Subgroups — #64901

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	D₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
5	#704	—	2	1	no	—
6	C₂×#170	—	2	1	no	—
7	#700	C₄	1	2	yes	—
8	D₅⋊C₈	—	2	1	no	—
9	D₅⋊C₈	—	2	1	no	—
10	D₅⋊C₈	C₄	1	2	yes	—
11	C₅×M₁₆	C₄	1	1	yes	characteristic
12	#681	C₄	1	1	yes	characteristic
13	#676	—	2	1	no	—
14	C₈×D₅	—	2	1	no	—
15	#341	—	5	1	no	maximal, Sylow
16	C₂²×D₅	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅	D₄	1	1	yes	characteristic
19	C₄×D₅	C₂×C₄	1	1	yes	characteristic
20	C₄×D₅	C₂×C₄	1	1	yes	characteristic
21	C₄×D₅	D₄	1	1	yes	characteristic
22	C₄×D₅	D₄	1	1	yes	characteristic
23	#170	—	2	2	no	—
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	C₂²×C₈	—	5	1	no	—
30	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₅	—	2	2	no	—
32	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₅	C₄²	1	1	yes	characteristic
35	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄	—	5	1	no	—
39	M₁₆	—	5	2	no	—
40	M₁₆	—	5	1	no	—
41	M₁₆	—	10	1	no	—
42	M₁₆	—	10	1	no	—
43	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₈	—	5	2	no	—
45	C₂×C₈	—	10	1	no	—
46	C₂×C₈	—	10	1	no	—
47	C₂×C₈	—	10	1	no	—
48	C₂×C₈	—	10	1	no	—
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	#95	1	1	yes	characteristic
51	D₅	—	2	1	no	—
52	D₅	#108	1	2	yes	—
53	C₂³	—	5	1	no	—
54	C₂×C₄	—	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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	10	2	no	—
62	C₈	—	10	2	no	—
63	C₈	—	10	1	no	—
64	C₅	#341	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	5	1	no	—
66	C₂²	—	5	1	no	—
67	C₂²	—	10	2	no	—
68	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₄	—	5	1	no	—
70	C₄	—	5	1	no	—
71	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	5	2	no	—
75	C₂	—	10	1	no	—
76	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×M₁₆	C₄
socle	C₂×C₅	#95
prime radical	C₂²×D₅	Q₈
cosocle	C₂×C₄×D₅	C₂²
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#341
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
64. C₅	#341	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊M₁₆	2
C₂×D₅⋊C₈	2
D₅×M₁₆	2
#341	5