Subgroups — C₃×#6729 #10623942

Order 720.

56 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, complemented
2	#6729	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×GL(2,𝔽₃)	—	5	1	no	maximal, Hall
4	C₃×#170	—	6	1	no	maximal
5	SL(2,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×#67	—	10	1	no	maximal
7	C₃×SL(2,𝔽₃)	—	5	1	no	—
8	C₃×Dic₅	—	6	1	no	—
9	GL(2,𝔽₃)	—	5	1	no	—
10	C₃×SD₁₆	—	15	1	no	—
11	#170	—	6	1	no	—
12	C₂²×C₃²	—	10	1	no	—
13	C₂×C₃×D₃	—	10	1	no	—
14	C₃×Dic₃	—	10	1	no	—
15	C₂×C₃×C₅	—	6	1	no	—
16	C₃×Q₈	—	5	1	no	—
17	C₃×D₄	—	15	1	no	—
18	#67	—	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	C₂×C₃²	—	20	1	no	—
25	C₃×D₃	—	20	1	no	—
26	SD₁₆	—	15	1	no	Sylow
27	C₃×C₅	—	6	1	no	—
28	C₂²×C₃	—	10	1	no	—
29	C₂²×C₃	—	10	1	no	—
30	C₂²×C₃	—	20	1	no	—
31	C₂×D₃	—	10	1	no	—
32	C₃×C₄	—	15	1	no	—
33	Dic₃	—	10	1	no	—
34	C₂×C₅	—	6	1	no	—
35	C₃²	—	10	1	no	Sylow
36	Q₈	—	5	1	no	—
37	D₄	—	15	1	no	—
38	C₈	—	15	1	no	—
39	C₂×C₃	—	10	1	no	—
40	C₂×C₃	—	20	1	no	—
41	C₂×C₃	—	20	2	no	—
42	C₂×C₃	—	20	1	no	—
43	C₂×C₃	—	20	1	no	—
44	C₂×C₃	S₅	1	1	yes	characteristic
45	D₃	—	20	1	no	—
46	C₅	—	6	1	no	Sylow
47	C₂²	—	10	1	no	—
48	C₄	—	15	1	no	—
49	C₃	—	10	1	no	—
50	C₃	—	20	1	no	—
51	C₃	#6729	1	1	yes	characteristic, direct product factor
52	C₂	—	20	1	no	—
53	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
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₃

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.