Subgroups — C₃²×#704 #10624083

Order 720.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×#170	C₂	1	2	yes	maximal, complemented
3	C₃×#704	C₃	1	4	yes	maximal, direct product factor
4	C₂²×C₃²×C₅	C₄	1	1	yes	characteristic
5	C₃²×Dic₅	C₄	1	1	yes	characteristic
6	C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	C₃²×M₁₆	—	5	1	no	maximal, Hall
8	C₂×C₃×Dic₅	C₂×C₃	1	4	yes	—
9	C₃×#170	C₂×C₃	1	8	yes	complemented
10	C₂×C₃²×C₅	—	2	1	no	—
11	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	#704	C₃²	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃²×C₄	—	5	1	no	—
14	C₃²×C₈	—	5	2	no	—
15	C₂²×C₃×C₅	C₃×C₄	1	4	yes	—
16	C₃×Dic₅	C₃×C₄	1	4	yes	—
17	C₃×Dic₅	C₂²×C₃	1	4	yes	—
18	C₃×M₁₆	—	5	4	no	—
19	C₃²×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
20	C₂×Dic₅	C₂×C₃²	1	1	yes	characteristic
21	#170	C₂×C₃²	1	2	yes	complemented
22	C₂²×C₃²	AGL(1,𝔽₅)	1	1	yes	characteristic
23	C₃²×C₄	—	5	1	no	—
24	C₃²×C₄	—	5	1	no	—
25	C₂×C₃×C₅	—	2	4	no	—
26	C₂×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
27	C₂×C₃×C₄	—	5	4	no	—
28	C₃×C₈	—	5	8	no	—
29	C₂²×C₅	C₃²×C₄	1	1	yes	characteristic
30	Dic₅	C₃²×C₄	1	1	yes	characteristic
31	Dic₅	C₂²×C₃²	1	1	yes	characteristic
32	C₂×C₃²	—	2	1	no	—
33	C₂×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
34	M₁₆	—	5	1	no	Sylow
35	C₃×C₅	C₃×M₁₆	1	4	yes	complemented
36	C₂²×C₃	C₃×AGL(1,𝔽₅)	1	4	yes	—
37	C₃×C₄	—	5	4	no	—
38	C₃×C₄	—	5	4	no	—
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
41	C₃²	#704	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₄	—	5	1	no	—
43	C₈	—	5	2	no	—
44	C₂×C₃	—	2	4	no	—
45	C₂×C₃	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
46	C₅	C₃²×M₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂²	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₄	—	5	1	no	—
49	C₄	—	5	1	no	—
50	C₃	C₃×#704	1	4	yes	direct product factor
51	C₂	—	2	1	no	—
52	C₂	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×AGL(1,𝔽₅)
Fitting, prime radical	C₂²×C₃²×C₅	C₄
socle	C₂×C₃²×C₅	C₂×C₄
cosocle	Dic₅	C₂²×C₃²
hypercenter	C₂²×C₃²	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
2. C₃²×#170	C₂	semidirect
3. C₃×#704	C₃	direct (N × H)
9. C₃×#170	C₂×C₃	semidirect
12. #704	C₃²	direct (N × H)
19. C₃²×C₅	M₁₆	semidirect
21. #170	C₂×C₃²	semidirect
35. C₃×C₅	C₃×M₁₆	semidirect
41. C₃²	#704	direct (N × H)
46. C₅	C₃²×M₁₆	semidirect
50. C₃	C₃×#704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₅	2
C₃²×#170	2
C₃×#704	3
C₃²×M₁₆	5

Subgroups — C32×#704 #10624083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#704 #10624083