Subgroups — C₃⋊#6729 #10623946

Order 720.

59 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	—	3	1	no	maximal
3	C₃⋊GL(2,𝔽₃)	—	5	1	no	maximal, Hall
4	C₃⋊#170	—	6	1	no	maximal
5	SL(2,𝔽₅)	D₃	1	1	yes	characteristic, complemented
6	C₃×SL(2,𝔽₃)	—	5	1	no	—
7	#626	—	10	1	no	maximal
8	C₃×Dic₅	—	6	1	no	—
9	GL(2,𝔽₃)	—	15	1	no	—
10	GL(2,𝔽₃)	—	15	2	no	—
11	#215	—	15	1	no	—
12	#170	—	18	1	no	—
13	C₂×C₃⋊D₃	—	10	1	no	—
14	C₂×C₃×D₃	—	10	1	no	—
15	C₃×Dic₃	—	10	1	no	—
16	C₂×C₃×C₅	—	6	1	no	—
17	C₃×Q₈	—	5	1	no	—
18	#67	—	30	1	no	—
19	D₁₂	—	15	1	no	—
20	SL(2,𝔽₃)	—	5	1	no	—
21	SL(2,𝔽₃)	—	5	2	no	—
22	C₃⋊C₈	—	15	1	no	—
23	Dic₅	—	6	1	no	—
24	C₂×C₃²	—	10	1	no	—
25	C₃⋊D₃	—	20	1	no	—
26	C₃×D₃	—	20	1	no	—
27	SD₁₆	—	45	1	no	Sylow
28	C₃×C₅	—	6	1	no	—
29	C₂²×C₃	—	30	1	no	—
30	C₂×D₃	—	10	1	no	—
31	C₂×D₃	—	30	1	no	—
32	C₂×D₃	—	60	1	no	—
33	C₃×C₄	—	15	1	no	—
34	Dic₃	—	10	1	no	—
35	C₂×C₅	—	6	1	no	—
36	C₃²	—	10	1	no	Sylow
37	Q₈	—	5	1	no	—
38	D₄	—	45	1	no	—
39	C₈	—	45	1	no	—
40	C₂×C₃	—	10	1	no	—
41	C₂×C₃	—	20	1	no	—
42	C₂×C₃	—	60	1	no	—
43	C₂×C₃	S₅	1	1	yes	characteristic
44	D₃	—	20	1	no	—
45	D₃	—	60	1	no	—
46	D₃	—	60	2	no	—
47	C₅	—	6	1	no	Sylow
48	C₂²	—	30	1	no	—
49	C₄	—	15	1	no	—
50	C₃	—	10	1	no	—
51	C₃	—	20	1	no	—
52	C₃	#6729	1	1	yes	characteristic, complemented
53	C₂	—	60	1	no	—
54	C₂	ΓL(2,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(2,𝔽₄)
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₅)	C₂
Fitting, socle	C₂×C₃	S₅
LCS	G ▷ C₃×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

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.