×C₉	—	2	2	no	—
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₅×C₉	D₄	1	2	yes	—
14	C₅×C₂²⋊C₄	C₉	1	1	yes	characteristic, direct product factor, Hall
15	C₂³×C₉	C₂×C₅	1	1	yes	characteristic
16	C₂×C₄×C₉	C₂×C₅	1	2	yes	complemented
17	C₂²×C₃×C₅	—	2	2	no	—
18	C₂²×C₃×C₅	C₃×C₄	1	2	yes	—
19	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
20	C₃×C₄×C₅	—	2	2	no	—
21	C₃×C₂²⋊C₄	C₃×C₅	1	1	yes	characteristic
22	C₅×C₉	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Hall
23	C₂³×C₅	C₂×C₉	1	1	yes	characteristic
24	C₂×C₄×C₅	C₂×C₉	1	2	yes	complemented
25	C₂²×C₉	—	2	2	no	—
26	C₂²×C₉	C₄×C₅	1	2	yes	complemented
27	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
28	C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₅	—	2	2	no	—
30	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅	C₃×D₄	1	2	yes	—
32	C₂³×C₃	C₂×C₃×C₅	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×C₃×C₅	1	2	yes	—
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	C₄×C₉	1	2	yes	complemented
36	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
37	C₄×C₅	—	2	2	no	—
38	C₂×C₉	—	2	2	no	—
39	C₂×C₉	C₂×C₄×C₅	1	1	yes	characteristic
40	C₂×C₉	C₅×D₄	1	2	yes	—
41	C₂²⋊C₄	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₃×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₃	—	2	2	no	—
44	C₂²×C₃	C₃×C₄×C₅	1	2	yes	—
45	C₂²×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
46	C₃×C₄	—	2	2	no	—
47	C₂×C₅	—	2	2	no	—
48	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
49	C₂×C₅	D₄×C₉	1	2	yes	—
50	C₉	C₅×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³	C₂×C₅×C₉	1	1	yes	characteristic
52	C₂×C₄	C₂×C₅×C₉	1	2	yes	complemented
53	C₂×C₃	—	2	2	no	—
54	C₂×C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
55	C₂×C₃	C₃×C₅×D₄	1	2	yes	—
56	C₅	C₉×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²	—	2	2	no	—
58	C₂²	C₄×C₅×C₉	1	2	yes	complemented
59	C₂²	C₂²×C₅×C₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₃	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
62	C₂	—	2	2	no	—
63	C₂	C₂×C₄×C₅×C₉	1	1	yes	characteristic
64	C₂	C₅×D₄×C₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₉	C₂²
commutator	C₂	C₂×C₄×C₅×C₉
Frattini, cosocle	C₂²×C₃	C₂²×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×C₃
prime radical	C₂³×C₃×C₅	C₂×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₉	C₂	semidirect
5. C₂²×C₅×C₉	C₄	semidirect
8. C₉×C₂²⋊C₄	C₅	direct (N × H)
14. C₅×C₂²⋊C₄	C₉	direct (N × H)
16. C₂×C₄×C₉	C₂×C₅	semidirect
22. C₅×C₉	C₂²⋊C₄	direct (N × H)
24. C₂×C₄×C₅	C₂×C₉	semidirect
26. C₂²×C₉	C₄×C₅	semidirect
35. C₂²×C₅	C₄×C₉	semidirect
41. C₂²⋊C₄	C₅×C₉	direct (N × H)
50. C₉	C₅×C₂²⋊C₄	direct (N × H)
52. C₂×C₄	C₂×C₅×C₉	semidirect
56. C₅	C₉×C₂²⋊C₄	direct (N × H)
58. C₂²	C₄×C₅×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₉	2
C₂×C₄×C₅×C₉	2
C₃×C₅×C₂²⋊C₄	3
C₉×C₂²⋊C₄	5

Subgroups — C5×C9×C22⋊C4 #10623613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₉×C₂²⋊C₄ #10623613