Subgroups — C₃×C₅×C₂²⋊C₄ #6721

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×C₃×C₄×C₅	C₂	1	2	yes	maximal, complemented
3	C₅×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, 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	C₃×C₄×C₅	—	2	2	no	—
8	C₃×C₂²⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₅	C₂×C₃	1	1	yes	characteristic
10	C₂×C₄×C₅	C₂×C₃	1	2	yes	complemented
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×C₃×C₄	C₂×C₅	1	2	yes	complemented
16	C₂²×C₅	—	2	2	no	—
17	C₂²×C₅	C₃×C₄	1	2	yes	complemented
18	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
19	C₄×C₅	—	2	2	no	—
20	C₂²⋊C₄	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
21	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, 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	C₃×C₄	—	2	2	no	—
26	C₂×C₅	—	2	2	no	—
27	C₂×C₅	C₂×C₃×C₄	1	1	yes	characteristic
28	C₂×C₅	C₃×D₄	1	2	yes	—
29	C₂³	C₂×C₃×C₅	1	1	yes	characteristic
30	C₂×C₄	C₂×C₃×C₅	1	2	yes	complemented
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	C₃×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²	—	2	2	no	—
36	C₂²	C₃×C₄×C₅	1	2	yes	complemented
37	C₂²	C₂²×C₃×C₅	1	1	yes	characteristic
38	C₄	—	2	2	no	—
39	C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	—	2	2	no	—
41	C₂	C₂×C₃×C₄×C₅	1	1	yes	characteristic
42	C₂	C₃×C₅×D₄	1	2	yes	—

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

Complements

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

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

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×C₃×C₄×C₅	2
C₅×C₂²⋊C₄	3
C₃×C₂²⋊C₄	5

Subgroups — C3×C5×C22⋊C4 #6721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₂²⋊C₄ #6721