Subgroups — C₄⋊Dic₄₅ #10623605

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	1	yes	maximal, characteristic
2	C₂×Dic₄₅	C₂	1	2	yes	maximal
3	C₄⋊Dic₁₅	—	3	1	no	maximal
4	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
5	C₄×C₅×C₉	C₄	1	2	yes	complemented
6	Dic₄₅	—	2	2	no	—
7	C₄⋊Dic₉	—	5	1	no	maximal, Hall
8	C₂×C₃×C₄×C₅	D₃	1	1	yes	characteristic
9	C₂×Dic₁₅	—	3	2	no	—
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	C₄⋊Dic₅	—	9	1	no	Hall
14	C₂×C₄×C₉	D₅	1	1	yes	characteristic
15	C₂×Dic₉	—	5	2	no	—
16	C₂²×C₃×C₅	C₂×D₃	1	1	yes	characteristic
17	C₃×C₄×C₅	Dic₃	1	2	yes	—
18	Dic₁₅	—	6	2	no	—
19	C₄⋊Dic₃	—	15	1	no	—
20	C₅×C₉	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
21	C₂×C₄×C₅	D₉	1	1	yes	characteristic
22	C₂×Dic₅	—	9	2	no	—
23	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
24	C₄×C₉	Dic₅	1	2	yes	complemented
25	Dic₉	—	10	2	no	—
26	C₂×C₃×C₅	Dic₆	1	1	yes	characteristic
27	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
28	C₂×C₃×C₅	C₂×Dic₃	1	1	yes	characteristic
29	C₂×C₃×C₄	D₁₅	1	1	yes	characteristic
30	C₂×Dic₃	—	15	2	no	—
31	C₂²×C₅	C₂×D₉	1	1	yes	characteristic
32	C₄×C₅	Dic₉	1	2	yes	complemented
33	Dic₅	—	18	2	no	—
34	C₂×C₉	Dic₁₀	1	1	yes	characteristic
35	C₂×C₉	D₂₀	1	1	yes	characteristic
36	C₂×C₉	C₂×Dic₅	1	1	yes	characteristic
37	C₄⋊C₄	—	45	1	no	Sylow
38	C₃×C₅	C₄⋊Dic₃	1	1	yes	characteristic
39	C₂²×C₃	C₂×D₁₅	1	1	yes	characteristic
40	C₃×C₄	Dic₁₅	1	2	yes	—
41	Dic₃	—	30	2	no	—
42	C₂×C₅	Dic₁₈	1	1	yes	characteristic
43	C₂×C₅	D₃₆	1	1	yes	characteristic
44	C₂×C₅	C₂×Dic₉	1	1	yes	characteristic
45	C₉	C₄⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	45	2	no	—
47	C₂×C₄	D₄₅	1	1	yes	characteristic
48	C₂×C₃	Dic₃₀	1	1	yes	characteristic
49	C₂×C₃	D₆₀	1	1	yes	characteristic
50	C₂×C₃	C₂×Dic₁₅	1	1	yes	characteristic
51	C₅	C₄⋊Dic₉	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂×D₄₅	1	1	yes	characteristic
53	C₄	—	90	2	no	—
54	C₄	Dic₄₅	1	2	yes	complemented
55	C₃	C₄⋊Dic₁₅	1	1	yes	characteristic
56	C₂	Dic₉₀	1	1	yes	characteristic
57	C₂	D₁₈₀	1	1	yes	characteristic
58	C₂	C₂×Dic₄₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₅
commutator	C₂×C₅×C₉	C₂×C₄
Frattini	C₂²×C₃	C₂×D₁₅
Fitting	C₂×C₄×C₅×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₃
cosocle	C₂²×C₅×C₉	C₂²
hypercenter	C₂×C₄	D₄₅
hypocenter	C₅×C₉	C₄⋊C₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
5. C₄×C₅×C₉	C₄	semidirect
20. C₅×C₉	C₄⋊C₄	semidirect
24. C₄×C₉	Dic₅	semidirect
32. C₄×C₅	Dic₉	semidirect
45. C₉	C₄⋊Dic₅	semidirect
51. C₅	C₄⋊Dic₉	semidirect
54. C₄	Dic₄₅	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
C₂×Dic₄₅	2
C₄⋊Dic₁₅	3
C₄⋊Dic₉	5

Subgroups — C4⋊Dic45 #10623605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊Dic₄₅ #10623605