Subgroups — #6668

Order 240.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#683	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	Dic₁₅	—	2	1	no	—
7	C₃×Dic₅	C₄	1	2	yes	complemented
8	C₅×Dic₃	—	2	1	no	—
9	C₄⋊Dic₃	—	5	1	no	maximal, Hall
10	C₂×C₄×C₅	—	3	1	no	—
11	C₂×Dic₅	—	3	1	no	—
12	C₂×Dic₅	D₃	1	1	yes	characteristic
13	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₅	D₄	1	1	yes	characteristic
15	C₂×C₃×C₅	Q₈	1	1	yes	characteristic
16	C₂×C₃×C₄	—	5	1	no	—
17	C₂×Dic₃	—	5	1	no	—
18	C₂×Dic₃	D₅	1	1	yes	characteristic
19	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
20	C₄×C₅	—	6	1	no	—
21	Dic₅	—	6	1	no	—
22	Dic₅	Dic₃	1	2	yes	complemented
23	C₄⋊C₄	—	15	1	no	Sylow
24	C₃×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
26	C₃×C₄	—	5	2	no	—
27	Dic₃	—	2	1	no	—
28	Dic₃	—	10	1	no	—
29	C₂×C₅	Dic₆	1	1	yes	characteristic
30	C₂×C₅	D₁₂	1	1	yes	characteristic
31	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
32	C₂×C₄	—	3	1	no	—
33	C₂×C₄	—	5	1	no	—
34	C₂×C₄	—	15	1	no	—
35	C₂×C₃	Dic₁₀	1	1	yes	characteristic
36	C₂×C₃	C₄×D₅	1	1	yes	characteristic
37	C₂×C₃	C₂²⋊D₅	1	1	yes	characteristic
38	C₅	C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
39	C₂²	D₃×D₅	1	1	yes	characteristic
40	C₄	—	5	2	no	—
41	C₄	—	6	1	no	—
42	C₄	—	30	1	no	—
43	C₃	#683	1	1	yes	characteristic, complemented, Sylow
44	C₂	D₅×Dic₃	1	1	yes	characteristic
45	C₂	#1203	1	1	yes	characteristic
46	C₂	#1204	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₃×Dic₅	C₄	semidirect
22. Dic₅	Dic₃	semidirect
24. C₃×C₅	C₄⋊C₄	semidirect
38. C₅	C₄⋊Dic₃	semidirect
43. C₃	#683	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₅	2
C₂×C₅×Dic₃	2
C₂×C₃×Dic₅	2
#683	3
C₄⋊Dic₃	5