Subgroups — #10623606

Order 720.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₄₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#6714	—	3	1	no	maximal
5	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
6	C₂×D₄₅	—	2	1	no	—
7	C₂×D₄₅	—	2	1	no	—
8	C₂×D₄₅	C₄	1	2	yes	complemented
9	C₄×C₅×C₉	—	2	1	no	—
10	Dic₄₅	—	2	1	no	—
11	(C₂×D₉)⋊C₄	—	5	1	no	maximal, Hall
12	C₂²×D₁₅	—	3	1	no	—
13	C₂×C₃×C₄×C₅	D₃	1	1	yes	characteristic
14	C₂×Dic₁₅	—	3	1	no	—
15	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×C₉	D₄	1	1	yes	characteristic
17	C₂×C₅×C₉	D₄	1	1	yes	characteristic
18	D₄₅	—	2	2	no	—
19	#685	—	9	1	no	Hall
20	C₂²×D₉	—	5	1	no	—
21	C₂×C₄×C₉	D₅	1	1	yes	characteristic, complemented
22	C₂×Dic₉	—	5	1	no	—
23	C₂²×C₃×C₅	C₂×D₃	1	1	yes	characteristic
24	C₂×D₁₅	—	3	2	no	—
25	C₂×D₁₅	—	6	1	no	—
26	C₂×D₁₅	—	6	1	no	—
27	C₃×C₄×C₅	—	2	1	no	—
28	Dic₁₅	—	6	1	no	—
29	(C₂×D₃)⋊C₄	—	15	1	no	—
30	C₅×C₉	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
31	C₂²×D₅	—	9	1	no	—
32	C₂×C₄×C₅	D₉	1	1	yes	characteristic, complemented
33	C₂×Dic₅	—	9	1	no	—
34	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
35	C₂×D₉	—	5	2	no	—
36	C₂×D₉	—	10	1	no	—
37	C₂×D₉	—	10	1	no	—
38	C₄×C₉	—	2	1	no	—
39	Dic₉	—	10	1	no	—
40	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
41	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
42	C₂×C₃×C₅	#67	1	1	yes	characteristic
43	D₁₅	—	6	2	no	—
44	C₂²×D₃	—	15	1	no	—
45	C₂×C₃×C₄	D₁₅	1	1	yes	characteristic
46	C₂×Dic₃	—	15	1	no	—
47	C₂²×C₅	C₂×D₉	1	1	yes	characteristic
48	C₂×D₅	—	9	2	no	—
49	C₂×D₅	—	18	1	no	—
50	C₂×D₅	—	18	1	no	—
51	C₄×C₅	—	2	1	no	—
52	Dic₅	—	18	1	no	—
53	C₂×C₉	C₄×D₅	1	1	yes	characteristic
54	C₂×C₉	D₂₀	1	1	yes	characteristic
55	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
56	D₉	—	10	2	no	—
57	C₂²⋊C₄	—	45	1	no	Sylow
58	C₃×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
59	C₂²×C₃	C₂×D₁₅	1	1	yes	characteristic
60	C₂×D₃	—	15	2	no	—
61	C₂×D₃	—	30	1	no	—
62	C₂×D₃	—	30	1	no	—
63	C₃×C₄	—	2	1	no	—
64	Dic₃	—	30	1	no	—
65	C₂×C₅	C₄×D₉	1	1	yes	characteristic
66	C₂×C₅	D₃₆	1	1	yes	characteristic
67	C₂×C₅	#611	1	1	yes	characteristic
68	D₅	—	18	2	no	—
69	C₉	#685	1	1	yes	characteristic, complemented, Sylow
70	C₂³	—	45	1	no	—
71	C₂×C₄	—	45	1	no	—
72	C₂×C₄	D₄₅	1	1	yes	characteristic, complemented
73	C₂×C₃	C₄×D₁₅	1	1	yes	characteristic
74	C₂×C₃	D₆₀	1	1	yes	characteristic
75	C₂×C₃	#1220	1	1	yes	characteristic
76	D₃	—	30	2	no	—
77	C₅	(C₂×D₉)⋊C₄	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	45	2	no	—
79	C₂²	—	90	1	no	—
80	C₂²	—	90	1	no	—
81	C₂²	C₂×D₄₅	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	90	1	no	—
84	C₃	#6714	1	1	yes	characteristic
85	C₂	—	90	2	no	—
86	C₂	C₄×D₄₅	1	1	yes	characteristic
87	C₂	D₁₈₀	1	1	yes	characteristic
88	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	C₂²×C₃×C₅	C₂×D₃
prime radical	C₂²×D₄₅	C₂
cosocle	C₂²×C₅×C₉	C₂²
hypercenter	C₂×C₄	D₄₅
hypocenter	C₅×C₉	C₂²⋊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
2. C₂×C₄×C₅×C₉	C₂	semidirect
3. C₂×Dic₄₅	C₂	semidirect
8. C₂×D₄₅	C₄	semidirect
21. C₂×C₄×C₉	D₅	semidirect
30. C₅×C₉	C₂²⋊C₄	semidirect
32. C₂×C₄×C₅	D₉	semidirect
69. C₉	#685	semidirect
72. C₂×C₄	D₄₅	semidirect
77. C₅	(C₂×D₉)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄₅	2
C₂×C₄×C₅×C₉	2
C₂×Dic₄₅	2
#6714	3
(C₂×D₉)⋊C₄	5