Subgroups — #10624311

Order 720.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×GO⁺(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	#67194	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₃²	—	2	1	no	—
5	#4379	—	2	1	no	—
6	C₅×#157	C₂²	1	1	yes	characteristic
7	AΓL(1,𝔽₉)	—	5	1	no	maximal, Hall
8	C₅×C₃⋊D₃	D₄	1	1	yes	characteristic
9	C₃×C₅×D₃	—	4	1	no	—
10	#687	—	9	1	no	maximal, Hall
11	PSU(3,𝔽₂)	—	5	1	no	—
12	GO⁺(4,𝔽₂)	D₅	1	1	yes	characteristic
13	AGL(1,𝔽₉)	—	5	1	no	—
14	C₂×C₅×D₃	—	12	1	no	—
15	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
16	C₅×D₄	—	9	1	no	—
17	Dic₁₀	—	9	1	no	—
18	#168	—	9	1	no	—
19	D₃²	—	2	1	no	—
20	#157	—	10	1	no	—
21	#157	C₂×D₅	1	1	yes	characteristic
22	C₂×C₃×C₅	—	12	1	no	—
23	C₅×D₃	—	4	1	no	—
24	C₅×D₃	—	12	1	no	—
25	C₂²×C₅	—	18	1	no	—
26	C₄×C₅	—	9	1	no	—
27	Dic₅	—	18	1	no	—
28	C₃⋊D₃	C₂²⋊D₅	1	1	yes	characteristic
29	C₃×D₃	—	4	1	no	—
30	SD₁₆	—	45	1	no	Sylow
31	C₃×C₅	—	4	1	no	—
32	C₂×D₃	—	12	1	no	—
33	C₂×C₅	—	9	1	no	—
34	C₂×C₅	—	12	1	no	—
35	C₃²	#687	1	1	yes	characteristic, complemented, Sylow
36	Q₈	—	45	1	no	—
37	D₄	—	9	1	no	—
38	C₈	—	45	1	no	—
39	C₂×C₃	—	12	1	no	—
40	D₃	—	4	1	no	—
41	D₃	—	12	1	no	—
42	C₅	AΓL(1,𝔽₉)	1	1	yes	characteristic, complemented, Sylow
43	C₂²	—	18	1	no	—
44	C₄	—	9	1	no	—
45	C₄	—	90	1	no	—
46	C₃	—	4	1	no	—
47	C₂	—	9	1	no	—
48	C₂	—	12	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₅×#157	C₂²
Fitting, socle, hypocenter	C₃²×C₅	SD₁₆
prime radical	C₅×GO⁺(4,𝔽₂)	C₂
LCS	G ▷ C₅×#157 ▷ C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁

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

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