Subgroups — #65718

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	D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₅	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	#704	C₄	1	2	yes	complemented
9	C₄⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	#700	C₄	1	2	yes	complemented
12	D₅⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×#168	C₄	1	1	yes	characteristic, complemented
15	C₈×D₅	C₄	1	1	yes	characteristic, complemented
16	C₈×D₅	C₄	1	1	yes	characteristic, complemented
17	C₈×D₅	C₂²	1	1	yes	characteristic
18	C₈×D₅	C₂²	1	1	yes	characteristic
19	#405	—	5	1	no	maximal, Sylow
20	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₅	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	C₂×C₄	1	1	yes	characteristic
27	C₄×D₅	C₂³	1	1	yes	characteristic
28	#170	C₂×C₄	1	4	yes	—
29	C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	#168	C₂×C₄	1	1	yes	characteristic
32	#168	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆	—	5	1	no	—
34	C₂²×C₈	—	5	1	no	—
35	#117	—	5	1	no	—
36	#97	—	5	2	no	—
37	C₄×C₈	—	5	2	no	—
38	C₂²×C₅	C₄²	1	1	yes	characteristic
39	C₂×D₅	—	2	2	no	—
40	C₂×D₅	C₄²	1	1	yes	characteristic
41	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
42	AGL(1,𝔽₅)	—	2	4	no	—
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	M₁₆	—	5	2	no	—
49	M₁₆	—	5	2	no	—
50	C₂×C₈	—	5	2	no	—
51	C₂×C₈	—	5	1	no	—
52	C₂×C₈	—	5	1	no	—
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	—	5	1	no	—
56	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
57	C₄⋊C₄	—	5	2	no	—
58	C₂²⋊C₄	—	5	2	no	—
59	C₄²	—	5	2	no	—
60	C₂×C₅	—	2	1	no	—
61	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
62	D₅	—	2	1	no	—
63	D₅	#131	1	2	yes	—
64	C₂³	—	5	1	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	4	no	—
71	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₈	—	5	4	no	—
73	C₈	—	5	1	no	—
74	C₈	—	5	1	no	—
75	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₅	#405	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	5	1	no	—
79	C₂²	—	5	1	no	—
80	C₂²	—	10	2	no	—
81	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₄	—	5	1	no	—
83	C₄	—	5	1	no	—
84	C₄	—	10	4	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	C₈	C₂×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂×C₄²
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈

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
11. #700	C₄	semidirect
13. C₂×C₅×C₈	C₄	semidirect
14. C₂×#168	C₄	semidirect
15. C₈×D₅	C₄	semidirect
16. C₈×D₅	C₄	semidirect
56. C₂×C₈	AGL(1,𝔽₅)	semidirect
77. 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
D₅⋊M₁₆	2
C₂×C₈×D₅	2
#3992	2
C₈×AGL(1,𝔽₅)	2
#405	5