Subgroups — #6817

Order 240.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	#1220	C₂	1	2	yes	maximal, complemented
3	D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₁₅	C₂	1	2	yes	maximal, complemented
5	Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
6	#709	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂×D₁₅	C₂²	1	2	yes	—
9	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	Dic₁₅	C₂²	1	2	yes	—
12	#235	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅	D₃	1	1	yes	characteristic, complemented
14	C₂²⋊D₅	—	3	2	no	—
15	D₂₀	—	3	1	no	—
16	C₄×D₅	—	3	2	no	—
17	Dic₁₀	—	3	1	no	—
18	C₂×C₃×C₅	—	2	1	no	—
19	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
20	D₁₅	—	2	2	no	—
21	C₂×C₃×C₄	D₅	1	1	yes	characteristic, complemented
22	#67	—	5	2	no	—
23	D₁₂	—	5	1	no	—
24	C₄×D₃	—	5	2	no	—
25	Dic₆	—	5	1	no	—
26	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
27	C₂×D₅	—	3	2	no	—
28	C₄×C₅	C₂×D₃	1	1	yes	characteristic
29	C₄×C₅	C₂×D₃	1	1	yes	characteristic
30	Dic₅	—	3	2	no	—
31	#41	—	15	1	no	Sylow
32	C₃×C₅	#41	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
34	C₂×D₃	—	5	2	no	—
35	C₃×C₄	C₂×D₅	1	1	yes	characteristic
36	C₃×C₄	C₂×D₅	1	1	yes	characteristic
37	Dic₃	—	5	2	no	—
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
40	D₅	—	6	2	no	—
41	Q₈	—	15	1	no	—
42	D₄	—	15	2	no	—
43	D₄	—	15	1	no	—
44	C₂×C₄	—	15	2	no	—
45	C₂×C₄	D₁₅	1	1	yes	characteristic, complemented
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
48	D₃	—	10	2	no	—
49	C₅	#235	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	15	2	no	—
51	C₂²	C₂×D₁₅	1	1	yes	characteristic
52	C₄	—	15	2	no	—
53	C₄	C₂×D₁₅	1	1	yes	characteristic
54	C₄	C₂×D₁₅	1	1	yes	characteristic
55	C₃	#709	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	30	2	no	—
58	C₂	C₂²×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Frattini	C₂	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅	C₂
hypercenter	C₂×C₄	D₁₅
hypocenter	C₃×C₅	#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅	C₂	semidirect
2. #1220	C₂	semidirect
3. D₆₀	C₂	semidirect
4. C₄×D₁₅	C₂	semidirect
5. Dic₃₀	C₂	semidirect
13. C₂×C₄×C₅	D₃	semidirect
21. C₂×C₃×C₄	D₅	semidirect
32. C₃×C₅	#41	semidirect
45. C₂×C₄	D₁₅	semidirect
49. C₅	#235	semidirect
55. C₃	#709	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅	2
#1220	2
D₆₀	2
C₄×D₁₅	2
Dic₃₀	2
#709	3
#235	5