Subgroups — C₅×C₉×C₄⋊C₄ #10623614

Order 720.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
5	C₄×C₅×C₉	—	2	2	no	—
6	C₄×C₅×C₉	C₄	1	2	yes	complemented
7	C₉×C₄⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₄×C₅	C₂×C₃	1	2	yes	—
9	C₂×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅×C₉	D₄	1	1	yes	characteristic
12	C₂×C₅×C₉	Q₈	1	1	yes	characteristic
13	C₅×C₄⋊C₄	C₉	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₄×C₉	C₂×C₅	1	2	yes	—
15	C₂×C₄×C₉	C₂×C₅	1	1	yes	characteristic
16	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
17	C₃×C₄×C₅	—	2	2	no	—
18	C₃×C₄×C₅	C₃×C₄	1	2	yes	—
19	C₃×C₄⋊C₄	C₃×C₅	1	1	yes	characteristic
20	C₅×C₉	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₄×C₅	C₂×C₉	1	2	yes	—
22	C₂×C₄×C₅	C₂×C₉	1	1	yes	characteristic
23	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
24	C₄×C₉	—	2	2	no	—
25	C₄×C₉	C₄×C₅	1	2	yes	complemented
26	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
28	C₂×C₃×C₅	C₃×Q₈	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂×C₃×C₅	1	2	yes	—
30	C₂×C₃×C₄	C₂×C₃×C₅	1	1	yes	characteristic
31	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
32	C₄×C₅	—	2	2	no	—
33	C₄×C₅	C₄×C₉	1	2	yes	complemented
34	C₂×C₉	C₂×C₄×C₅	1	1	yes	characteristic
35	C₂×C₉	C₅×D₄	1	1	yes	characteristic
36	C₂×C₉	C₅×Q₈	1	1	yes	characteristic
37	C₄⋊C₄	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
38	C₃×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
40	C₃×C₄	—	2	2	no	—
41	C₃×C₄	C₃×C₄×C₅	1	2	yes	—
42	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
43	C₂×C₅	D₄×C₉	1	1	yes	characteristic
44	C₂×C₅	Q₈×C₉	1	1	yes	characteristic
45	C₉	C₅×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×C₄	C₂×C₅×C₉	1	2	yes	—
47	C₂×C₄	C₂×C₅×C₉	1	1	yes	characteristic
48	C₂×C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
49	C₂×C₃	C₃×C₅×D₄	1	1	yes	characteristic
50	C₂×C₃	C₃×C₅×Q₈	1	1	yes	characteristic
51	C₅	C₉×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²	C₂²×C₅×C₉	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	C₄×C₅×C₉	1	2	yes	complemented
55	C₃	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
56	C₂	C₂×C₄×C₅×C₉	1	1	yes	characteristic
57	C₂	C₅×D₄×C₉	1	1	yes	characteristic
58	C₂	C₅×Q₈×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₉	C₂²
commutator	C₂	C₂×C₄×C₅×C₉
Frattini, cosocle	C₂²×C₃	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂²×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
6. C₄×C₅×C₉	C₄	semidirect
7. C₉×C₄⋊C₄	C₅	direct (N × H)
13. C₅×C₄⋊C₄	C₉	direct (N × H)
20. C₅×C₉	C₄⋊C₄	direct (N × H)
25. C₄×C₉	C₄×C₅	semidirect
33. C₄×C₅	C₄×C₉	semidirect
37. C₄⋊C₄	C₅×C₉	direct (N × H)
45. C₉	C₅×C₄⋊C₄	direct (N × H)
51. C₅	C₉×C₄⋊C₄	direct (N × H)
54. C₄	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
C₂×C₄×C₅×C₉	2
C₃×C₅×C₄⋊C₄	3
C₉×C₄⋊C₄	5

Subgroups — C5×C9×C4⋊C4 #10623614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₉×C₄⋊C₄ #10623614