Subgroups — C₃×C₅×C₄⋊C₄ #6722

Order 240.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
5	C₃×C₄×C₅	—	2	2	no	—
6	C₃×C₄×C₅	C₄	1	2	yes	complemented
7	C₃×C₄⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅	C₂×C₃	1	2	yes	—
9	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₅	D₄	1	1	yes	characteristic
12	C₂×C₃×C₅	Q₈	1	1	yes	characteristic
13	C₂×C₃×C₄	C₂×C₅	1	2	yes	—
14	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic
15	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
16	C₄×C₅	—	2	2	no	—
17	C₄×C₅	C₃×C₄	1	2	yes	complemented
18	C₄⋊C₄	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×C₅	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
21	C₃×C₄	—	2	2	no	—
22	C₃×C₄	C₄×C₅	1	2	yes	complemented
23	C₂×C₅	C₂×C₃×C₄	1	1	yes	characteristic
24	C₂×C₅	C₃×D₄	1	1	yes	characteristic
25	C₂×C₅	C₃×Q₈	1	1	yes	characteristic
26	C₂×C₄	C₂×C₃×C₅	1	2	yes	—
27	C₂×C₄	C₂×C₃×C₅	1	1	yes	characteristic
28	C₂×C₃	C₂×C₄×C₅	1	1	yes	characteristic
29	C₂×C₃	C₅×D₄	1	1	yes	characteristic
30	C₂×C₃	C₅×Q₈	1	1	yes	characteristic
31	C₅	C₃×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²	C₂²×C₃×C₅	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	C₃×C₄×C₅	1	2	yes	complemented
35	C₃	C₅×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
36	C₂	C₂×C₃×C₄×C₅	1	1	yes	characteristic
37	C₂	C₃×C₅×D₄	1	1	yes	characteristic
38	C₂	C₃×C₅×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂²
commutator	C₂	C₂×C₃×C₄×C₅
Frattini, cosocle	C₂²	C₂²×C₃×C₅
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
3. C₅×C₄⋊C₄	C₃	direct (N × H)
6. C₃×C₄×C₅	C₄	semidirect
7. C₃×C₄⋊C₄	C₅	direct (N × H)
17. C₄×C₅	C₃×C₄	semidirect
18. C₄⋊C₄	C₃×C₅	direct (N × H)
19. C₃×C₅	C₄⋊C₄	direct (N × H)
22. C₃×C₄	C₄×C₅	semidirect
31. C₅	C₃×C₄⋊C₄	direct (N × H)
34. C₄	C₃×C₄×C₅	semidirect
35. C₃	C₅×C₄⋊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
C₂×C₃×C₄×C₅	2
C₅×C₄⋊C₄	3
C₃×C₄⋊C₄	5

Subgroups — C3×C5×C4⋊C4 #6722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₄⋊C₄ #6722