Subgroups — #13083699

Order 1440.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₉×D₅	C₂	1	1	yes	maximal, characteristic
2	C₉⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#90934	—	3	1	no	maximal
5	C₄×C₉×D₅	C₄	1	1	yes	characteristic
6	C₅×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₉×#168	C₄	1	1	yes	characteristic
8	#63371	—	5	1	no	maximal, Hall
9	C₃×C₈×D₅	D₃	1	1	yes	characteristic
10	C₃⋊#672	—	3	1	no	—
11	C₅×C₃⋊C₁₆	—	3	1	no	—
12	C₂×C₉×D₅	C₈	1	1	yes	characteristic
13	C₄×C₅×C₉	C₂×C₄	1	1	yes	characteristic
14	C₉×Dic₅	C₈	1	1	yes	characteristic
15	#3930	—	9	1	no	Hall
16	C₂×C₈×C₉	—	5	1	no	—
17	C₉⋊C₁₆	—	5	1	no	—
18	C₉⋊C₁₆	D₅	1	1	yes	characteristic, complemented
19	C₃×C₄×D₅	Dic₃	1	1	yes	characteristic
20	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
21	C₃×#168	Dic₃	1	1	yes	characteristic
22	#812	—	15	1	no	—
23	C₂×C₅×C₉	C₂×C₈	1	1	yes	characteristic
24	C₉×D₅	—	2	1	no	—
25	C₈×D₅	D₉	1	1	yes	characteristic
26	C₅×C₁₆	—	9	1	no	—
27	#672	—	9	1	no	—
28	C₂×C₄×C₉	—	5	1	no	—
29	C₈×C₉	—	5	1	no	—
30	C₈×C₉	C₂×D₅	1	1	yes	characteristic
31	C₂×C₃×D₅	C₃⋊C₈	1	1	yes	characteristic
32	C₃×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
33	C₃×Dic₅	C₃⋊C₈	1	1	yes	characteristic
34	C₂×C₃×C₈	—	5	1	no	—
35	C₃⋊C₁₆	—	3	1	no	—
36	C₃⋊C₁₆	—	15	1	no	—
37	C₅×C₉	M₃₂	1	1	yes	characteristic, complemented, Hall
38	C₄×D₅	Dic₉	1	1	yes	characteristic
39	C₅×C₈	C₂×D₉	1	1	yes	characteristic
40	#168	Dic₉	1	1	yes	characteristic
41	C₂²×C₉	—	5	1	no	—
42	C₄×C₉	—	5	1	no	—
43	C₄×C₉	C₄×D₅	1	1	yes	characteristic
44	M₃₂	—	45	1	no	Sylow
45	C₂×C₃×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
46	C₃×D₅	—	2	1	no	—
47	C₂×C₃×C₄	—	5	1	no	—
48	C₃×C₈	—	5	1	no	—
49	C₃×C₈	D₃×D₅	1	1	yes	characteristic
50	C₂×D₅	C₉⋊C₈	1	1	yes	characteristic
51	C₄×C₅	C₂×Dic₉	1	1	yes	characteristic
52	Dic₅	C₉⋊C₈	1	1	yes	characteristic
53	C₂×C₉	—	10	1	no	—
54	C₂×C₉	C₈×D₅	1	1	yes	characteristic
55	C₂×C₈	—	5	1	no	—
56	C₁₆	—	9	1	no	—
57	C₁₆	—	45	1	no	—
58	C₃×C₅	#812	1	1	yes	characteristic
59	C₂²×C₃	—	5	1	no	—
60	C₃×C₄	—	5	1	no	—
61	C₃×C₄	D₅×Dic₃	1	1	yes	characteristic
62	C₂×C₅	C₂×C₉⋊C₈	1	1	yes	characteristic
63	D₅	—	2	1	no	—
64	C₉	#3930	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄	—	5	1	no	—
66	C₈	—	5	1	no	—
67	C₈	D₅×D₉	1	1	yes	characteristic
68	C₂×C₃	—	10	1	no	—
69	C₂×C₃	D₅×C₃⋊C₈	1	1	yes	characteristic
70	C₅	#63371	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	5	1	no	—
72	C₄	—	5	1	no	—
73	C₄	D₅×Dic₉	1	1	yes	characteristic
74	C₃	#90934	1	1	yes	characteristic
75	C₂	—	10	1	no	—
76	C₂	D₅×C₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	D₅×D₉
commutator	C₂×C₅×C₉	C₂×C₈
Frattini	C₃×C₈	D₃×D₅
Fitting, cosocle	C₅×C₈×C₉	C₂²
socle	C₂×C₃×C₅	C₂×C₃⋊C₈
prime radical	C₂×C₃×D₅	C₃⋊C₈
hypocenter	C₅×C₉	M₃₂
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₉⋊#672	C₂	semidirect
3. C₅×C₉⋊C₁₆	C₂	semidirect
18. C₉⋊C₁₆	D₅	semidirect
37. C₅×C₉	M₃₂	semidirect
64. C₉	#3930	semidirect
70. C₅	#63371	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₉×D₅	2
C₉⋊#672	2
C₅×C₉⋊C₁₆	2
#90934	3
#63371	5