Subgroups — #10623989

Order 720.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	#67126	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#622	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₅	C₄	1	1	yes	characteristic
5	C₃⋊Dic₁₅	C₄	1	1	yes	characteristic
6	C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	#3792	—	5	1	no	maximal, Hall
8	D₅×Dic₃	—	6	2	no	—
9	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
10	C₃²×D₅	—	2	1	no	—
11	#676	—	9	1	no	maximal, Hall
12	C₂×C₃⋊Dic₃	—	5	1	no	—
13	#622	—	5	1	no	—
14	#622	D₅	1	1	yes	characteristic, complemented
15	C₂×C₃×D₅	—	2	2	no	—
16	Dic₁₅	—	6	2	no	—
17	C₅×Dic₃	—	6	2	no	—
18	C₃²×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
19	C₄×D₅	—	9	1	no	—
20	C₅×C₈	—	9	1	no	—
21	#168	—	9	1	no	—
22	C₂²×C₃²	—	5	1	no	—
23	C₃⋊Dic₃	—	5	1	no	—
24	C₃⋊Dic₃	C₂×D₅	1	1	yes	characteristic
25	C₂×C₃×C₅	—	2	2	no	—
26	C₃×D₅	—	2	4	no	—
27	C₂×Dic₃	—	30	2	no	—
28	C₂×D₅	#157	1	1	yes	characteristic
29	C₄×C₅	—	9	1	no	—
30	Dic₅	—	9	1	no	—
31	C₂×C₃²	—	10	1	no	—
32	C₂×C₃²	C₄×D₅	1	1	yes	characteristic
33	M₁₆	—	45	1	no	Sylow
34	C₃×C₅	—	2	2	no	—
35	C₂²×C₃	—	10	2	no	—
36	Dic₃	—	6	2	no	—
37	Dic₃	—	30	2	no	—
38	C₂×C₅	C₂×#157	1	1	yes	characteristic
39	D₅	—	2	1	no	—
40	C₃²	#676	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₄	—	45	1	no	—
42	C₈	—	9	1	no	—
43	C₈	—	45	1	no	—
44	C₂×C₃	—	2	2	no	—
45	C₂×C₃	—	10	4	no	—
46	C₅	#3792	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	5	1	no	—
48	C₄	—	9	1	no	—
49	C₄	—	45	1	no	—
50	C₃	—	2	2	no	—
51	C₂	—	10	1	no	—
52	C₂	D₅×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×#157
commutator, Fitting, socle	C₂×C₃²×C₅	C₂×C₄
prime radical	C₂×C₃²×D₅	C₄
cosocle	C₅×C₃⋊Dic₃	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. #67126	C₂	semidirect
3. C₅×#622	C₂	semidirect
14. #622	D₅	semidirect
18. C₃²×C₅	M₁₆	semidirect
40. C₃²	#676	semidirect
46. C₅	#3792	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₃⋊Dic₃	2
#67126	2
C₅×#622	2
#3792	5
#676	9