Subgroups — C₉×#689 #10623578

Order 720.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₉×#168	C₂	1	1	yes	maximal, characteristic
4	C₃×#689	C₃	1	1	yes	maximal, characteristic
5	C₄×C₅×C₉	—	2	1	no	—
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₉×Dic₅	—	2	1	no	—
8	C₉×Q₁₆	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×Dic₁₀	C₂×C₃	1	1	yes	characteristic
11	C₃×#168	C₂×C₃	1	1	yes	characteristic
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	#689	C₉	1	1	yes	characteristic, direct product factor, Hall
14	Q₈×C₉	—	5	1	no	—
15	Q₈×C₉	D₅	1	1	yes	characteristic
16	C₈×C₉	—	5	1	no	—
17	C₃×C₄×C₅	—	2	1	no	—
18	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
19	C₃×Dic₅	—	2	1	no	—
20	C₃×Q₁₆	—	5	1	no	—
21	C₅×C₉	Q₁₆	1	1	yes	characteristic, complemented, Hall
22	C₅×Q₈	C₂×C₉	1	1	yes	characteristic
23	Dic₁₀	C₂×C₉	1	1	yes	characteristic
24	#168	C₂×C₉	1	1	yes	characteristic
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	—	10	1	no	—
27	C₄×C₉	C₂×D₅	1	1	yes	characteristic
28	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
29	C₃×Q₈	—	5	1	no	—
30	C₃×Q₈	C₃×D₅	1	1	yes	characteristic
31	C₃×C₈	—	5	1	no	—
32	C₄×C₅	—	2	1	no	—
33	C₄×C₅	C₂²×C₉	1	1	yes	characteristic
34	Dic₅	—	2	1	no	—
35	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
36	Q₁₆	—	5	1	no	Sylow
37	C₃×C₅	C₃×Q₁₆	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	10	1	no	—
40	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
41	C₂×C₅	D₄×C₉	1	1	yes	characteristic
42	C₉	#689	1	1	yes	characteristic, direct product factor, Sylow
43	Q₈	—	5	1	no	—
44	Q₈	C₉×D₅	1	1	yes	characteristic
45	C₈	—	5	1	no	—
46	C₂×C₃	C₃×C₂²⋊D₅	1	1	yes	characteristic
47	C₅	C₉×Q₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₄	—	2	1	no	—
49	C₄	—	10	1	no	—
50	C₄	C₂×C₉×D₅	1	1	yes	characteristic
51	C₃	C₃×#689	1	1	yes	characteristic
52	C₂	C₉×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×D₅
Fitting	C₅×Q₈×C₉	C₂
socle, prime radical	C₂×C₃×C₅	C₃×D₄
cosocle	C₃×C₄×C₅	C₂²×C₃
hypercenter	Q₈×C₉	D₅
hypocenter	C₅	C₉×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
13. #689	C₉	direct (N × H)
21. C₅×C₉	Q₁₆	semidirect
42. C₉	#689	direct (N × H)
47. C₅	C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×C₉	2
C₉×Dic₁₀	2
C₉×#168	2
C₃×#689	3
C₉×Q₁₆	5

Subgroups — C9×#689 #10623578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#689 #10623578