Subgroups — #66264

Order 320.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#41	C₂	1	1	yes	maximal, characteristic
2	#4133	C₂	1	2	yes	maximal, complemented
3	#4131	C₂	1	6	yes	maximal, complemented
4	D₅⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	C₂×#700	C₂	1	3	yes	maximal, complemented
6	C₅×#41	C₄	1	1	yes	characteristic
7	#713	C₂²	1	1	yes	characteristic
8	Q₈×D₅	C₄	1	1	yes	characteristic
9	#711	C₂²	1	3	yes	—
10	D₄×D₅	C₄	1	3	yes	—
11	#709	C₄	1	3	yes	—
12	C₂×C₄×D₅	C₂²	1	3	yes	—
13	#704	C₂²	1	6	yes	complemented
14	C₂×#170	C₂²	1	6	yes	complemented
15	#700	C₂²	1	3	yes	—
16	#700	C₂²	1	1	yes	characteristic
17	#700	C₂²	1	6	yes	complemented
18	D₅⋊C₈	C₂²	1	6	yes	complemented
19	#568	—	5	1	no	maximal, Sylow
20	C₂²×D₅	C₂×C₄	1	3	yes	—
21	C₅×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₅×D₄	C₂×C₄	1	3	yes	—
23	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
24	C₂²⋊D₅	C₂×C₄	1	6	yes	—
25	C₂×Dic₅	C₂³	1	3	yes	—
26	D₂₀	C₂×C₄	1	3	yes	—
27	C₄×D₅	C₂×C₄	1	3	yes	—
28	C₄×D₅	C₂×C₄	1	3	yes	—
29	C₄×D₅	C₂³	1	3	yes	—
30	C₄×D₅	C₂³	1	1	yes	characteristic
31	Dic₁₀	C₂×C₄	1	3	yes	—
32	#170	C₂³	1	2	yes	—
33	#170	C₂³	1	6	yes	—
34	C₂×#41	—	5	1	no	—
35	#131	—	5	6	no	—
36	#131	—	5	2	no	—
37	C₂×M₁₆	—	5	3	no	—
38	C₂×M₁₆	—	5	3	no	—
39	C₂²×C₅	C₂²×C₄	1	3	yes	—
40	C₂×D₅	—	4	3	no	—
41	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×D₅	C₂²×C₄	1	3	yes	—
43	C₄×C₅	C₂²×C₄	1	3	yes	—
44	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	Dic₅	C₂²×C₄	1	3	yes	—
46	Dic₅	C₂⁴	1	1	yes	characteristic
47	#41	—	5	1	no	—
48	#41	—	5	3	no	—
49	#41	—	5	3	no	—
50	#41	AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×Q₈	—	5	1	no	—
52	C₂×D₄	—	5	3	no	—
53	C₂²×C₄	—	5	3	no	—
54	M₁₆	—	5	3	no	—
55	M₁₆	—	5	1	no	—
56	M₁₆	—	5	6	no	—
57	M₁₆	—	5	6	no	—
58	C₂×C₈	—	5	6	no	—
59	C₂×C₈	—	5	6	no	—
60	C₂×C₅	—	2	3	no	—
61	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
62	D₅	—	2	1	no	—
63	D₅	—	2	3	no	—
64	C₂³	—	5	3	no	—
65	Q₈	—	5	3	no	—
66	Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
67	D₄	—	5	6	no	—
68	D₄	—	5	3	no	—
69	D₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
70	C₂×C₄	—	5	3	no	—
71	C₂×C₄	—	5	3	no	—
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	5	3	no	—
74	C₂×C₄	—	5	3	no	—
75	C₂×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
76	C₈	—	5	2	no	—
77	C₈	—	5	6	no	—
78	C₅	#568	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	5	1	no	—
80	C₂²	—	5	3	no	—
81	C₂²	—	20	3	no	—
82	C₂²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
83	C₄	—	5	1	no	—
84	C₄	—	5	3	no	—
85	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	10	1	no	—
89	C₂	—	10	3	no	—
90	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Fitting	C₅×#41	C₄
prime radical	D₅×#41	C₂
cosocle	Dic₅	C₂⁴
hypercenter	#41	AGL(1,𝔽₅)
hypocenter	C₅	#568
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #4133	C₂	semidirect
3. #4131	C₂	semidirect
4. D₅⋊M₁₆	C₂	semidirect
5. C₂×#700	C₂	semidirect
13. #704	C₂²	semidirect
14. C₂×#170	C₂²	semidirect
17. #700	C₂²	semidirect
18. D₅⋊C₈	C₂²	semidirect
78. C₅	#568	semidirect

Maximal subgroups

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

subgroup	index
D₅×#41	2
#4133	2
#4131	2
D₅⋊M₁₆	2
C₂×#700	2
#568	5