Subgroups — #65727

Order 320.

118 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×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#3994	C₂	1	2	yes	maximal, complemented
5	#3993	C₂	1	2	yes	maximal, complemented
6	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
7	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
9	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
10	C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
12	C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
13	C₄×AGL(1,𝔽₅)	—	2	1	no	—
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	#428	—	5	1	no	maximal, Sylow
19	C₂²×D₅	D₄	1	1	yes	characteristic
20	C₂×AGL(1,𝔽₅)	—	2	1	no	—
21	C₂×AGL(1,𝔽₅)	—	2	1	no	—
22	C₂×AGL(1,𝔽₅)	—	2	2	no	—
23	C₂×AGL(1,𝔽₅)	—	4	2	no	—
24	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₅	Q₈	1	1	yes	characteristic
26	C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	C₄×D₅	D₄	1	1	yes	characteristic
28	C₄×D₅	Q₈	1	1	yes	characteristic
29	C₄×D₅	C₂³	1	1	yes	characteristic
30	C₅×C₈	C₂×C₄	1	2	yes	complemented
31	#168	C₂×C₄	1	2	yes	complemented
32	C₂×M₁₆	—	5	1	no	—
33	#117	—	5	1	no	—
34	C₂×C₄⋊C₄	—	5	1	no	—
35	#107	—	5	2	no	—
36	#106	—	5	2	no	—
37	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
38	C₂×D₅	—	2	1	no	—
39	C₂×D₅	—	2	1	no	—
40	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂×D₅	C₂×D₄	1	1	yes	characteristic
42	AGL(1,𝔽₅)	—	4	1	no	—
43	AGL(1,𝔽₅)	—	4	2	no	—
44	AGL(1,𝔽₅)	—	4	1	no	—
45	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	Dic₅	C₄⋊C₄	1	1	yes	characteristic
48	Dic₅	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄	—	5	1	no	—
50	C₂²×C₄	—	10	1	no	—
51	M₁₆	—	5	1	no	—
52	M₁₆	—	5	2	no	—
53	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
54	C₂×C₈	—	5	2	no	—
55	C₄⋊C₄	—	5	1	no	—
56	C₄⋊C₄	—	5	2	no	—
57	C₄⋊C₄	—	5	1	no	—
58	C₄⋊C₄	—	10	1	no	—
59	C₂²⋊C₄	—	10	1	no	—
60	C₄²	—	10	1	no	—
61	C₂×C₅	—	2	1	no	—
62	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
63	D₅	—	2	1	no	—
64	D₅	AGL(1,ℤ/8)	1	1	yes	characteristic
65	D₅	#137	1	1	yes	characteristic
66	C₂³	—	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	1	no	—
71	C₂×C₄	—	5	1	no	—
72	C₂×C₄	—	10	1	no	—
73	C₂×C₄	—	10	1	no	—
74	C₂×C₄	—	10	2	no	—
75	C₂×C₄	—	20	2	no	—
76	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₈	—	5	2	no	—
78	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
79	C₅	#428	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	1	no	—
81	C₂²	—	5	1	no	—
82	C₂²	—	10	1	no	—
83	C₂²	—	10	1	no	—
84	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₄	—	5	1	no	—
86	C₄	—	5	1	no	—
87	C₄	—	20	2	no	—
88	C₄	—	20	1	no	—
89	C₄	—	20	1	no	—
90	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	5	1	no	—
94	C₂	—	5	1	no	—
95	C₂	—	10	1	no	—
96	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×M₁₆	C₄
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₄×D₅	C₂³
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#428
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
4. #3994	C₂	semidirect
5. #3993	C₂	semidirect
14. C₅×M₁₆	C₄	semidirect
15. #681	C₄	semidirect
16. #676	C₄	semidirect
30. C₅×C₈	C₂×C₄	semidirect
31. #168	C₂×C₄	semidirect
53. M₁₆	AGL(1,𝔽₅)	semidirect
78. C₈	C₂×AGL(1,𝔽₅)	semidirect
79. C₅	#428	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×C₄⋊AGL(1,𝔽₅)	2
D₅×M₁₆	2
#3994	2
#3993	2
#428	5