Subgroups — #6650

Order 240.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#681	—	3	1	no	maximal, Hall
5	C₂×C₅×D₃	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×Dic₃	C₄	1	1	yes	characteristic
8	ΓL(1,𝔽₂₅)	—	5	1	no	maximal, Hall
9	C₂×C₄×C₅	—	3	1	no	—
10	#168	—	3	1	no	—
11	#168	D₃	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
13	C₅×D₃	—	2	1	no	—
14	C₄×D₃	D₅	1	1	yes	characteristic
15	C₃×C₈	—	5	1	no	—
16	C₃⋊C₈	—	5	1	no	—
17	C₂²×C₅	—	3	1	no	—
18	C₄×C₅	—	3	1	no	—
19	C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	M₁₆	—	15	1	no	Sylow
21	C₃×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂×D₃	Dic₅	1	1	yes	characteristic
23	C₃×C₄	C₂×D₅	1	1	yes	characteristic
24	Dic₃	Dic₅	1	1	yes	characteristic
25	C₂×C₅	—	6	1	no	—
26	C₂×C₅	C₄×D₃	1	1	yes	characteristic
27	C₂×C₄	—	3	1	no	—
28	C₈	—	5	1	no	—
29	C₈	—	15	1	no	—
30	C₂×C₃	C₂×Dic₅	1	1	yes	characteristic
31	D₃	—	2	1	no	—
32	C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented, Sylow
33	C₂²	—	3	1	no	—
34	C₄	—	3	1	no	—
35	C₄	D₃×D₅	1	1	yes	characteristic
36	C₃	#681	1	1	yes	characteristic, complemented, Sylow
37	C₂	—	6	1	no	—
38	C₂	D₃×Dic₅	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×C₅×D₃	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.