Subgroups — #10623913

Order 720.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
3	#6719	—	3	4	no	maximal
4	C₂²×C₃²×C₅	—	2	2	no	—
5	C₂²×C₃²×C₅	C₄	1	2	yes	complemented
6	C₂²×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊Dic₁₅	—	2	2	no	—
8	#3757	—	5	1	no	maximal, Hall
9	C₂³×C₃×C₅	D₃	1	4	yes	—
10	C₂×Dic₁₅	—	3	8	no	—
11	C₂×C₃²×C₅	—	2	2	no	—
12	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃²×C₅	D₄	1	2	yes	—
14	C₂²⋊Dic₅	—	9	1	no	Hall
15	C₂³×C₃²	D₅	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	—	5	2	no	—
17	C₂²×C₃×C₅	—	2	8	no	—
18	C₂²×C₃×C₅	Dic₃	1	8	yes	complemented
19	C₂²×C₃×C₅	C₂×D₃	1	4	yes	—
20	Dic₁₅	—	6	8	no	—
21	#217	—	15	4	no	—
22	C₃²×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
23	C₂³×C₅	C₃⋊D₃	1	1	yes	characteristic
24	C₂×Dic₅	—	9	2	no	—
25	C₂²×C₃²	—	2	2	no	—
26	C₂²×C₃²	Dic₅	1	2	yes	complemented
27	C₂²×C₃²	C₂×D₅	1	1	yes	characteristic
28	C₃⋊Dic₃	—	10	2	no	—
29	C₂×C₃×C₅	—	2	8	no	—
30	C₂×C₃×C₅	C₂×Dic₃	1	4	yes	—
31	C₂×C₃×C₅	#67	1	8	yes	—
32	C₂³×C₃	D₁₅	1	4	yes	—
33	C₂×Dic₃	—	15	8	no	—
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	C₃⋊Dic₃	1	2	yes	complemented
36	C₂²×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
37	Dic₅	—	18	2	no	—
38	C₂×C₃²	—	2	2	no	—
39	C₂×C₃²	C₂×Dic₅	1	1	yes	characteristic
40	C₂×C₃²	C₂²⋊D₅	1	2	yes	—
41	C₂²⋊C₄	—	45	1	no	Sylow
42	C₃×C₅	#217	1	4	yes	complemented
43	C₂²×C₃	—	2	8	no	—
44	C₂²×C₃	Dic₁₅	1	8	yes	complemented
45	C₂²×C₃	C₂×D₁₅	1	4	yes	—
46	Dic₃	—	30	8	no	—
47	C₂×C₅	—	2	2	no	—
48	C₂×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
49	C₂×C₅	#638	1	2	yes	—
50	C₃²	C₂²⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
51	C₂³	C₃⋊D₁₅	1	1	yes	characteristic
52	C₂×C₄	—	45	2	no	—
53	C₂×C₃	—	2	8	no	—
54	C₂×C₃	C₂×Dic₁₅	1	4	yes	—
55	C₂×C₃	#1220	1	8	yes	—
56	C₅	#3757	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	2	2	no	—
58	C₂²	C₃⋊Dic₁₅	1	2	yes	complemented
59	C₂²	C₂×C₃⋊D₁₅	1	1	yes	characteristic
60	C₄	—	90	2	no	—
61	C₃	#6719	1	4	yes	complemented
62	C₂	—	2	2	no	—
63	C₂	C₂×C₃⋊Dic₁₅	1	1	yes	characteristic
64	C₂	#67184	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊D₁₅
commutator	C₂×C₃²×C₅	C₂×C₄
Fitting, prime radical	C₂³×C₃²×C₅	C₂
socle, cosocle	C₂²×C₃²×C₅	C₂²
hypercenter	C₂³	C₃⋊D₁₅
hypocenter	C₃²×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊Dic₁₅	C₂	semidirect
5. C₂²×C₃²×C₅	C₄	semidirect
18. C₂²×C₃×C₅	Dic₃	semidirect
22. C₃²×C₅	C₂²⋊C₄	semidirect
26. C₂²×C₃²	Dic₅	semidirect
35. C₂²×C₅	C₃⋊Dic₃	semidirect
42. C₃×C₅	#217	semidirect
44. C₂²×C₃	Dic₁₅	semidirect
50. C₃²	C₂²⋊Dic₅	semidirect
56. C₅	#3757	semidirect
58. C₂²	C₃⋊Dic₁₅	semidirect
61. C₃	#6719	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₅	2
C₂×C₃⋊Dic₁₅	2
#6719	3
#3757	5