Subgroups — #6665

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×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#685	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×D₅	—	2	1	no	—
7	C₂×C₃×D₅	—	2	1	no	—
8	C₂×C₃×D₅	C₄	1	2	yes	complemented
9	Dic₁₅	—	2	1	no	—
10	C₅×Dic₃	—	2	1	no	—
11	#217	—	5	1	no	maximal, Hall
12	C₂²×D₅	D₃	1	1	yes	characteristic
13	C₂×C₄×C₅	—	3	1	no	—
14	C₂×Dic₅	—	3	1	no	—
15	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₅	D₄	1	1	yes	characteristic
17	C₂×C₃×C₅	D₄	1	1	yes	characteristic
18	C₃×D₅	—	2	2	no	—
19	C₂³×C₃	—	5	1	no	—
20	C₂×Dic₃	—	5	1	no	—
21	C₂×Dic₃	D₅	1	1	yes	characteristic, complemented
22	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×D₅	—	2	1	no	—
24	C₂×D₅	—	2	1	no	—
25	C₂×D₅	Dic₃	1	2	yes	complemented
26	C₄×C₅	—	6	1	no	—
27	Dic₅	—	6	1	no	—
28	C₂²⋊C₄	—	15	1	no	Sylow
29	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃	—	5	2	no	—
31	C₂²×C₃	—	10	1	no	—
32	C₂²×C₃	—	10	1	no	—
33	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
34	Dic₃	—	2	1	no	—
35	Dic₃	—	10	1	no	—
36	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
37	C₂×C₅	#67	1	1	yes	characteristic
38	C₂×C₅	#67	1	1	yes	characteristic
39	D₅	—	2	2	no	—
40	C₂³	—	5	1	no	—
41	C₂×C₄	—	3	1	no	—
42	C₂×C₄	—	15	1	no	—
43	C₂×C₃	—	10	2	no	—
44	C₂×C₃	C₄×D₅	1	1	yes	characteristic
45	C₂×C₃	D₂₀	1	1	yes	characteristic
46	C₂×C₃	C₂²⋊D₅	1	1	yes	characteristic
47	C₅	#217	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	5	2	no	—
49	C₂²	—	10	1	no	—
50	C₂²	—	10	1	no	—
51	C₂²	D₃×D₅	1	1	yes	characteristic
52	C₄	—	6	1	no	—
53	C₄	—	30	1	no	—
54	C₃	#685	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	10	2	no	—
56	C₂	D₅×Dic₃	1	1	yes	characteristic
57	C₂	#1201	1	1	yes	characteristic
58	C₂	#1202	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄
Fitting, socle, cosocle	C₂²×C₃×C₅	C₂²
prime radical	C₂²×C₃×D₅	C₂
hypocenter	C₃×C₅	C₂²⋊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
2. C₂×Dic₁₅	C₂	semidirect
3. C₂×C₅×Dic₃	C₂	semidirect
8. C₂×C₃×D₅	C₄	semidirect
21. C₂×Dic₃	D₅	semidirect
25. C₂×D₅	Dic₃	semidirect
29. C₃×C₅	C₂²⋊C₄	semidirect
47. C₅	#217	semidirect
54. C₃	#685	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×Dic₁₅	2
C₂×C₅×Dic₃	2
#685	3
#217	5