Subgroups — #65728

Order 320.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4130	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₈	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	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂²⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
8	C₂×#170	C₄	1	2	yes	complemented
9	C₄⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	D₅⋊C₈	C₄	1	2	yes	complemented
12	D₅⋊C₈	C₂²	1	1	yes	characteristic
13	D₅⋊C₈	C₂²	1	1	yes	characteristic
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	#405	—	5	1	no	maximal, Sylow
19	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₄×D₅	C₂×C₄	1	1	yes	characteristic
26	C₄×D₅	C₂³	1	1	yes	characteristic
27	#170	C₂×C₄	1	2	yes	—
28	#170	C₂×C₄	1	2	yes	—
29	C₅×C₈	C₂×C₄	1	2	yes	—
30	#168	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	—	5	1	no	—
32	C₂²×C₈	—	5	1	no	—
33	#117	—	5	1	no	—
34	#97	—	5	2	no	—
35	C₄×C₈	—	5	2	no	—
36	C₂²×C₅	C₄²	1	1	yes	characteristic
37	C₂×D₅	—	2	2	no	—
38	C₂×D₅	C₄²	1	1	yes	characteristic
39	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
40	AGL(1,𝔽₅)	—	2	4	no	—
41	C₄×C₅	C₄²	1	1	yes	characteristic
42	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	Dic₅	C₄²	1	1	yes	characteristic
44	Dic₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄	—	5	1	no	—
46	M₁₆	—	5	1	no	—
47	M₁₆	—	5	2	no	—
48	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
49	C₂×C₈	—	5	1	no	—
50	C₂×C₈	—	5	1	no	—
51	C₂×C₈	—	5	2	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	—	5	2	no	—
54	C₄⋊C₄	—	5	2	no	—
55	C₂²⋊C₄	—	5	2	no	—
56	C₄²	—	5	2	no	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
59	D₅	—	2	1	no	—
60	D₅	#131	1	2	yes	—
61	C₂³	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	—	5	4	no	—
68	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₈	—	5	2	no	—
70	C₈	—	5	2	no	—
71	C₈	—	5	2	no	—
72	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
73	C₅	#405	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	1	no	—
75	C₂²	—	5	1	no	—
76	C₂²	—	10	2	no	—
77	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₄	—	5	1	no	—
79	C₄	—	5	1	no	—
80	C₄	—	10	4	no	—
81	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	5	2	no	—
85	C₂	—	10	1	no	—
86	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	#405
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. C₂×#170	C₄	semidirect
11. D₅⋊C₈	C₄	semidirect
14. C₅×M₁₆	C₄	semidirect
15. #681	C₄	semidirect
16. #676	C₄	semidirect
48. M₁₆	AGL(1,𝔽₅)	semidirect
73. C₅	#405	semidirect

Maximal subgroups

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

subgroup	index
#4130	2
C₂×D₅⋊C₈	2
D₅×M₁₆	2
#3992	2
C₈×AGL(1,𝔽₅)	2
#405	5