Subgroups — C₅×(C₂×D₃)⋊C₄ #6698

Order 240.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊C₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₃	—	2	1	no	—
7	C₂×C₅×D₃	—	2	1	no	—
8	C₂×C₅×D₃	C₄	1	2	yes	complemented
9	C₃×C₄×C₅	—	2	1	no	—
10	C₅×Dic₃	—	2	1	no	—
11	(C₂×D₃)⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂³×C₅	—	3	1	no	—
13	C₂×C₄×C₅	—	3	1	no	—
14	C₂×C₄×C₅	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₅	D₄	1	1	yes	characteristic
17	C₂×C₃×C₅	D₄	1	1	yes	characteristic
18	C₅×D₃	—	2	2	no	—
19	C₂²×D₃	C₂×C₅	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂²×C₅	—	3	2	no	—
23	C₂²×C₅	—	6	1	no	—
24	C₂²×C₅	—	6	1	no	—
25	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
26	C₄×C₅	—	2	1	no	—
27	C₄×C₅	—	6	1	no	—
28	C₂²⋊C₄	—	3	1	no	Sylow
29	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
31	C₂×D₃	—	2	1	no	—
32	C₂×D₃	—	2	1	no	—
33	C₂×D₃	C₄×C₅	1	2	yes	complemented
34	C₃×C₄	—	2	1	no	—
35	Dic₃	—	2	1	no	—
36	C₂×C₅	—	6	2	no	—
37	C₂×C₅	C₄×D₃	1	1	yes	characteristic
38	C₂×C₅	D₁₂	1	1	yes	characteristic
39	C₂×C₅	#67	1	1	yes	characteristic
40	C₂³	—	3	1	no	—
41	C₂×C₄	—	3	1	no	—
42	C₂×C₄	C₅×D₃	1	1	yes	characteristic, complemented
43	C₂×C₃	C₂×C₄×C₅	1	1	yes	characteristic
44	C₂×C₃	C₅×D₄	1	1	yes	characteristic
45	C₂×C₃	C₅×D₄	1	1	yes	characteristic
46	D₃	—	2	2	no	—
47	C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²	—	3	2	no	—
49	C₂²	—	6	1	no	—
50	C₂²	—	6	1	no	—
51	C₂²	C₂×C₅×D₃	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	6	1	no	—
54	C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	6	2	no	—
56	C₂	C₄×C₅×D₃	1	1	yes	characteristic
57	C₂	C₅×D₁₂	1	1	yes	characteristic
58	C₂	C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅
Frattini	C₂²	C₂×C₅×D₃
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
prime radical	C₂²×C₅×D₃	C₂
cosocle	C₂²×C₃	C₂²×C₅
hypercenter	C₂×C₄×C₅	D₃
hypocenter	C₃	C₅×C₂²⋊C₄
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
2. C₂×C₃×C₄×C₅	C₂	semidirect
3. C₂×C₅×Dic₃	C₂	semidirect
8. C₂×C₅×D₃	C₄	semidirect
11. (C₂×D₃)⋊C₄	C₅	direct (N × H)
14. C₂×C₄×C₅	D₃	semidirect
20. C₂×C₃×C₄	C₂×C₅	semidirect
21. C₂×Dic₃	C₂×C₅	semidirect
29. C₃×C₅	C₂²⋊C₄	semidirect
33. C₂×D₃	C₄×C₅	semidirect
42. C₂×C₄	C₅×D₃	semidirect
47. C₅	(C₂×D₃)⋊C₄	direct (N × H)
54. 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
C₂²×C₅×D₃	2
C₂×C₃×C₄×C₅	2
C₂×C₅×Dic₃	2
C₅×C₂²⋊C₄	3
(C₂×D₃)⋊C₄	5

Subgroups — C5×(C2×D3)⋊C4 #6698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×(C₂×D₃)⋊C₄ #6698