Subgroups — #10623906

Order 720.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	#6712	—	3	4	no	maximal
5	C₂²×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₅	—	2	1	no	—
7	C₃⋊Dic₁₅	—	2	1	no	—
8	C₃⋊Dic₁₅	C₄	1	2	yes	complemented
9	#3750	—	5	1	no	maximal, Hall
10	C₂×C₃×C₄×C₅	D₃	1	4	yes	—
11	C₂×Dic₁₅	—	3	4	no	—
12	C₂×Dic₁₅	—	3	4	no	—
13	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
15	C₂×C₃²×C₅	Q₈	1	1	yes	characteristic
16	#683	—	9	1	no	Hall
17	C₂×C₃²×C₄	D₅	1	1	yes	characteristic
18	C₂×C₃⋊Dic₃	—	5	1	no	—
19	C₂×C₃⋊Dic₃	—	5	1	no	—
20	C₂²×C₃×C₅	C₂×D₃	1	4	yes	—
21	C₃×C₄×C₅	—	2	4	no	—
22	Dic₁₅	—	3	8	no	—
23	Dic₁₅	—	6	4	no	—
24	Dic₃⋊C₄	—	15	4	no	—
25	C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
26	C₂×C₄×C₅	C₃⋊D₃	1	1	yes	characteristic
27	C₂×Dic₅	—	9	1	no	—
28	C₂×Dic₅	—	9	1	no	—
29	C₂²×C₃²	C₂×D₅	1	1	yes	characteristic
30	C₃²×C₄	—	2	1	no	—
31	C₃⋊Dic₃	—	5	2	no	—
32	C₃⋊Dic₃	—	10	1	no	—
33	C₂×C₃×C₅	Dic₆	1	4	yes	—
34	C₂×C₃×C₅	C₄×D₃	1	4	yes	—
35	C₂×C₃×C₅	#67	1	4	yes	—
36	C₂×C₃×C₄	D₁₅	1	4	yes	—
37	C₂×Dic₃	—	15	4	no	—
38	C₂×Dic₃	—	15	4	no	—
39	C₂²×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
40	C₄×C₅	—	2	1	no	—
41	Dic₅	—	9	2	no	—
42	Dic₅	—	18	1	no	—
43	C₂×C₃²	Dic₁₀	1	1	yes	characteristic
44	C₂×C₃²	C₄×D₅	1	1	yes	characteristic
45	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
46	C₄⋊C₄	—	45	1	no	Sylow
47	C₃×C₅	Dic₃⋊C₄	1	4	yes	complemented
48	C₂²×C₃	C₂×D₁₅	1	4	yes	—
49	C₃×C₄	—	2	4	no	—
50	Dic₃	—	15	8	no	—
51	Dic₃	—	30	4	no	—
52	C₂×C₅	#634	1	1	yes	characteristic
53	C₂×C₅	C₄×C₃⋊D₃	1	1	yes	characteristic
54	C₂×C₅	#638	1	1	yes	characteristic
55	C₃²	#683	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄	—	45	1	no	—
57	C₂×C₄	—	45	1	no	—
58	C₂×C₄	C₃⋊D₁₅	1	1	yes	characteristic
59	C₂×C₃	Dic₃₀	1	4	yes	—
60	C₂×C₃	C₄×D₁₅	1	4	yes	—
61	C₂×C₃	#1220	1	4	yes	—
62	C₅	#3750	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₂×C₃⋊D₁₅	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	45	2	no	—
66	C₄	—	90	1	no	—
67	C₃	#6712	1	4	yes	complemented
68	C₂	#67180	1	1	yes	characteristic
69	C₂	C₄×C₃⋊D₁₅	1	1	yes	characteristic
70	C₂	#67184	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₃⋊Dic₁₅	C₄	semidirect
25. C₃²×C₅	C₄⋊C₄	semidirect
47. C₃×C₅	Dic₃⋊C₄	semidirect
55. C₃²	#683	semidirect
62. C₅	#3750	semidirect
67. C₃	#6712	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₅	2
C₂×C₃⋊Dic₁₅	2
#6712	3
#3750	5