Subgroups — #64915

Order 320.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#685	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#170	C₂	1	2	yes	maximal, complemented
3	C₂³×D₅	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	C₂²	1	1	yes	characteristic
6	C₂×#170	—	2	2	no	—
7	#685	—	2	2	no	—
8	#323	—	5	1	no	maximal, Sylow
9	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
10	C₂²×D₅	—	2	1	no	—
11	C₂²×D₅	—	2	1	no	—
12	C₂²×D₅	—	4	1	no	—
13	C₂²×D₅	—	4	1	no	—
14	C₂²×D₅	C₈	1	2	yes	complemented
15	C₂×C₄×C₅	—	2	1	no	—
16	C₂×C₄×C₅	—	4	1	no	—
17	C₂×Dic₅	—	4	1	no	—
18	C₂×Dic₅	D₄	1	2	yes	—
19	#170	—	4	2	no	—
20	C₂×C₂²⋊C₄	—	5	1	no	—
21	C₂²⋊C₈	—	5	2	no	—
22	C₂²×C₅	—	2	1	no	—
23	C₂²×C₅	—	2	1	no	—
24	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₅	M₁₆	1	1	yes	characteristic
27	C₂×D₅	—	2	2	no	—
28	C₂×D₅	—	4	2	no	—
29	C₂×D₅	—	4	1	no	—
30	C₂×D₅	—	4	1	no	—
31	C₂×D₅	—	4	2	no	—
32	C₄×C₅	—	4	1	no	—
33	Dic₅	—	2	2	no	—
34	C₂⁴	—	5	1	no	—
35	C₂²×C₄	—	5	1	no	—
36	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×C₈	—	10	2	no	—
38	C₂²⋊C₄	—	10	2	no	—
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	—	2	1	no	—
41	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₅	#99	1	1	yes	characteristic
43	C₂×C₅	#100	1	1	yes	characteristic
44	D₅	—	4	2	no	—
45	C₂³	—	5	2	no	—
46	C₂³	—	10	1	no	—
47	C₂³	—	10	1	no	—
48	C₂³	—	20	1	no	—
49	C₂³	—	20	1	no	—
50	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	5	2	no	—
54	C₂×C₄	—	20	1	no	—
55	C₈	—	20	2	no	—
56	C₅	#323	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	10	2	no	—
60	C₂²	—	20	2	no	—
61	C₂²	—	20	2	no	—
62	C₂²	—	20	1	no	—
63	C₂²	—	20	1	no	—
64	C₂²	D₅⋊C₈	1	1	yes	characteristic
65	C₂²	#700	1	1	yes	characteristic
66	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₄	—	4	1	no	—
68	C₄	—	10	2	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	20	2	no	—
72	C₂	#3999	1	1	yes	characteristic
73	C₂	#4003	1	1	yes	characteristic
74	C₂	#4013	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₈
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂³×D₅	C₄
cosocle	C₂²×Dic₅	C₂²
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#323
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²⋊#170	C₂	semidirect
14. C₂²×D₅	C₈	semidirect
56. C₅	#323	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#685	2
C₂²⋊#170	2
#323	5