Subgroups — #10623604

Order 720.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₅	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₄₅	C₂	1	1	yes	maximal, characteristic
4	#6712	—	3	1	no	maximal
5	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
6	C₄×C₅×C₉	—	2	1	no	—
7	Dic₄₅	—	2	1	no	—
8	Dic₄₅	C₄	1	2	yes	complemented
9	Dic₉⋊C₄	—	5	1	no	maximal, Hall
10	C₂×C₃×C₄×C₅	D₃	1	1	yes	characteristic
11	C₂×Dic₁₅	—	3	1	no	—
12	C₂×Dic₁₅	—	3	1	no	—
13	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₅×C₉	D₄	1	1	yes	characteristic
15	C₂×C₅×C₉	Q₈	1	1	yes	characteristic
16	#683	—	9	1	no	Hall
17	C₂×C₄×C₉	D₅	1	1	yes	characteristic
18	C₂×Dic₉	—	5	1	no	—
19	C₂×Dic₉	—	5	1	no	—
20	C₂²×C₃×C₅	C₂×D₃	1	1	yes	characteristic
21	C₃×C₄×C₅	—	2	1	no	—
22	Dic₁₅	—	3	2	no	—
23	Dic₁₅	—	6	1	no	—
24	Dic₃⋊C₄	—	15	1	no	—
25	C₅×C₉	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
26	C₂×C₄×C₅	D₉	1	1	yes	characteristic
27	C₂×Dic₅	—	9	1	no	—
28	C₂×Dic₅	—	9	1	no	—
29	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
30	C₄×C₉	—	2	1	no	—
31	Dic₉	—	5	2	no	—
32	Dic₉	—	10	1	no	—
33	C₂×C₃×C₅	Dic₆	1	1	yes	characteristic
34	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
35	C₂×C₃×C₅	#67	1	1	yes	characteristic
36	C₂×C₃×C₄	D₁₅	1	1	yes	characteristic
37	C₂×Dic₃	—	15	1	no	—
38	C₂×Dic₃	—	15	1	no	—
39	C₂²×C₅	C₂×D₉	1	1	yes	characteristic
40	C₄×C₅	—	2	1	no	—
41	Dic₅	—	9	2	no	—
42	Dic₅	—	18	1	no	—
43	C₂×C₉	Dic₁₀	1	1	yes	characteristic
44	C₂×C₉	C₄×D₅	1	1	yes	characteristic
45	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
46	C₄⋊C₄	—	45	1	no	Sylow
47	C₃×C₅	Dic₃⋊C₄	1	1	yes	characteristic
48	C₂²×C₃	C₂×D₁₅	1	1	yes	characteristic
49	C₃×C₄	—	2	1	no	—
50	Dic₃	—	15	2	no	—
51	Dic₃	—	30	1	no	—
52	C₂×C₅	Dic₁₈	1	1	yes	characteristic
53	C₂×C₅	C₄×D₉	1	1	yes	characteristic
54	C₂×C₅	#611	1	1	yes	characteristic
55	C₉	#683	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄	—	45	1	no	—
57	C₂×C₄	—	45	1	no	—
58	C₂×C₄	D₄₅	1	1	yes	characteristic
59	C₂×C₃	Dic₃₀	1	1	yes	characteristic
60	C₂×C₃	C₄×D₁₅	1	1	yes	characteristic
61	C₂×C₃	#1220	1	1	yes	characteristic
62	C₅	Dic₉⋊C₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₂×D₄₅	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	45	2	no	—
66	C₄	—	90	1	no	—
67	C₃	#6712	1	1	yes	characteristic
68	C₂	Dic₉₀	1	1	yes	characteristic
69	C₂	C₄×D₄₅	1	1	yes	characteristic
70	C₂	#67099	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₅
commutator	C₂×C₅×C₉	C₂×C₄
Frattini	C₂²×C₃	C₂×D₁₅
Fitting	C₂×C₄×C₅×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₃
cosocle	C₂²×C₅×C₉	C₂²
hypercenter	C₂×C₄	D₄₅
hypocenter	C₅×C₉	C₄⋊C₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. Dic₄₅	C₄	semidirect
25. C₅×C₉	C₄⋊C₄	semidirect
55. C₉	#683	semidirect
62. C₅	Dic₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₉	2
C₂×Dic₄₅	2
#6712	3
Dic₉⋊C₄	5