Subgroups — M₁₆×AGL(1,𝔽₅) #65726

Order 320.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	D₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#3992	C₂	1	2	yes	maximal, complemented
5	C₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	C₂²×AGL(1,𝔽₅)	C₄	1	2	yes	—
7	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
8	#704	C₄	1	2	yes	complemented
9	C₄×AGL(1,𝔽₅)	C₄	1	2	yes	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
11	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
12	#700	C₄	1	2	yes	complemented
13	D₅⋊C₈	C₂²	1	2	yes	—
14	C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
15	#681	C₄	1	1	yes	characteristic, complemented
16	#676	C₄	1	2	yes	complemented
17	C₈×D₅	C₂²	1	2	yes	—
18	C₄×M₁₆	—	5	1	no	maximal, Sylow
19	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×AGL(1,𝔽₅)	—	2	4	no	—
21	C₂×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
22	C₂×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₄×D₅	C₂×C₄	1	1	yes	characteristic
26	C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	C₄×D₅	C₂³	1	1	yes	characteristic
29	#170	C₂×C₄	1	4	yes	complemented
30	C₅×C₈	C₂×C₄	1	2	yes	complemented
31	#168	C₂×C₄	1	2	yes	complemented
32	C₂×M₁₆	—	5	1	no	—
33	C₂×M₁₆	—	5	1	no	—
34	C₂×C₄²	—	5	1	no	—
35	#97	—	5	2	no	—
36	C₄×C₈	—	5	2	no	—
37	C₂²×C₅	C₄²	1	1	yes	characteristic
38	C₂×D₅	—	2	2	no	—
39	C₂×D₅	C₄²	1	1	yes	characteristic
40	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
41	AGL(1,𝔽₅)	—	2	2	no	—
42	AGL(1,𝔽₅)	M₁₆	1	4	yes	direct product factor
43	C₄×C₅	C₄²	1	1	yes	characteristic
44	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	Dic₅	C₄²	1	1	yes	characteristic
46	Dic₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄	—	5	1	no	—
48	C₂²×C₄	—	5	2	no	—
49	M₁₆	—	5	2	no	—
50	M₁₆	—	5	2	no	—
51	M₁₆	—	5	1	no	—
52	M₁₆	—	5	2	no	—
53	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
54	C₂×C₈	—	5	2	no	—
55	C₂×C₈	—	5	2	no	—
56	C₄²	—	5	1	no	—
57	C₄²	—	5	1	no	—
58	C₄²	—	5	2	no	—
59	C₂×C₅	—	2	1	no	—
60	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
61	D₅	—	2	1	no	—
62	D₅	C₂×M₁₆	1	2	yes	—
63	C₂³	—	5	1	no	—
64	C₂×C₄	—	5	2	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	—	5	1	no	—
68	C₂×C₄	—	5	1	no	—
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	5	2	no	—
71	C₂×C₄	—	10	4	no	—
72	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₈	—	5	4	no	—
74	C₈	—	5	2	no	—
75	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
76	C₅	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	5	1	no	—
78	C₂²	—	5	1	no	—
79	C₂²	—	10	2	no	—
80	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₄	—	5	4	no	—
82	C₄	—	5	1	no	—
83	C₄	—	5	1	no	—
84	C₄	—	10	2	no	—
85	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	5	2	no	—
89	C₂	—	10	1	no	—
90	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂×C₄²
Fitting	C₅×M₁₆	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
4. #3992	C₂	semidirect
5. C₈×AGL(1,𝔽₅)	C₂	semidirect
8. #704	C₄	semidirect
12. #700	C₄	semidirect
14. C₅×M₁₆	C₄	semidirect
15. #681	C₄	semidirect
16. #676	C₄	semidirect
29. #170	C₂×C₄	semidirect
30. C₅×C₈	C₂×C₄	semidirect
31. #168	C₂×C₄	semidirect
42. AGL(1,𝔽₅)	M₁₆	direct (N × H)
53. M₁₆	AGL(1,𝔽₅)	direct (N × H)
75. C₈	C₂×AGL(1,𝔽₅)	semidirect
76. C₅	C₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×AGL(1,𝔽₅)	2
D₅⋊M₁₆	2
D₅×M₁₆	2
#3992	2
C₈×AGL(1,𝔽₅)	2
C₄×M₁₆	5

Subgroups — M16×AGL(1,𝔽5) #65726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×AGL(1,𝔽₅) #65726