Subgroups — #64894

Order 320.

90 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×#676	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
5	C₂²⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
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	C₂×C₅×C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×#168	C₄	1	1	yes	characteristic, complemented
13	#676	—	2	1	no	—
14	#676	—	2	1	no	—
15	#344	—	5	1	no	maximal, Sylow
16	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₄×D₅	D₄	1	1	yes	characteristic
21	C₄×D₅	D₄	1	1	yes	characteristic
22	C₄×D₅	D₄	1	1	yes	characteristic
23	C₄×D₅	Q₈	1	1	yes	characteristic
24	#170	—	2	2	no	—
25	C₅×C₈	—	2	1	no	—
26	#168	—	2	1	no	—
27	C₂×M₁₆	—	5	1	no	—
28	C₂×M₁₆	—	5	1	no	—
29	#117	—	5	1	no	—
30	C₂²×C₅	C₄²	1	1	yes	characteristic
31	C₂×D₅	—	4	1	no	—
32	C₂×D₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
34	AGL(1,𝔽₅)	—	4	2	no	—
35	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
37	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
38	Dic₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄	—	5	1	no	—
40	M₁₆	—	5	2	no	—
41	M₁₆	—	10	1	no	—
42	M₁₆	—	10	1	no	—
43	M₁₆	—	10	1	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	10	1	no	—
46	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
47	C₄⋊C₄	—	10	1	no	—
48	C₂²⋊C₄	—	5	2	no	—
49	C₄²	—	10	1	no	—
50	C₂×C₅	—	2	1	no	—
51	C₂×C₅	#95	1	1	yes	characteristic
52	D₅	—	2	1	no	—
53	D₅	—	2	1	no	—
54	C₂³	—	5	1	no	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	10	2	no	—
64	C₈	—	10	1	no	—
65	C₅	#344	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	5	1	no	—
67	C₂²	—	5	1	no	—
68	C₂²	—	20	1	no	—
69	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	5	1	no	—
71	C₄	—	5	1	no	—
72	C₄	—	20	2	no	—
73	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	10	1	no	—
77	C₂	—	10	1	no	—
78	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂×C₅	#95
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#344
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
6. #704	C₄	semidirect
11. C₂×C₅×C₈	C₄	semidirect
12. C₂×#168	C₄	semidirect
46. C₂×C₈	AGL(1,𝔽₅)	semidirect
65. C₅	#344	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×#676	2
#344	5