Subgroups — #6649

Order 240.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#676	—	3	1	no	maximal, Hall
5	C₂×C₃×D₅	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×Dic₅	C₄	1	1	yes	characteristic
8	#208	—	5	1	no	maximal, Hall
9	C₄×D₅	D₃	1	1	yes	characteristic
10	C₅×C₈	—	3	1	no	—
11	#168	—	3	1	no	—
12	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
13	C₃×D₅	—	2	1	no	—
14	C₂×C₃×C₄	—	5	1	no	—
15	C₃⋊C₈	—	5	1	no	—
16	C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
17	C₂×D₅	Dic₃	1	1	yes	characteristic
18	C₄×C₅	C₂×D₃	1	1	yes	characteristic
19	Dic₅	Dic₃	1	1	yes	characteristic
20	M₁₆	—	15	1	no	Sylow
21	C₃×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	—	5	1	no	—
23	C₃×C₄	—	5	1	no	—
24	C₃×C₄	C₂×D₅	1	1	yes	characteristic
25	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
26	D₅	—	2	1	no	—
27	C₂×C₄	—	5	1	no	—
28	C₈	—	3	1	no	—
29	C₈	—	15	1	no	—
30	C₂×C₃	—	10	1	no	—
31	C₂×C₃	C₄×D₅	1	1	yes	characteristic
32	C₅	#208	1	1	yes	characteristic, complemented, Sylow
33	C₂²	—	5	1	no	—
34	C₄	—	5	1	no	—
35	C₄	D₃×D₅	1	1	yes	characteristic
36	C₃	#676	1	1	yes	characteristic, complemented, Sylow
37	C₂	—	10	1	no	—
38	C₂	D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×D₅
commutator, socle	C₂×C₃×C₅	C₂×C₄
Fitting, cosocle	C₃×C₄×C₅	C₂²
prime radical	C₂×C₃×D₅	C₄
hypocenter	C₃×C₅	M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄

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

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