Subgroups — #10624011

Order 720.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#157	C₂	1	1	yes	maximal, characteristic
2	#67155	C₂	1	2	yes	maximal
3	C₂×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
4	C₅×#157	C₄	1	2	yes	complemented
5	C₃×Dic₁₅	—	2	2	no	—
6	#3773	—	5	1	no	maximal, Hall
7	D₃×Dic₅	—	6	2	no	—
8	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
9	C₅×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
10	C₅×C₃⋊D₃	Q₈	1	1	yes	characteristic
11	C₄⋊Dic₅	—	9	1	no	maximal, Hall
12	C₂×#157	D₅	1	1	yes	characteristic
13	#624	—	5	2	no	—
14	C₂×C₅×D₃	—	6	2	no	—
15	Dic₁₅	—	2	2	no	—
16	C₃×Dic₅	—	6	2	no	—
17	C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
18	C₂×C₄×C₅	—	9	1	no	—
19	C₂×Dic₅	—	9	2	no	—
20	C₂×C₃⋊D₃	C₂×D₅	1	1	yes	characteristic
21	#157	Dic₅	1	2	yes	complemented
22	C₃×Dic₃	—	10	2	no	—
23	C₂×C₃×C₅	—	2	2	no	—
24	C₅×D₃	—	6	2	no	—
25	C₅×D₃	—	6	2	no	—
26	C₄×D₃	—	30	2	no	—
27	C₂²×C₅	—	9	1	no	—
28	C₄×C₅	—	9	2	no	—
29	Dic₅	—	6	2	no	—
30	C₂×C₃²	D₂₀	1	1	yes	characteristic
31	C₃⋊D₃	Dic₁₀	1	1	yes	characteristic
32	C₃⋊D₃	C₂×Dic₅	1	1	yes	characteristic
33	C₄⋊C₄	—	45	1	no	Sylow
34	C₃×C₅	—	2	2	no	—
35	C₂×D₃	—	6	2	no	—
36	C₃×C₄	—	30	2	no	—
37	Dic₃	—	10	2	no	—
38	C₂×C₅	—	9	1	no	—
39	C₂×C₅	—	9	1	no	—
40	C₂×C₅	GO⁺(4,𝔽₂)	1	1	yes	characteristic
41	C₃²	C₄⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	—	9	1	no	—
43	C₂×C₄	—	45	2	no	—
44	C₂×C₃	—	2	2	no	—
45	D₃	—	6	2	no	—
46	D₃	—	6	2	no	—
47	C₅	#3773	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	9	1	no	—
49	C₄	—	9	2	no	—
50	C₄	—	30	2	no	—
51	C₃	—	2	2	no	—
52	C₂	—	9	1	no	—
53	C₂	—	9	1	no	—
54	C₂	#157⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#157⋊D₅
commutator	C₅×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃²×C₅	D₄
prime radical, cosocle	C₂×C₅×C₃⋊D₃	C₂²
hypocenter	C₃²×C₅	C₄⋊C₄
LCS	G ▷ C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
4. C₅×#157	C₄	semidirect
17. C₃²×C₅	C₄⋊C₄	semidirect
21. #157	Dic₅	semidirect
41. C₃²	C₄⋊Dic₅	semidirect
47. C₅	#3773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#157	2
#67155	2
#3773	5
C₄⋊Dic₅	9