Subgroups — #64899

Order 320.

98 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	C₂²×AGL(1,𝔽₅)	C₄	1	2	yes	—
5	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
6	#704	C₄	1	2	yes	complemented
7	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
8	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
9	#700	—	2	1	no	—
10	D₅⋊C₈	—	2	1	no	—
11	C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
12	#681	C₄	1	1	yes	characteristic, complemented
13	#676	—	2	1	no	—
14	C₈×D₅	—	2	1	no	—
15	#343	—	5	1	no	maximal, Sylow
16	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×AGL(1,𝔽₅)	—	4	2	no	—
19	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₄×D₅	D₄	1	1	yes	characteristic
22	C₄×D₅	D₄	1	1	yes	characteristic
23	C₄×D₅	D₄	1	1	yes	characteristic
24	C₄×D₅	Q₈	1	1	yes	characteristic
25	#170	—	2	2	no	—
26	C₅×C₈	—	2	1	no	—
27	#168	—	2	1	no	—
28	C₂×M₁₆	—	5	1	no	—
29	C₂×M₁₆	—	5	1	no	—
30	C₂×C₄⋊C₄	—	5	1	no	—
31	C₂²×C₅	C₄²	1	1	yes	characteristic
32	C₂×D₅	—	2	1	no	—
33	C₂×D₅	—	2	1	no	—
34	C₂×D₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
36	AGL(1,𝔽₅)	—	4	2	no	—
37	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
40	Dic₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄	—	5	1	no	—
42	C₂²×C₄	—	5	2	no	—
43	M₁₆	—	5	2	no	—
44	M₁₆	—	5	1	no	—
45	M₁₆	—	10	1	no	—
46	M₁₆	—	10	1	no	—
47	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
48	C₂×C₈	—	10	1	no	—
49	C₂×C₈	—	10	1	no	—
50	C₄⋊C₄	—	10	1	no	—
51	C₄⋊C₄	—	10	1	no	—
52	C₂×C₅	—	2	1	no	—
53	C₂×C₅	#95	1	1	yes	characteristic
54	D₅	—	2	1	no	—
55	D₅	#100	1	1	yes	characteristic
56	D₅	#101	1	1	yes	characteristic
57	C₂³	—	5	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	10	2	no	—
64	C₂×C₄	—	20	2	no	—
65	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	10	2	no	—
68	C₈	—	10	1	no	—
69	C₅	#343	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	5	1	no	—
71	C₂²	—	5	1	no	—
72	C₂²	—	10	1	no	—
73	C₂²	—	10	1	no	—
74	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₄	—	5	1	no	—
76	C₄	—	5	1	no	—
77	C₄	—	20	2	no	—
78	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	5	1	no	—
82	C₂	—	5	1	no	—
83	C₂	—	10	1	no	—
84	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×AGL(1,𝔽₅))⋊C₄
commutator	C₂²×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×M₁₆	C₄
socle	C₂×C₅	#95
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#343
LCS	G ▷ C₂²×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
6. #704	C₄	semidirect
11. C₅×M₁₆	C₄	semidirect
12. #681	C₄	semidirect
47. M₁₆	AGL(1,𝔽₅)	semidirect
69. C₅	#343	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
#343	5