Subgroups — #10623983

Order 720.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#67137	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊#170	C₂	1	2	yes	maximal, complemented
3	#6740	—	3	2	no	maximal
4	C₂×C₃⋊D₁₅	C₄	1	1	yes	characteristic
5	C₅×C₃⋊Dic₃	C₄	1	1	yes	characteristic
6	C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	#3712	—	5	1	no	maximal, Hall
8	#1200	—	3	2	no	—
9	#1200	—	6	1	no	—
10	C₃⋊#170	D₃	1	2	yes	complemented
11	C₃×#170	—	3	2	no	—
12	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₃⋊D₁₅	—	2	1	no	—
14	#700	—	9	1	no	Hall
15	C₄×C₃⋊D₃	—	5	1	no	—
16	C₃×C₃⋊C₈	—	5	2	no	—
17	C₂×D₁₅	—	3	2	no	—
18	C₂×D₁₅	—	6	1	no	—
19	C₃×Dic₅	—	2	1	no	—
20	C₃×Dic₅	C₂×D₃	1	2	yes	—
21	C₅×Dic₃	—	3	2	no	—
22	C₅×Dic₃	—	6	1	no	—
23	ΓL(1,𝔽₂₅)	—	15	2	no	—
24	C₃²×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
25	C₄×D₅	—	9	1	no	—
26	#170	—	3	2	no	—
27	C₂×C₃⋊D₃	—	5	1	no	—
28	C₃²×C₄	—	5	1	no	—
29	C₃⋊Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
30	C₂×C₃×C₅	—	2	1	no	—
31	C₂×C₃×C₅	C₄×D₃	1	2	yes	—
32	D₁₅	—	6	2	no	—
33	D₁₅	—	6	2	no	—
34	C₄×D₃	—	15	2	no	—
35	C₄×D₃	—	30	1	no	—
36	C₃×C₈	—	15	2	no	—
37	C₃⋊C₈	—	5	2	no	—
38	C₂×D₅	—	9	1	no	—
39	C₄×C₅	—	9	1	no	—
40	Dic₅	D₃²	1	1	yes	characteristic
41	C₂×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₃⋊D₃	—	10	1	no	—
43	M₁₆	—	45	1	no	Sylow
44	C₃×C₅	—	2	1	no	—
45	C₃×C₅	ΓL(1,𝔽₂₅)	1	2	yes	complemented
46	C₂×D₃	—	15	2	no	—
47	C₂×D₃	—	30	1	no	—
48	C₃×C₄	—	5	2	no	—
49	C₃×C₄	—	10	1	no	—
50	Dic₃	—	3	2	no	—
51	Dic₃	—	6	1	no	—
52	C₂×C₅	#624	1	1	yes	characteristic
53	D₅	—	18	1	no	—
54	C₃²	#700	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄	—	45	1	no	—
56	C₈	—	15	2	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	D₃×AGL(1,𝔽₅)	1	2	yes	—
59	D₃	—	30	2	no	—
60	D₃	—	30	2	no	—
61	C₅	#3712	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	45	1	no	—
63	C₄	—	5	1	no	—
64	C₄	—	9	1	no	—
65	C₃	—	2	1	no	—
66	C₃	#6740	1	2	yes	complemented
67	C₂	—	90	1	no	—
68	C₂	ΓL(1,𝔽₁₆)⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₁₆)⋊D₃
commutator, Fitting, socle	C₂×C₃²×C₅	C₂×C₄
prime radical	C₂×C₃⋊D₁₅	C₄
cosocle	C₃²×Dic₅	C₂²
hypocenter	C₃²×C₅	M₁₆
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊#170	C₂	semidirect
10. C₃⋊#170	D₃	semidirect
24. C₃²×C₅	M₁₆	semidirect
45. C₃×C₅	ΓL(1,𝔽₂₅)	semidirect
54. C₃²	#700	semidirect
61. C₅	#3712	semidirect
66. C₃	#6740	semidirect

Maximal subgroups

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

subgroup	index
#67137	2
C₃×C₃⋊#170	2
#6740	3
#3712	5