Subgroups — #11746723

Order 960.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91225	C₂	1	2	yes	maximal
2	C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#64852	—	3	1	no	maximal, Hall
4	#6758	—	2	2	no	—
5	C₃×C₈×D₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×C₁₆	C₄	1	1	yes	characteristic
7	C₃×#672	C₄	1	1	yes	characteristic
8	#4506	—	5	1	no	maximal, Hall
9	#3996	—	3	2	no	—
10	D₅×C₁₆	D₃	1	1	yes	characteristic
11	C₃×C₄×D₅	D₄	1	1	yes	characteristic
12	C₃⋊#170	—	4	2	no	—
13	C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₃×#168	Q₈	1	1	yes	characteristic
15	C₂×C₃×C₁₆	—	5	1	no	—
16	#819	—	5	2	no	—
17	#700	—	6	2	no	—
18	C₈×D₅	C₂×D₃	1	1	yes	characteristic
19	C₅×C₁₆	Dic₃	1	1	yes	characteristic
20	#672	Dic₃	1	1	yes	characteristic
21	#368	—	15	1	no	Sylow
22	C₂×C₃×D₅	Q₁₆	1	1	yes	characteristic
23	C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
24	C₃×Dic₅	D₈	1	1	yes	characteristic
25	C₂×C₃×C₈	—	5	1	no	—
26	#208	—	10	2	no	—
27	C₃×C₁₆	—	5	1	no	—
28	C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
29	C₄×D₅	D₁₂	1	1	yes	characteristic
30	#170	—	12	2	no	—
31	C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
32	#168	Dic₆	1	1	yes	characteristic
33	C₂×C₁₆	—	5	1	no	—
34	#108	—	15	2	no	—
35	C₂×C₃×C₅	#107	1	1	yes	characteristic
36	C₃×D₅	—	2	1	no	—
37	C₂×C₃×C₄	—	5	1	no	—
38	C₃×C₈	—	5	1	no	—
39	C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₃⋊C₈	—	20	2	no	—
41	C₂×D₅	Dic₁₂	1	1	yes	characteristic
42	C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
43	Dic₅	D₂₄	1	1	yes	characteristic
44	M₁₆	—	30	2	no	—
45	C₂×C₈	—	5	1	no	—
46	C₁₆	—	5	1	no	—
47	C₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
48	C₃×C₅	#368	1	1	yes	characteristic, complemented, Hall
49	C₂²×C₃	—	5	1	no	—
50	C₃×C₄	—	5	1	no	—
51	C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂×C₅	#818	1	1	yes	characteristic
53	D₅	—	2	1	no	—
54	C₂×C₄	—	5	1	no	—
55	C₈	—	5	1	no	—
56	C₈	—	60	2	no	—
57	C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
58	C₂×C₃	—	10	1	no	—
59	C₂×C₃	#3994	1	1	yes	characteristic
60	C₅	#4506	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	5	1	no	—
62	C₄	—	5	1	no	—
63	C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
64	C₃	#64852	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	10	1	no	—
66	C₂	#91223	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#91223
commutator	C₃×C₅×C₈	C₂×C₄
Frattini	C₈	C₂×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×C₁₆	C₄
socle	C₂×C₃×C₅	#107
prime radical	C₂×C₃×D₅	Q₁₆
cosocle	C₃×C₈×D₅	C₂²
hypercenter	C₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#368
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
48. C₃×C₅	#368	semidirect
60. C₅	#4506	semidirect
64. C₃	#64852	semidirect

Maximal subgroups

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

subgroup	index
#91225	2
C₃×D₅×C₁₆	2
#64852	3
#4506	5