Subgroups — #10624009

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#67158	C₂	1	1	yes	maximal, characteristic
2	C₅×#627	C₂	1	1	yes	maximal, characteristic
3	#67126	C₂	1	1	yes	maximal, characteristic
4	C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
5	C₃×Dic₁₅	—	2	1	no	—
6	C₃×C₅×Dic₃	—	2	1	no	—
7	#3776	—	5	1	no	maximal, Hall
8	C₅×Dic₆	—	6	1	no	—
9	#1204	—	6	1	no	—
10	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
11	#689	—	9	1	no	maximal, Hall
12	#627	—	5	1	no	—
13	#627	D₅	1	1	yes	characteristic
14	#622	—	5	1	no	—
15	C₃×C₄×C₅	—	6	1	no	—
16	Dic₁₅	—	2	1	no	—
17	C₃×Dic₅	—	6	1	no	—
18	C₅×Dic₃	—	2	1	no	—
19	C₅×Dic₃	—	6	1	no	—
20	C₅×Dic₃	—	6	1	no	—
21	C₃²×C₅	Q₁₆	1	1	yes	characteristic, complemented, Hall
22	C₅×Q₈	—	9	1	no	—
23	Dic₁₀	—	9	1	no	—
24	#168	—	9	1	no	—
25	C₃⋊Dic₃	C₂×D₅	1	1	yes	characteristic
26	C₃×Dic₃	—	2	1	no	—
27	C₃×Dic₃	—	10	1	no	—
28	C₂×C₃×C₅	—	2	1	no	—
29	C₂×C₃×C₅	—	2	1	no	—
30	Dic₆	—	6	1	no	—
31	Dic₆	—	30	1	no	—
32	C₄×C₅	—	6	1	no	—
33	C₄×C₅	—	9	1	no	—
34	Dic₅	—	6	1	no	—
35	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
36	Q₁₆	—	45	1	no	Sylow
37	C₃×C₅	—	2	1	no	—
38	C₃×C₅	—	2	1	no	—
39	C₃×C₄	—	6	1	no	—
40	C₃×C₄	—	30	1	no	—
41	Dic₃	—	2	1	no	—
42	Dic₃	—	6	1	no	—
43	Dic₃	—	6	1	no	—
44	Dic₃	—	10	1	no	—
45	C₂×C₅	GO⁺(4,𝔽₂)	1	1	yes	characteristic
46	C₃²	#689	1	1	yes	characteristic, complemented, Sylow
47	Q₈	—	9	1	no	—
48	Q₈	—	45	1	no	—
49	C₈	—	45	1	no	—
50	C₂×C₃	—	2	1	no	—
51	C₂×C₃	—	2	1	no	—
52	C₅	#3776	1	1	yes	characteristic, complemented, Sylow
53	C₄	—	6	1	no	—
54	C₄	—	9	1	no	—
55	C₄	—	30	1	no	—
56	C₃	—	2	1	no	—
57	C₃	—	2	1	no	—
58	C₂	D₃²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊D₅
commutator, cosocle	C₅×C₃⋊Dic₃	C₂²
Fitting, socle, prime radical	C₂×C₃²×C₅	D₄
hypocenter	C₃²×C₅	Q₁₆
LCS	G ▷ C₅×C₃⋊Dic₃ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
21. C₃²×C₅	Q₁₆	semidirect
46. C₃²	#689	semidirect
52. C₅	#3776	semidirect

Maximal subgroups

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

subgroup	index
#67158	2
C₅×#627	2
#67126	2
#3776	5
#689	9