Subgroups — #6763

Order 240.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂²⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃×D₅	—	2	2	no	—
6	C₂×C₃×D₅	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
8	ΓL(1,𝔽₁₆)	—	2	2	no	—
9	#217	—	5	1	no	maximal, Hall
10	C₂²×D₅	D₃	1	1	yes	characteristic
11	C₂×AGL(1,𝔽₅)	—	3	2	no	—
12	C₂×C₃×C₅	—	2	1	no	—
13	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
14	C₃×D₅	—	2	1	no	—
15	C₃×D₅	D₄	1	2	yes	—
16	C₂³×C₃	—	5	1	no	—
17	C₂×Dic₃	—	5	2	no	—
18	C₂²×C₅	Dic₃	1	1	yes	characteristic, complemented
19	C₂×D₅	—	2	2	no	—
20	C₂×D₅	Dic₃	1	1	yes	characteristic, complemented
21	C₂×D₅	C₂×D₃	1	1	yes	characteristic
22	AGL(1,𝔽₅)	—	6	2	no	—
23	C₂²⋊C₄	—	15	1	no	Sylow
24	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃	—	5	1	no	—
26	C₂²×C₃	—	5	1	no	—
27	C₂²×C₃	—	10	2	no	—
28	C₂²×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
29	Dic₃	—	10	2	no	—
30	C₂×C₅	—	2	1	no	—
31	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
32	D₅	—	2	1	no	—
33	D₅	#67	1	2	yes	—
34	C₂³	—	5	1	no	—
35	C₂×C₄	—	15	2	no	—
36	C₂×C₃	—	2	1	no	—
37	C₂×C₃	—	5	2	no	—
38	C₂×C₃	—	10	1	no	—
39	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₅	#217	1	1	yes	characteristic, complemented, Sylow
41	C₂²	—	5	1	no	—
42	C₂²	—	5	1	no	—
43	C₂²	—	10	2	no	—
44	C₂²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
45	C₄	—	30	2	no	—
46	C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	2	1	no	—
48	C₂	—	5	2	no	—
49	C₂	—	10	1	no	—
50	C₂	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂×C₄
Fitting	C₂²×C₃×C₅	C₄
prime radical	C₂²×C₃×D₅	C₂
cosocle	C₂×C₃×D₅	C₂²
hypercenter	C₂²	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×ΓL(1,𝔽₁₆)	C₂	semidirect
4. C₂²×C₃×C₅	C₄	semidirect
6. C₂×C₃×D₅	C₄	semidirect
18. C₂²×C₅	Dic₃	semidirect
20. C₂×D₅	Dic₃	semidirect
24. C₃×C₅	C₂²⋊C₄	semidirect
28. C₂²×C₃	AGL(1,𝔽₅)	semidirect
40. C₅	#217	semidirect
44. C₂²	ΓL(1,𝔽₁₆)	semidirect
46. C₃	C₂²⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₅	2
C₂×ΓL(1,𝔽₁₆)	2
C₂²⋊AGL(1,𝔽₅)	3
#217	5