Subgroups — #6777

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₅	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#1201	C₂	1	2	yes	maximal, complemented
6	D₄×D₅	—	3	1	no	maximal, Hall
7	C₂×D₁₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×D₃	C₂²	1	2	yes	complemented
9	C₂×C₃×D₅	C₂²	1	2	yes	complemented
10	D₃×D₅	—	2	4	no	—
11	C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
12	Dic₁₅	C₂²	1	1	yes	characteristic, complemented
13	D₃×D₄	—	5	1	no	maximal, Hall
14	C₂²×D₅	—	3	2	no	—
15	C₅×D₄	—	3	1	no	—
16	C₂²⋊D₅	—	3	2	no	—
17	D₂₀	D₃	1	1	yes	characteristic, complemented
18	C₄×D₅	—	3	1	no	—
19	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
20	D₁₅	D₄	1	2	yes	complemented
21	C₃×D₅	—	2	2	no	—
22	C₅×D₃	—	2	2	no	—
23	C₂²×D₃	—	5	2	no	—
24	C₃×D₄	—	5	1	no	—
25	#67	—	5	2	no	—
26	D₁₂	D₅	1	1	yes	characteristic, complemented
27	C₄×D₃	—	5	1	no	—
28	C₂²×C₅	—	3	2	no	—
29	C₂×D₅	—	3	1	no	—
30	C₂×D₅	—	6	4	no	—
31	C₂×D₅	C₂×D₃	1	2	yes	complemented
32	C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
33	Dic₅	—	3	1	no	—
34	C₂×D₄	—	15	1	no	Sylow
35	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₃	—	5	2	no	—
37	C₂×D₃	—	5	1	no	—
38	C₂×D₃	—	10	4	no	—
39	C₂×D₃	C₂×D₅	1	2	yes	complemented
40	C₃×C₄	C₂×D₅	1	1	yes	characteristic, complemented
41	Dic₃	—	5	1	no	—
42	C₂×C₅	—	6	2	no	—
43	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
44	D₅	—	2	2	no	—
45	D₅	—	3	2	no	—
46	C₂³	—	15	2	no	—
47	D₄	—	3	1	no	—
48	D₄	—	5	1	no	—
49	D₄	—	15	2	no	—
50	C₂×C₄	—	15	1	no	—
51	C₂×C₃	—	10	2	no	—
52	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
53	D₃	—	2	2	no	—
54	D₃	—	5	2	no	—
55	C₅	D₃×D₄	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	3	2	no	—
57	C₂²	—	5	2	no	—
58	C₂²	—	15	1	no	—
59	C₂²	—	30	4	no	—
60	C₄	—	15	1	no	—
61	C₄	D₃×D₅	1	1	yes	characteristic, complemented
62	C₃	D₄×D₅	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	6	2	no	—
64	C₂	—	10	2	no	—
65	C₂	—	15	2	no	—
66	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₄×C₅	C₂²
hypercenter	C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₅	C₂	semidirect
2. C₄×D₁₅	C₂	semidirect
3. C₅×D₁₂	C₂	semidirect
4. C₃×D₂₀	C₂	semidirect
5. #1201	C₂	semidirect
8. C₂×C₅×D₃	C₂²	semidirect
9. C₂×C₃×D₅	C₂²	semidirect
11. C₃×C₄×C₅	C₂²	semidirect
12. Dic₁₅	C₂²	semidirect
17. D₂₀	D₃	semidirect
20. D₁₅	D₄	semidirect
26. D₁₂	D₅	semidirect
31. C₂×D₅	C₂×D₃	semidirect
32. C₄×C₅	C₂×D₃	semidirect
35. C₃×C₅	C₂×D₄	semidirect
39. C₂×D₃	C₂×D₅	semidirect
40. C₃×C₄	C₂×D₅	semidirect
55. C₅	D₃×D₄	semidirect
61. C₄	D₃×D₅	semidirect
62. C₃	D₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₅	2
C₄×D₁₅	2
C₅×D₁₂	2
C₃×D₂₀	2
#1201	2
D₄×D₅	3
D₃×D₄	5