Subgroups — C₂×C₅×Dic₆ #6804

Order 240.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₃	C₂	1	2	yes	maximal
3	C₅×Dic₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×Q₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅	C₂²	1	2	yes	—
7	C₅×Dic₃	C₂²	1	4	yes	—
8	C₂×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×Q₈	—	3	4	no	—
10	C₂×C₄×C₅	—	3	2	no	—
11	C₂×C₄×C₅	D₃	1	1	yes	characteristic
12	C₂×C₃×C₅	Q₈	1	2	yes	complemented
13	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
14	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic
15	C₂×Dic₃	C₂×C₅	1	2	yes	—
16	Dic₆	C₂×C₅	1	4	yes	direct product factor
17	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
18	C₄×C₅	—	3	4	no	—
19	C₄×C₅	C₂×D₃	1	2	yes	—
20	C₂×Q₈	—	3	1	no	Sylow
21	C₃×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
23	C₃×C₄	C₂²×C₅	1	2	yes	—
24	Dic₃	C₂²×C₅	1	4	yes	—
25	C₂×C₅	Dic₆	1	2	yes	direct product factor
26	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
27	Q₈	—	3	4	no	—
28	C₂×C₄	—	3	2	no	—
29	C₂×C₄	C₅×D₃	1	1	yes	characteristic
30	C₂×C₃	C₅×Q₈	1	2	yes	complemented
31	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
32	C₅	C₂×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²	C₂×C₅×D₃	1	1	yes	characteristic
34	C₄	—	3	4	no	—
35	C₄	C₂×C₅×D₃	1	2	yes	—
36	C₃	C₂×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
37	C₂	C₅×Dic₆	1	2	yes	direct product factor
38	C₂	C₂²×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅	C₂
socle, prime radical	C₂²×C₃×C₅	C₂²
hypercenter	C₂×C₄×C₅	D₃
hypocenter	C₃	C₂×C₅×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₅×Dic₆	C₂	direct (N × H)
8. C₂×Dic₆	C₅	direct (N × H)
12. C₂×C₃×C₅	Q₈	semidirect
16. Dic₆	C₂×C₅	direct (N × H)
21. C₃×C₅	C₂×Q₈	semidirect
25. C₂×C₅	Dic₆	direct (N × H)
30. C₂×C₃	C₅×Q₈	semidirect
32. C₅	C₂×Dic₆	direct (N × H)
36. C₃	C₂×C₅×Q₈	semidirect
37. C₂	C₅×Dic₆	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
C₂×C₅×Dic₃	2
C₅×Dic₆	2
C₂×C₅×Q₈	3
C₂×Dic₆	5

Subgroups — C2×C5×Dic6 #6804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×Dic₆ #6804