Subgroups — #10623609

Order 720.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	Dic₉₀	C₂	1	1	yes	maximal, characteristic
3	C₉⋊#168	C₂	1	1	yes	maximal, characteristic
4	#6718	—	3	1	no	maximal
5	C₄×C₅×C₉	—	2	1	no	—
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	Dic₄₅	—	2	1	no	—
8	#3674	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₈	D₃	1	1	yes	characteristic
10	Dic₃₀	—	3	1	no	—
11	C₃⋊#168	—	3	1	no	—
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	#689	—	9	1	no	Hall
14	Q₈×C₉	D₅	1	1	yes	characteristic
15	Dic₁₈	—	5	1	no	—
16	C₉⋊C₈	—	5	1	no	—
17	C₃×C₄×C₅	—	2	1	no	—
18	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
19	Dic₁₅	—	6	1	no	—
20	#216	—	15	1	no	—
21	C₅×C₉	Q₁₆	1	1	yes	characteristic, complemented, Hall
22	C₅×Q₈	D₉	1	1	yes	characteristic
23	Dic₁₀	—	9	1	no	—
24	#168	—	9	1	no	—
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	C₂×D₅	1	1	yes	characteristic
27	Dic₉	—	10	1	no	—
28	C₂×C₃×C₅	#67	1	1	yes	characteristic
29	C₃×Q₈	D₁₅	1	1	yes	characteristic
30	Dic₆	—	15	1	no	—
31	C₃⋊C₈	—	15	1	no	—
32	C₄×C₅	—	2	1	no	—
33	C₄×C₅	C₂×D₉	1	1	yes	characteristic
34	Dic₅	—	18	1	no	—
35	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
36	Q₁₆	—	45	1	no	Sylow
37	C₃×C₅	#216	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
40	Dic₃	—	30	1	no	—
41	C₂×C₅	#611	1	1	yes	characteristic
42	C₉	#689	1	1	yes	characteristic, complemented, Sylow
43	Q₈	—	45	1	no	—
44	Q₈	D₄₅	1	1	yes	characteristic
45	C₈	—	45	1	no	—
46	C₂×C₃	#1220	1	1	yes	characteristic
47	C₅	#3674	1	1	yes	characteristic, complemented, Sylow
48	C₄	—	2	1	no	—
49	C₄	—	90	1	no	—
50	C₄	C₂×D₄₅	1	1	yes	characteristic
51	C₃	#6718	1	1	yes	characteristic
52	C₂	#67099	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67099
commutator, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	C₂×D₁₅
Fitting	C₅×Q₈×C₉	C₂
socle, prime radical	C₂×C₃×C₅	#67
hypercenter	Q₈	D₄₅
hypocenter	C₅×C₉	Q₁₆
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
21. C₅×C₉	Q₁₆	semidirect
42. C₉	#689	semidirect
47. C₅	#3674	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×C₉	2
Dic₉₀	2
C₉⋊#168	2
#6718	3
#3674	5