Subgroups — C₃×#6728 #10623941

Order 720.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×SL(2,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	#6728	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×#226	—	5	1	no	maximal, Hall
4	C₃×#170	—	6	1	no	maximal
5	SL(2,𝔽₅)	C₂×C₃	1	1	yes	characteristic
6	C₃×Dic₆	—	10	1	no	maximal
7	C₃×SL(2,𝔽₃)	—	5	1	no	—
8	C₃×Dic₅	—	6	1	no	—
9	#226	—	5	1	no	—
10	C₃×Q₁₆	—	15	1	no	—
11	#170	—	6	1	no	—
12	C₃²×C₄	—	10	1	no	—
13	C₃×Dic₃	—	10	1	no	—
14	C₃×Dic₃	—	10	1	no	—
15	C₂×C₃×C₅	—	6	1	no	—
16	C₃×Q₈	—	5	1	no	—
17	C₃×Q₈	—	15	1	no	—
18	Dic₆	—	10	1	no	—
19	SL(2,𝔽₃)	—	5	1	no	—
20	SL(2,𝔽₃)	—	10	1	no	—
21	C₃×C₈	—	15	1	no	—
22	Dic₅	—	6	1	no	—
23	C₂×C₃²	—	10	1	no	—
24	Q₁₆	—	15	1	no	Sylow
25	C₃×C₅	—	6	1	no	—
26	C₃×C₄	—	10	1	no	—
27	C₃×C₄	—	10	1	no	—
28	C₃×C₄	—	15	1	no	—
29	C₃×C₄	—	20	1	no	—
30	Dic₃	—	10	1	no	—
31	Dic₃	—	10	1	no	—
32	C₂×C₅	—	6	1	no	—
33	C₃²	—	10	1	no	Sylow
34	Q₈	—	5	1	no	—
35	Q₈	—	15	1	no	—
36	C₈	—	15	1	no	—
37	C₂×C₃	—	10	1	no	—
38	C₂×C₃	—	20	1	no	—
39	C₂×C₃	S₅	1	1	yes	characteristic
40	C₅	—	6	1	no	Sylow
41	C₄	—	10	1	no	—
42	C₄	—	15	1	no	—
43	C₃	—	10	1	no	—
44	C₃	—	20	1	no	—
45	C₃	#6728	1	1	yes	characteristic, direct product factor
46	C₂	C₃×S₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₃	S₅
commutator, cosocle, hypocenter	SL(2,𝔽₅)	C₂×C₃
Frattini	C₂	C₃×S₅
prime radical	C₃×SL(2,𝔽₅)	C₂
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₃

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

N	H ≅ G/N	type
2. #6728	C₃	direct (N × H)
45. C₃	#6728	direct (N × H)

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