Subgroups — #64919

Order 320.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#704	C₂	1	2	yes	maximal
2	C₂×#683	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
4	C₂²×Dic₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	C₂²	1	1	yes	characteristic
6	#704	—	2	2	no	—
7	#704	C₄	1	4	yes	complemented
8	C₂×#170	—	2	2	no	—
9	#683	—	2	2	no	—
10	#343	—	5	1	no	maximal, Sylow
11	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅	—	2	1	no	—
13	C₂×C₄×C₅	—	4	1	no	—
14	C₂×Dic₅	—	2	1	no	—
15	C₂×Dic₅	—	4	1	no	—
16	C₂×Dic₅	C₂×C₄	1	2	yes	—
17	C₂×Dic₅	D₄	1	1	yes	characteristic
18	C₂×Dic₅	D₄	1	2	yes	—
19	C₂×Dic₅	Q₈	1	1	yes	characteristic
20	#170	—	2	4	no	—
21	C₂×M₁₆	—	5	2	no	—
22	C₂×C₄⋊C₄	—	5	1	no	—
23	C₂²×C₅	—	2	1	no	—
24	C₂²×C₅	—	2	1	no	—
25	C₂²×C₅	C₄²	1	1	yes	characteristic
26	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₅	—	4	1	no	—
29	Dic₅	—	4	1	no	—
30	Dic₅	C₂²⋊C₄	1	2	yes	—
31	Dic₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	1	no	—
34	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
35	M₁₆	—	5	4	no	—
36	M₁₆	—	10	2	no	—
37	C₂×C₈	—	10	2	no	—
38	C₄⋊C₄	—	10	2	no	—
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	—	2	1	no	—
41	C₂×C₅	#95	1	1	yes	characteristic
42	C₂×C₅	#100	1	1	yes	characteristic
43	C₂×C₅	#101	1	1	yes	characteristic
44	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	5	2	no	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	5	2	no	—
51	C₂×C₄	—	10	1	no	—
52	C₂×C₄	—	20	1	no	—
53	C₈	—	10	4	no	—
54	C₅	#343	1	1	yes	characteristic, complemented, Sylow
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄	—	4	1	no	—
61	C₄	—	5	2	no	—
62	C₄	—	5	2	no	—
63	C₄	—	20	1	no	—
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	#4005	1	1	yes	characteristic
67	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
68	C₂	#4013	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₄²
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂²
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#343
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
7. #704	C₄	semidirect
54. 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×#704	2
C₂×#683	2
#343	5