Subgroups — C₅×#3754 #10623894

Order 720.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#214	—	3	4	no	maximal
5	C₂²×C₃²×C₅	—	2	1	no	—
6	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×C₃⋊Dic₃	—	2	1	no	—
8	#3754	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×D₄	D₃	1	4	yes	—
10	C₅×Dic₆	—	3	4	no	—
11	C₅×C₃⋊C₈	—	3	4	no	—
12	C₂×C₃²×C₅	—	4	1	no	—
13	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
14	C₅×SD₁₆	—	9	1	no	Hall
15	C₃²×D₄	C₂×C₅	1	1	yes	characteristic
16	#634	C₂×C₅	1	1	yes	characteristic, complemented
17	C₃⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂²×C₃×C₅	—	2	4	no	—
19	C₃×C₄×C₅	C₂×D₃	1	4	yes	—
20	C₅×Dic₃	—	6	4	no	—
21	#214	—	3	4	no	—
22	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
23	C₅×Q₈	—	9	1	no	—
24	C₅×D₄	C₃⋊D₃	1	1	yes	characteristic
25	C₅×C₈	—	9	1	no	—
26	C₂²×C₃²	—	2	1	no	—
27	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
28	C₃⋊Dic₃	—	2	1	no	—
29	C₂×C₃×C₅	—	4	4	no	—
30	C₂×C₃×C₅	#67	1	4	yes	—
31	C₃×D₄	C₅×D₃	1	4	yes	—
32	Dic₆	—	3	4	no	—
33	C₃⋊C₈	—	3	4	no	—
34	C₂²×C₅	—	2	1	no	—
35	C₄×C₅	—	18	1	no	—
36	C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	C₅×D₄	1	1	yes	characteristic
39	SD₁₆	—	9	1	no	Sylow
40	C₃×C₅	#214	1	4	yes	complemented
41	C₂²×C₃	—	2	4	no	—
42	C₃×C₄	C₂×C₅×D₃	1	4	yes	—
43	Dic₃	—	6	4	no	—
44	C₂×C₅	—	4	1	no	—
45	C₂×C₅	#638	1	1	yes	characteristic
46	C₃²	C₅×SD₁₆	1	1	yes	characteristic, complemented, Sylow
47	Q₈	—	9	1	no	—
48	D₄	C₅×C₃⋊D₃	1	1	yes	characteristic
49	C₈	—	9	1	no	—
50	C₂×C₃	—	4	4	no	—
51	C₂×C₃	C₅×#67	1	4	yes	—
52	C₅	#3754	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	2	1	no	—
54	C₄	—	18	1	no	—
55	C₄	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
56	C₃	C₅×#214	1	4	yes	complemented
57	C₂	—	4	1	no	—
58	C₂	C₅×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#638
commutator, cosocle	C₃²×C₄	C₂²×C₅
Frattini	C₄	C₂×C₅×C₃⋊D₃
Fitting, prime radical	C₃²×C₅×D₄	C₂
socle	C₂×C₃²×C₅	D₄
hypercenter	C₅×D₄	C₃⋊D₃
hypocenter	C₃²	C₅×SD₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₅×#634	C₂	semidirect
3. C₅×C₃⋊(C₃⋊C₈)	C₂	semidirect
8. #3754	C₅	direct (N × H)
16. #634	C₂×C₅	semidirect
17. C₃⋊(C₃⋊C₈)	C₂×C₅	semidirect
22. C₃²×C₅	SD₁₆	semidirect
40. C₃×C₅	#214	semidirect
46. C₃²	C₅×SD₁₆	semidirect
52. C₅	#3754	direct (N × H)
56. C₃	C₅×#214	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×D₄	2
C₅×#634	2
C₅×C₃⋊(C₃⋊C₈)	2
C₅×#214	3
#3754	5

Subgroups — C5×#3754 #10623894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3754 #10623894