Subgroups — C₅×#3673 #10623592

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#213	—	3	1	no	maximal
5	C₂²×C₅×C₉	—	2	1	no	—
6	C₂×C₅×D₉	—	2	1	no	—
7	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
8	#3673	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×D₄	D₃	1	1	yes	characteristic
10	C₅×D₁₂	—	3	1	no	—
11	C₅×C₃⋊C₈	—	3	1	no	—
12	C₂×C₅×C₉	—	4	1	no	—
13	C₂×C₅×C₉	D₄	1	1	yes	characteristic
14	C₅×D₉	—	4	1	no	—
15	C₅×D₈	—	9	1	no	Hall
16	D₄×C₉	C₂×C₅	1	1	yes	characteristic, complemented
17	D₃₆	C₂×C₅	1	1	yes	characteristic, complemented
18	C₉⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
19	C₂²×C₃×C₅	—	2	1	no	—
20	C₂×C₅×D₃	—	6	1	no	—
21	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
22	#213	—	3	1	no	—
23	C₅×C₉	D₈	1	1	yes	characteristic, complemented, Hall
24	C₅×D₄	—	9	1	no	—
25	C₅×D₄	D₉	1	1	yes	characteristic, complemented
26	C₅×C₈	—	9	1	no	—
27	C₂²×C₉	—	2	1	no	—
28	C₂×D₉	—	2	1	no	—
29	C₄×C₉	C₂²×C₅	1	1	yes	characteristic
30	C₂×C₃×C₅	—	4	1	no	—
31	C₂×C₃×C₅	#67	1	1	yes	characteristic
32	C₅×D₃	—	12	1	no	—
33	C₃×D₄	C₅×D₃	1	1	yes	characteristic
34	D₁₂	—	3	1	no	—
35	C₃⋊C₈	—	3	1	no	—
36	C₂²×C₅	—	2	1	no	—
37	C₂²×C₅	—	18	1	no	—
38	C₄×C₅	C₂×D₉	1	1	yes	characteristic
39	C₂×C₉	—	4	1	no	—
40	C₂×C₉	C₅×D₄	1	1	yes	characteristic
41	D₉	—	4	1	no	—
42	D₈	—	9	1	no	Sylow
43	C₃×C₅	#213	1	1	yes	characteristic
44	C₂²×C₃	—	2	1	no	—
45	C₂×D₃	—	6	1	no	—
46	C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
47	C₂×C₅	—	4	1	no	—
48	C₂×C₅	—	36	1	no	—
49	C₂×C₅	#611	1	1	yes	characteristic
50	C₉	C₅×D₈	1	1	yes	characteristic, complemented, Sylow
51	D₄	—	9	1	no	—
52	D₄	C₅×D₉	1	1	yes	characteristic, complemented
53	C₈	—	9	1	no	—
54	C₂×C₃	—	4	1	no	—
55	C₂×C₃	C₅×#67	1	1	yes	characteristic
56	D₃	—	12	1	no	—
57	C₅	#3673	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	2	1	no	—
59	C₂²	—	18	1	no	—
60	C₄	C₂×C₅×D₉	1	1	yes	characteristic
61	C₃	C₅×#213	1	1	yes	characteristic
62	C₂	—	4	1	no	—
63	C₂	—	36	1	no	—
64	C₂	C₅×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#611
commutator, cosocle	C₄×C₉	C₂²×C₅
Frattini	C₃×C₄	C₂×C₅×D₃
Fitting	C₅×D₄×C₉	C₂
socle	C₂×C₃×C₅	#67
hypercenter	C₅×D₄	D₉
hypocenter	C₉	C₅×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₅×D₄×C₉	C₂	semidirect
2. C₅×D₃₆	C₂	semidirect
3. C₅×C₉⋊C₈	C₂	semidirect
8. #3673	C₅	direct (N × H)
16. D₄×C₉	C₂×C₅	semidirect
17. D₃₆	C₂×C₅	semidirect
18. C₉⋊C₈	C₂×C₅	semidirect
23. C₅×C₉	D₈	semidirect
25. C₅×D₄	D₉	semidirect
50. C₉	C₅×D₈	semidirect
52. D₄	C₅×D₉	semidirect
57. C₅	#3673	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×D₄×C₉	2
C₅×D₃₆	2
C₅×C₉⋊C₈	2
C₅×#213	3
#3673	5

Subgroups — C5×#3673 #10623592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3673 #10623592