Subgroups — #11762065

Order 1080.

336 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊Dic₁₅)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₃³×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#67141	—	3	13	no	maximal
5	C₂×C₃³×C₅	C₂²	1	1	yes	characteristic
6	#6308	—	5	1	no	maximal, Hall
7	C₃⋊Dic₁₅	—	3	13	no	—
8	C₅×C₃⋊Dic₃	—	3	13	no	—
9	C₃²×Dic₅	D₃	1	13	yes	—
10	C₃³×C₅	Q₈	1	1	yes	characteristic, complemented, Hall
11	#1204	—	9	13	no	—
12	C₃³×C₄	—	5	1	no	—
13	C₃⋊(C₃⋊Dic₃)	—	5	1	no	—
14	C₃⋊(C₃⋊Dic₃)	D₅	1	1	yes	characteristic
15	C₂×C₃²×C₅	C₂×D₃	1	13	yes	—
16	#634	—	15	13	no	—
17	Dic₁₅	—	9	13	no	—
18	C₃×Dic₅	C₃⋊D₃	1	13	yes	—
19	C₅×Dic₃	—	9	13	no	—
20	C₂×C₃³	C₂×D₅	1	1	yes	characteristic
21	C₃²×C₅	Dic₆	1	13	yes	complemented
22	Dic₁₀	—	27	1	no	Hall
23	C₃²×C₄	—	5	13	no	—
24	C₃⋊Dic₃	—	3	13	no	—
25	C₃⋊Dic₃	—	15	13	no	—
26	C₂×C₃×C₅	C₂×C₃⋊D₃	1	13	yes	—
27	C₃³	Dic₁₀	1	1	yes	characteristic, complemented, Sylow
28	Dic₆	—	45	13	no	—
29	C₄×C₅	—	27	1	no	—
30	Dic₅	—	27	1	no	—
31	Dic₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
32	C₂×C₃²	D₃×D₅	1	13	yes	—
33	C₃×C₅	#634	1	13	yes	complemented
34	C₃×C₄	—	5	13	no	—
35	Dic₃	—	9	13	no	—
36	Dic₃	—	45	13	no	—
37	C₂×C₅	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
38	C₃²	#1204	1	13	yes	complemented
39	Q₈	—	135	1	no	Sylow
40	C₂×C₃	D₅×C₃⋊D₃	1	13	yes	—
41	C₅	#6308	1	1	yes	characteristic, complemented, Sylow
42	C₄	—	5	1	no	—
43	C₄	—	27	1	no	—
44	C₄	—	135	1	no	—
45	C₃	#67141	1	13	yes	complemented
46	C₂	D₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×C₃⋊(C₃⋊D₃)
commutator, Fitting, socle, prime radical, cosocle	C₂×C₃³×C₅	C₂²
hypocenter	C₃³×C₅	Q₈
LCS	G ▷ C₂×C₃³×C₅ ▷ C₃³×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
10. C₃³×C₅	Q₈	semidirect
21. C₃²×C₅	Dic₆	semidirect
27. C₃³	Dic₁₀	semidirect
33. C₃×C₅	#634	semidirect
38. C₃²	#1204	semidirect
41. C₅	#6308	semidirect
45. C₃	#67141	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊Dic₁₅)	2
C₅×C₃⋊(C₃⋊Dic₃)	2
C₃³×Dic₅	2
#67141	3
#6308	5