Subgroups — #13086294

Order 1440.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	#91221	—	3	4	no	maximal
5	C₂×C₃⋊ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
6	C₃²×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊#170)	C₄	1	2	yes	complemented
8	C₃²×C₅×C₈	C₄	1	1	yes	characteristic, complemented
9	C₃²×#168	C₄	1	1	yes	characteristic, complemented
10	#63642	—	5	1	no	maximal, Hall
11	C₄×ΓL(1,𝔽₁₆)	—	3	4	no	—
12	D₅⋊(C₃⋊C₈)	—	3	4	no	—
13	C₃×C₈×D₅	D₃	1	4	yes	—
14	C₂×C₃²×D₅	C₂×C₄	1	1	yes	characteristic
15	C₃⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
16	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃²×Dic₅	C₂×C₄	1	1	yes	characteristic
18	#3992	—	9	1	no	Hall
19	C₂×C₃²×C₈	—	5	1	no	—
20	C₄×C₃⋊Dic₃	—	5	1	no	—
21	C₂×C₃⋊(C₃⋊C₈)	—	5	1	no	—
22	C₂×ΓL(1,𝔽₁₆)	—	3	8	no	—
23	C₃×C₄×D₅	C₂×D₃	1	4	yes	—
24	C₃⋊#170	—	3	8	no	—
25	C₃×C₅×C₈	Dic₃	1	4	yes	complemented
26	C₃×#168	Dic₃	1	4	yes	complemented
27	#815	—	15	4	no	—
28	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
29	C₃²×D₅	M₁₆	1	2	yes	—
30	C₄×AGL(1,𝔽₅)	—	9	1	no	—
31	D₅⋊C₈	—	9	1	no	—
32	C₈×D₅	C₃⋊D₃	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	5	1	no	—
34	C₂×C₃⋊Dic₃	—	5	2	no	—
35	C₃²×C₈	—	5	1	no	—
36	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
37	C₃⋊(C₃⋊C₈)	—	5	2	no	—
38	C₂×C₃×D₅	C₄×D₃	1	4	yes	—
39	ΓL(1,𝔽₁₆)	—	6	8	no	—
40	C₃×C₄×C₅	C₂×Dic₃	1	4	yes	—
41	C₃×Dic₅	C₄×D₃	1	4	yes	—
42	C₂×C₃×C₈	—	5	4	no	—
43	C₄×Dic₃	—	15	4	no	—
44	C₂×C₃⋊C₈	—	15	4	no	—
45	C₃²×C₅	#97	1	1	yes	characteristic, complemented, Hall
46	C₂×AGL(1,𝔽₅)	—	9	2	no	—
47	C₄×D₅	C₂×C₃⋊D₃	1	1	yes	characteristic
48	#170	—	9	2	no	—
49	C₅×C₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
50	#168	C₃⋊Dic₃	1	1	yes	characteristic, complemented
51	C₂²×C₃²	—	5	1	no	—
52	C₃²×C₄	—	5	1	no	—
53	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₃⋊Dic₃	—	10	2	no	—
55	#97	—	45	1	no	Sylow
56	C₂×C₃×C₅	C₄×Dic₃	1	4	yes	—
57	C₃×D₅	ΓL(1,𝔽₂₅)	1	8	yes	—
58	C₂×C₃×C₄	—	5	4	no	—
59	C₂×Dic₃	—	15	8	no	—
60	C₃×C₈	—	5	4	no	—
61	C₃×C₈	ΓL(1,𝔽₁₆)	1	4	yes	complemented
62	C₃⋊C₈	—	15	8	no	—
63	C₂×D₅	C₄×C₃⋊D₃	1	1	yes	characteristic
64	AGL(1,𝔽₅)	—	18	2	no	—
65	C₄×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
66	Dic₅	C₄×C₃⋊D₃	1	1	yes	characteristic
67	C₂×C₃²	—	5	2	no	—
68	C₂×C₃²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₈	—	5	1	no	—
70	C₂×C₈	—	45	1	no	—
71	C₄²	—	45	1	no	—
72	C₃×C₅	#815	1	4	yes	complemented
73	C₂²×C₃	—	5	4	no	—
74	C₃×C₄	—	5	4	no	—
75	C₃×C₄	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
76	Dic₃	—	30	8	no	—
77	C₂×C₅	C₄×C₃⋊Dic₃	1	1	yes	characteristic
78	D₅	#3743	1	2	yes	—
79	C₃²	#3992	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄	—	5	1	no	—
81	C₂×C₄	—	45	2	no	—
82	C₈	—	5	1	no	—
83	C₈	—	45	2	no	—
84	C₈	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
85	C₂×C₃	—	5	8	no	—
86	C₂×C₃	C₄×ΓL(1,𝔽₁₆)	1	4	yes	—
87	C₅	#63642	1	1	yes	characteristic, complemented, Sylow
88	C₂²	—	5	1	no	—
89	C₄	—	5	1	no	—
90	C₄	—	90	2	no	—
91	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
92	C₃	#91221	1	4	yes	complemented
93	C₂	—	5	2	no	—
94	C₂	C₄×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃²×C₅	C₄²
Fitting	C₃²×C₅×C₈	C₄
prime radical	C₂×C₃²×D₅	C₂×C₄
cosocle	C₃²×C₄×D₅	C₂²
hypercenter	C₈	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	#97
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
7. C₃⋊(C₃⋊#170)	C₄	semidirect
8. C₃²×C₅×C₈	C₄	semidirect
9. C₃²×#168	C₄	semidirect
25. C₃×C₅×C₈	Dic₃	semidirect
26. C₃×#168	Dic₃	semidirect
36. C₃²×C₈	AGL(1,𝔽₅)	semidirect
45. C₃²×C₅	#97	semidirect
49. C₅×C₈	C₃⋊Dic₃	semidirect
50. #168	C₃⋊Dic₃	semidirect
61. C₃×C₈	ΓL(1,𝔽₁₆)	semidirect
72. C₃×C₅	#815	semidirect
79. C₃²	#3992	semidirect
84. C₈	C₃⋊ΓL(1,𝔽₁₆)	semidirect
87. C₅	#63642	semidirect
92. C₃	#91221	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
D₅⋊(C₃⋊(C₃⋊C₈))	2
C₃²×C₈×D₅	2
#91221	3
#63642	5