Subgroups — C₅×#217 #6703

Order 240.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₅×C₂²⋊C₄	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅	—	2	2	no	—
5	C₂²×C₃×C₅	C₄	1	2	yes	complemented
6	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₅×Dic₃	—	2	2	no	—
8	#217	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₅	D₃	1	1	yes	characteristic
10	C₂×C₄×C₅	—	3	2	no	—
11	C₂×C₃×C₅	—	2	2	no	—
12	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₅	D₄	1	2	yes	—
14	C₂³×C₃	C₂×C₅	1	1	yes	characteristic
15	C₂×Dic₃	C₂×C₅	1	2	yes	complemented
16	C₂²×C₅	—	2	2	no	—
17	C₂²×C₅	Dic₃	1	2	yes	complemented
18	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
19	C₄×C₅	—	6	2	no	—
20	C₂²⋊C₄	—	3	1	no	Sylow
21	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	—	2	2	no	—
23	C₂²×C₃	C₄×C₅	1	2	yes	complemented
24	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
25	Dic₃	—	2	2	no	—
26	C₂×C₅	—	2	2	no	—
27	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
28	C₂×C₅	#67	1	2	yes	—
29	C₂³	C₅×D₃	1	1	yes	characteristic
30	C₂×C₄	—	3	2	no	—
31	C₂×C₃	—	2	2	no	—
32	C₂×C₃	C₂×C₄×C₅	1	1	yes	characteristic
33	C₂×C₃	C₅×D₄	1	2	yes	—
34	C₅	#217	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²	—	2	2	no	—
36	C₂²	C₅×Dic₃	1	2	yes	complemented
37	C₂²	C₂×C₅×D₃	1	1	yes	characteristic
38	C₄	—	6	2	no	—
39	C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
40	C₂	—	2	2	no	—
41	C₂	C₂×C₅×Dic₃	1	1	yes	characteristic
42	C₂	C₅×#67	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅
Frattini	C₂²	C₂×C₅×D₃
Fitting, prime radical	C₂³×C₃×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
cosocle	C₂²×C₃	C₂²×C₅
hypercenter	C₂³×C₅	D₃
hypocenter	C₃	C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×Dic₃	C₂	semidirect
5. C₂²×C₃×C₅	C₄	semidirect
8. #217	C₅	direct (N × H)
15. C₂×Dic₃	C₂×C₅	semidirect
17. C₂²×C₅	Dic₃	semidirect
21. C₃×C₅	C₂²⋊C₄	semidirect
23. C₂²×C₃	C₄×C₅	semidirect
34. C₅	#217	direct (N × H)
36. C₂²	C₅×Dic₃	semidirect
39. C₃	C₅×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅	2
C₂×C₅×Dic₃	2
C₅×C₂²⋊C₄	3
#217	5

Subgroups — C5×#217 #6703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#217 #6703