Subgroups — C₃⋊#6732 #10623949

Order 720.

66 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	#6732	—	3	1	no	maximal
3	C₃⋊(#41⋊C₃)	—	5	1	no	maximal, Hall
4	#1200	—	6	1	no	maximal
5	SL(2,𝔽₅)	D₃	1	1	yes	characteristic, complemented
6	C₃×SL(2,𝔽₃)	—	5	1	no	—
7	#624	—	10	1	no	maximal
8	C₂×D₁₅	—	6	1	no	—
9	C₃×Dic₅	—	6	1	no	—
10	C₅×Dic₃	—	6	1	no	—
11	#239	—	5	1	no	—
12	#41⋊C₃	—	15	1	no	—
13	C₄×D₅	—	18	1	no	—
14	C₂×C₃⋊D₃	—	10	1	no	—
15	C₃×Dic₃	—	10	1	no	—
16	C₃×Dic₃	—	10	1	no	—
17	C₂×C₃×C₅	—	6	1	no	—
18	D₁₅	—	6	2	no	—
19	C₃×Q₈	—	5	1	no	—
20	D₁₂	—	15	1	no	—
21	C₄×D₃	—	15	1	no	—
22	C₄×D₃	—	30	1	no	—
23	SL(2,𝔽₃)	—	5	1	no	—
24	SL(2,𝔽₃)	—	10	1	no	—
25	C₂×D₅	—	18	1	no	—
26	C₄×C₅	—	18	1	no	—
27	Dic₅	—	6	1	no	—
28	C₂×C₃²	—	10	1	no	—
29	C₃⋊D₃	—	10	2	no	—
30	#41	—	15	1	no	Sylow
31	C₃×C₅	—	6	1	no	—
32	C₂×D₃	—	15	1	no	—
33	C₂×D₃	—	30	1	no	—
34	C₂×D₃	—	60	1	no	—
35	C₃×C₄	—	15	1	no	—
36	C₃×C₄	—	30	1	no	—
37	Dic₃	—	10	1	no	—
38	Dic₃	A₅	1	1	yes	maximal normal, characteristic
39	C₂×C₅	—	6	1	no	—
40	D₅	—	18	2	no	—
41	C₃²	—	10	1	no	Sylow
42	Q₈	—	5	1	no	—
43	D₄	—	45	1	no	—
44	C₂×C₄	—	45	1	no	—
45	C₂×C₃	—	10	1	no	—
46	C₂×C₃	—	20	1	no	—
47	C₂×C₃	C₂×A₅	1	1	yes	characteristic
48	D₃	—	30	2	no	—
49	D₃	—	30	1	no	—
50	D₃	—	60	2	no	—
51	C₅	—	6	1	no	Sylow
52	C₂²	—	45	1	no	—
53	C₄	—	3	1	no	—
54	C₄	—	15	1	no	—
55	C₃	—	10	1	no	—
56	C₃	—	20	1	no	—
57	C₃	#6732	1	1	yes	characteristic, complemented
58	C₂	—	90	1	no	—
59	C₂	D₃×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×A₅
commutator, hypocenter	C₃×SL(2,𝔽₅)	C₂
Fitting, socle, cosocle	C₂×C₃	C₂×A₅
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.