Subgroups — #65721

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4130	C₂	1	2	yes	maximal
2	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#3994	C₂	1	2	yes	maximal, complemented
4	#3993	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
7	C₄⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
8	C₄×AGL(1,𝔽₅)	—	2	2	no	—
9	C₂×C₅×C₈	C₄	1	1	yes	characteristic, complemented
10	C₂×#168	C₄	1	1	yes	characteristic, complemented
11	C₈×D₅	C₄	1	1	yes	characteristic, complemented
12	C₈×D₅	C₄	1	1	yes	characteristic, complemented
13	C₈×D₅	C₂²	1	1	yes	characteristic
14	C₈×D₅	C₂²	1	1	yes	characteristic
15	#427	—	5	1	no	maximal, Sylow
16	C₂²×D₅	D₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	2	4	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	Q₈	1	1	yes	characteristic
20	C₄×D₅	C₂×C₄	1	1	yes	characteristic
21	C₄×D₅	D₄	1	1	yes	characteristic
22	C₄×D₅	Q₈	1	1	yes	characteristic
23	C₄×D₅	C₂³	1	1	yes	characteristic
24	C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	#168	C₂×C₄	1	1	yes	characteristic
27	#168	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	—	5	1	no	—
29	#117	—	5	2	no	—
30	#107	—	5	2	no	—
31	#106	—	5	2	no	—
32	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₅	C₂×D₄	1	1	yes	characteristic
36	AGL(1,𝔽₅)	—	4	4	no	—
37	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	Dic₅	C₄⋊C₄	1	1	yes	characteristic
40	Dic₅	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄	—	5	1	no	—
42	C₂×C₈	—	5	1	no	—
43	C₂×C₈	—	5	1	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	5	1	no	—
46	C₂×C₈	—	5	1	no	—
47	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
48	C₄⋊C₄	—	5	4	no	—
49	C₂²⋊C₄	—	10	2	no	—
50	C₄²	—	10	2	no	—
51	C₂×C₅	—	2	1	no	—
52	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
53	D₅	—	2	1	no	—
54	D₅	#135	1	2	yes	—
55	C₂³	—	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	—	5	1	no	—
61	C₂×C₄	—	10	4	no	—
62	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₈	—	5	1	no	—
64	C₈	—	5	1	no	—
65	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₅	#427	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	5	1	no	—
69	C₂²	—	5	1	no	—
70	C₂²	—	10	2	no	—
71	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	—	5	1	no	—
73	C₄	—	5	1	no	—
74	C₄	—	20	4	no	—
75	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	5	2	no	—
79	C₂	—	10	1	no	—
80	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#427
LCS	G ▷ C₄×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
3. #3994	C₂	semidirect
4. #3993	C₂	semidirect
9. C₂×C₅×C₈	C₄	semidirect
10. C₂×#168	C₄	semidirect
11. C₈×D₅	C₄	semidirect
12. C₈×D₅	C₄	semidirect
47. C₂×C₈	AGL(1,𝔽₅)	semidirect
67. C₅	#427	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×C₈×D₅	2
#3994	2
#3993	2
#427	5