Subgroups — #13086296

Order 1440.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃⋊ΓL(1,𝔽₁₆))	C₂	1	2	yes	maximal
2	C₃²×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#91223	—	3	4	no	maximal
4	C₂×C₃⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
5	C₃²×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₃²×C₅×C₈	C₄	1	1	yes	characteristic, complemented
7	C₃²×#168	C₄	1	1	yes	characteristic, complemented
8	#63645	—	5	1	no	maximal, Hall
9	C₄⋊ΓL(1,𝔽₁₆)	—	3	8	no	—
10	C₃×C₈×D₅	D₃	1	4	yes	—
11	C₂×C₃²×D₅	D₄	1	1	yes	characteristic
12	C₃⋊ΓL(1,𝔽₁₆)	—	4	2	no	—
13	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₃²×Dic₅	Q₈	1	1	yes	characteristic
15	#3994	—	9	1	no	Hall
16	C₂×C₃²×C₈	—	5	1	no	—
17	C₄⋊(C₃⋊Dic₃)	—	5	2	no	—
18	C₂×ΓL(1,𝔽₁₆)	—	6	8	no	—
19	C₃×C₄×D₅	C₂×D₃	1	4	yes	—
20	C₃×C₅×C₈	Dic₃	1	4	yes	complemented
21	C₃×#168	Dic₃	1	4	yes	complemented
22	#818	—	15	4	no	—
23	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
24	C₃²×D₅	D₈	1	1	yes	characteristic
25	C₃²×D₅	Q₁₆	1	1	yes	characteristic
26	C₄⋊AGL(1,𝔽₅)	—	9	2	no	—
27	C₈×D₅	C₃⋊D₃	1	1	yes	characteristic
28	C₂×C₃²×C₄	—	5	1	no	—
29	C₂×C₃⋊Dic₃	—	10	2	no	—
30	C₃²×C₈	—	5	1	no	—
31	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
32	C₂×C₃×D₅	D₁₂	1	4	yes	—
33	ΓL(1,𝔽₁₆)	—	12	8	no	—
34	C₃×C₄×C₅	C₂×Dic₃	1	4	yes	—
35	C₃×Dic₅	Dic₆	1	4	yes	—
36	C₂×C₃×C₈	—	5	4	no	—
37	C₄⋊Dic₃	—	15	8	no	—
38	C₃²×C₅	#107	1	1	yes	characteristic, complemented, Hall
39	C₂×AGL(1,𝔽₅)	—	18	2	no	—
40	C₄×D₅	C₂×C₃⋊D₃	1	1	yes	characteristic
41	C₅×C₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
42	#168	C₃⋊Dic₃	1	1	yes	characteristic, complemented
43	C₂²×C₃²	—	5	1	no	—
44	C₃²×C₄	—	5	1	no	—
45	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃⋊Dic₃	—	20	2	no	—
47	#107	—	45	1	no	Sylow
48	C₂×C₃×C₅	C₄⋊Dic₃	1	4	yes	—
49	C₃×D₅	D₂₄	1	4	yes	—
50	C₃×D₅	Dic₁₂	1	4	yes	—
51	C₂×C₃×C₄	—	5	4	no	—
52	C₂×Dic₃	—	30	8	no	—
53	C₃×C₈	—	5	4	no	—
54	C₃×C₈	ΓL(1,𝔽₁₆)	1	4	yes	complemented
55	C₂×D₅	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
56	AGL(1,𝔽₅)	—	36	2	no	—
57	C₄×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
58	Dic₅	#634	1	1	yes	characteristic
59	C₂×C₃²	—	5	1	no	—
60	C₂×C₃²	—	5	1	no	—
61	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂×C₈	—	5	1	no	—
63	C₄⋊C₄	—	45	2	no	—
64	C₃×C₅	#818	1	4	yes	complemented
65	C₂²×C₃	—	5	4	no	—
66	C₃×C₄	—	5	4	no	—
67	C₃×C₄	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
68	Dic₃	—	60	8	no	—
69	C₂×C₅	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
70	D₅	#3745	1	1	yes	characteristic
71	D₅	#3746	1	1	yes	characteristic
72	C₃²	#3994	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	90	2	no	—
75	C₈	—	5	1	no	—
76	C₈	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
77	C₂×C₃	—	5	4	no	—
78	C₂×C₃	—	5	4	no	—
79	C₂×C₃	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	—
80	C₅	#63645	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	5	1	no	—
82	C₄	—	5	1	no	—
83	C₄	—	180	2	no	—
84	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
85	C₃	#91223	1	4	yes	complemented
86	C₂	—	5	1	no	—
87	C₂	—	5	1	no	—
88	C₂	C₄⋊(C₃⋊ΓL(1,𝔽₁₆))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊(C₃⋊ΓL(1,𝔽₁₆))
commutator	C₃²×C₄×C₅	C₂×C₄
Frattini	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₃²×C₅×C₈	C₄
socle	C₂×C₃²×C₅	C₄⋊C₄
prime radical	C₂×C₃²×D₅	D₄
cosocle	C₃²×C₄×D₅	C₂²
hypercenter	C₈	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	#107
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
6. C₃²×C₅×C₈	C₄	semidirect
7. C₃²×#168	C₄	semidirect
20. C₃×C₅×C₈	Dic₃	semidirect
21. C₃×#168	Dic₃	semidirect
31. C₃²×C₈	AGL(1,𝔽₅)	semidirect
38. C₃²×C₅	#107	semidirect
41. C₅×C₈	C₃⋊Dic₃	semidirect
42. #168	C₃⋊Dic₃	semidirect
54. C₃×C₈	ΓL(1,𝔽₁₆)	semidirect
64. C₃×C₅	#818	semidirect
72. C₃²	#3994	semidirect
76. C₈	C₃⋊ΓL(1,𝔽₁₆)	semidirect
80. C₅	#63645	semidirect
85. C₃	#91223	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃⋊ΓL(1,𝔽₁₆))	2
C₃²×C₈×D₅	2
#91223	3
#63645	5