Subgroups — C₂×C₅×C₃⋊(C₃⋊C₈) #10623887

Order 720.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×C₃⋊C₈	—	3	4	no	maximal
4	C₂²×C₃²×C₅	C₄	1	1	yes	characteristic
5	C₃²×C₄×C₅	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₄×C₅	D₃	1	4	yes	—
9	C₅×C₃⋊C₈	—	3	8	no	—
10	C₂×C₃²×C₅	C₈	1	2	yes	complemented
11	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	C₂×C₅×C₈	—	9	1	no	Hall
13	C₂×C₃²×C₄	C₂×C₅	1	1	yes	characteristic
14	C₃⋊(C₃⋊C₈)	C₂×C₅	1	2	yes	direct product factor
15	C₂²×C₃×C₅	Dic₃	1	4	yes	—
16	C₃×C₄×C₅	Dic₃	1	4	yes	—
17	C₃×C₄×C₅	C₂×D₃	1	4	yes	—
18	C₂×C₃⋊C₈	—	3	4	no	—
19	C₃²×C₅	C₂×C₈	1	1	yes	characteristic, complemented, Hall
20	C₂×C₄×C₅	C₃⋊D₃	1	1	yes	characteristic
21	C₅×C₈	—	9	2	no	—
22	C₂²×C₃²	C₄×C₅	1	1	yes	characteristic
23	C₃²×C₄	C₄×C₅	1	1	yes	characteristic
24	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
25	C₂×C₃×C₅	C₃⋊C₈	1	8	yes	complemented
26	C₂×C₃×C₅	C₂×Dic₃	1	4	yes	—
27	C₂×C₃×C₄	C₅×D₃	1	4	yes	—
28	C₃⋊C₈	—	3	8	no	—
29	C₂²×C₅	C₃⋊Dic₃	1	1	yes	characteristic
30	C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
31	C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
32	C₂×C₃²	C₅×C₈	1	2	yes	complemented
33	C₂×C₃²	C₂×C₄×C₅	1	1	yes	characteristic
34	C₂×C₈	—	9	1	no	Sylow
35	C₃×C₅	C₂×C₃⋊C₈	1	4	yes	complemented
36	C₂²×C₃	C₅×Dic₃	1	4	yes	—
37	C₃×C₄	C₅×Dic₃	1	4	yes	—
38	C₃×C₄	C₂×C₅×D₃	1	4	yes	—
39	C₂×C₅	C₃⋊(C₃⋊C₈)	1	2	yes	direct product factor
40	C₂×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
41	C₃²	C₂×C₅×C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	C₅×C₃⋊D₃	1	1	yes	characteristic
43	C₈	—	9	2	no	—
44	C₂×C₃	C₅×C₃⋊C₈	1	8	yes	complemented
45	C₂×C₃	C₂×C₅×Dic₃	1	4	yes	—
46	C₅	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²	C₅×C₃⋊Dic₃	1	1	yes	characteristic
48	C₄	C₅×C₃⋊Dic₃	1	1	yes	characteristic
49	C₄	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
50	C₃	C₂×C₅×C₃⋊C₈	1	4	yes	complemented
51	C₂	C₅×C₃⋊(C₃⋊C₈)	1	2	yes	direct product factor
52	C₂	C₂×C₅×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₅×C₈
Frattini	C₄	C₂×C₅×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₅	C₂
socle, prime radical	C₂²×C₃²×C₅	C₄
cosocle	C₃²×C₄	C₂²×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
7. C₂×C₃⋊(C₃⋊C₈)	C₅	direct (N × H)
10. C₂×C₃²×C₅	C₈	semidirect
14. C₃⋊(C₃⋊C₈)	C₂×C₅	direct (N × H)
19. C₃²×C₅	C₂×C₈	semidirect
25. C₂×C₃×C₅	C₃⋊C₈	semidirect
32. C₂×C₃²	C₅×C₈	semidirect
35. C₃×C₅	C₂×C₃⋊C₈	semidirect
39. C₂×C₅	C₃⋊(C₃⋊C₈)	direct (N × H)
41. C₃²	C₂×C₅×C₈	semidirect
44. C₂×C₃	C₅×C₃⋊C₈	semidirect
46. C₅	C₂×C₃⋊(C₃⋊C₈)	direct (N × H)
50. C₃	C₂×C₅×C₃⋊C₈	semidirect
51. C₂	C₅×C₃⋊(C₃⋊C₈)	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×C₃²×C₄×C₅	2
C₅×C₃⋊(C₃⋊C₈)	2
C₂×C₅×C₃⋊C₈	3
C₂×C₃⋊(C₃⋊C₈)	5

Subgroups — C2×C5×C3⋊(C3⋊C8) #10623887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₃⋊(C₃⋊C₈) #10623887