Subgroups — D₅×Dic₆ #6764

Order 240.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
4	#1204	C₂	1	2	yes	maximal, complemented
5	D₅×Dic₃	C₂	1	2	yes	maximal
6	Q₈×D₅	—	3	1	no	maximal, Hall
7	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	Dic₁₅	C₂²	1	2	yes	—
10	C₃×Dic₅	C₂²	1	1	yes	characteristic
11	C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×Dic₆	—	5	1	no	maximal, Hall
13	C₅×Q₈	—	3	1	no	—
14	C₄×D₅	—	3	2	no	—
15	C₄×D₅	D₃	1	1	yes	characteristic
16	Dic₁₀	—	3	1	no	—
17	Dic₁₀	—	3	2	no	—
18	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₃×D₅	Q₈	1	2	yes	complemented
20	C₂×C₃×C₄	—	5	1	no	—
21	C₂×Dic₃	—	5	2	no	—
22	Dic₆	—	5	1	no	—
23	Dic₆	—	5	2	no	—
24	Dic₆	D₅	1	1	yes	characteristic, direct product factor
25	C₂×D₅	C₂×D₃	1	1	yes	characteristic
26	C₄×C₅	—	3	2	no	—
27	C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	Dic₅	—	3	2	no	—
29	Dic₅	C₂×D₃	1	1	yes	characteristic
30	C₂×Q₈	—	15	1	no	Sylow
31	C₃×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₃	—	5	1	no	—
33	C₃×C₄	—	5	1	no	—
34	C₃×C₄	C₂×D₅	1	1	yes	characteristic
35	Dic₃	—	5	2	no	—
36	Dic₃	C₂×D₅	1	2	yes	—
37	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
38	D₅	Dic₆	1	2	yes	direct product factor
39	Q₈	—	3	1	no	—
40	Q₈	—	15	2	no	—
41	Q₈	—	15	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	15	2	no	—
44	C₂×C₃	—	5	2	no	—
45	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
46	C₅	C₂×Dic₆	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	5	1	no	—
48	C₄	—	3	2	no	—
49	C₄	—	5	1	no	—
50	C₄	—	15	2	no	—
51	C₄	D₃×D₅	1	1	yes	characteristic
52	C₃	Q₈×D₅	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	5	2	no	—
54	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₄×C₅	C₂²
prime radical	C₂×C₃×D₅	C₂²
hypercenter	C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. Dic₃₀	C₂	semidirect
2. C₅×Dic₆	C₂	semidirect
4. #1204	C₂	semidirect
19. C₃×D₅	Q₈	semidirect
24. Dic₆	D₅	direct (N × H)
31. C₃×C₅	C₂×Q₈	semidirect
38. D₅	Dic₆	direct (N × H)
46. C₅	C₂×Dic₆	semidirect
52. C₃	Q₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₀	2
C₅×Dic₆	2
C₃×C₄×D₅	2
#1204	2
D₅×Dic₃	2
Q₈×D₅	3
C₂×Dic₆	5

Subgroups — D5×Dic6 #6764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×Dic₆ #6764