Subgroups — C₂×C₉×#168 #10623569

Order 720.

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

	subgroup	G/H
center, hypercenter	C₂×C₄×C₉	D₅
commutator, hypocenter	C₅	C₂×C₈×C₉
Frattini	C₃×C₄	C₂×C₃×D₅
Fitting	C₂×C₄×C₅×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	C₃×C₄
cosocle	C₃×C₄×C₅	C₂²×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₉×#168	C₂	direct (N × H)
10. C₂×C₅×C₉	C₈	semidirect
12. C₂×#168	C₉	direct (N × H)
19. C₅×C₉	C₂×C₈	semidirect
21. #168	C₂×C₉	direct (N × H)
32. C₂×C₉	#168	direct (N × H)
39. C₂×C₅	C₈×C₉	semidirect
41. C₉	C₂×#168	direct (N × H)
46. C₅	C₂×C₈×C₉	semidirect
51. C₂	C₉×#168	direct (N × H)

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
C₉×#168	2
C₂×C₃×#168	3
C₂×C₈×C₉	5

Subgroups — C2×C9×#168 #10623569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#168 #10623569