Subgroups — C₅×#3718 #10623785

Order 720.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Dic₆	C₂	1	2	yes	maximal
2	C₅×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₅×#216	—	3	2	no	maximal
4	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
5	C₃×C₅×Dic₃	—	2	2	no	—
6	#3718	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×Q₈	—	3	2	no	—
8	C₅×Dic₆	D₃	1	2	yes	—
9	C₅×C₃⋊C₈	—	3	2	no	—
10	C₅×C₃⋊C₈	—	6	1	no	—
11	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
12	C₅×Q₁₆	—	9	1	no	Hall
13	C₃×Dic₆	C₂×C₅	1	2	yes	—
14	C₃⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
15	C₃×C₄×C₅	—	2	1	no	—
16	C₃×C₄×C₅	—	6	2	no	—
17	C₃×C₄×C₅	C₂×D₃	1	2	yes	—
18	C₅×Dic₃	—	2	2	no	—
19	#216	—	3	2	no	—
20	C₃²×C₅	Q₁₆	1	1	yes	characteristic, complemented, Hall
21	C₅×Q₈	—	3	2	no	—
22	C₅×C₈	—	9	1	no	—
23	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
24	C₃×Dic₃	—	2	2	no	—
25	C₂×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅	#67	1	2	yes	—
27	C₃×Q₈	—	3	2	no	—
28	Dic₆	C₅×D₃	1	2	yes	—
29	C₃⋊C₈	—	3	2	no	—
30	C₃⋊C₈	—	6	1	no	—
31	C₄×C₅	—	6	2	no	—
32	C₄×C₅	D₃²	1	1	yes	characteristic
33	C₂×C₃²	C₅×D₄	1	1	yes	characteristic
34	Q₁₆	—	9	1	no	Sylow
35	C₃×C₅	—	2	1	no	—
36	C₃×C₅	#216	1	2	yes	complemented
37	C₃×C₄	—	2	1	no	—
38	C₃×C₄	—	6	2	no	—
39	C₃×C₄	C₂×C₅×D₃	1	2	yes	—
40	Dic₃	—	2	2	no	—
41	C₂×C₅	#625	1	1	yes	characteristic
42	C₃²	C₅×Q₁₆	1	1	yes	characteristic, complemented, Sylow
43	Q₈	—	3	2	no	—
44	C₈	—	9	1	no	—
45	C₂×C₃	—	2	1	no	—
46	C₂×C₃	C₅×#67	1	2	yes	—
47	C₅	#3718	1	1	yes	characteristic, direct product factor, Sylow
48	C₄	—	6	2	no	—
49	C₄	C₅×D₃²	1	1	yes	characteristic
50	C₃	—	2	1	no	—
51	C₃	C₅×#216	1	2	yes	complemented
52	C₂	C₅×#625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#625
commutator, cosocle	C₃²×C₄	C₂²×C₅
Frattini	C₄	C₅×D₃²
Fitting	C₃²×C₄×C₅	C₂²
socle, prime radical	C₂×C₃²×C₅	D₄
hypercenter	C₄×C₅	D₃²
hypocenter	C₃²	C₅×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅

Complements

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

N	H ≅ G/N	type
6. #3718	C₅	direct (N × H)
20. C₃²×C₅	Q₁₆	semidirect
36. C₃×C₅	#216	semidirect
42. C₃²	C₅×Q₁₆	semidirect
47. C₅	#3718	direct (N × H)
51. C₃	C₅×#216	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×Dic₆	2
C₅×C₃⋊(C₃⋊C₈)	2
C₅×#216	3
#3718	5

Subgroups — C5×#3718 #10623785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3718 #10623785