Subgroups — #41×AGL(1,𝔽₅) #66265

Order 320.

310 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	Q₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
3	D₄×AGL(1,𝔽₅)	C₂	1	6	yes	maximal, complemented
4	#4130	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×AGL(1,𝔽₅)	C₂	1	3	yes	maximal, complemented
6	C₂²×AGL(1,𝔽₅)	C₂²	1	6	yes	—
7	C₅×#41	C₄	1	1	yes	characteristic, complemented
8	#713	C₄	1	1	yes	characteristic, complemented
9	Q₈×D₅	C₂²	1	1	yes	characteristic
10	#711	C₄	1	3	yes	complemented
11	D₄×D₅	C₂²	1	3	yes	—
12	#709	C₄	1	3	yes	complemented
13	C₂×C₄×D₅	C₂²	1	3	yes	—
14	C₂²⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
15	C₄⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
16	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
17	C₄×AGL(1,𝔽₅)	C₂²	1	3	yes	—
18	C₄×AGL(1,𝔽₅)	C₂²	1	6	yes	—
19	C₄×#41	—	5	1	no	maximal, Sylow
20	C₂²×D₅	C₂³	1	3	yes	—
21	C₂×AGL(1,𝔽₅)	—	2	12	no	—
22	C₂×AGL(1,𝔽₅)	C₂³	1	6	yes	—
23	C₂×AGL(1,𝔽₅)	C₂³	1	2	yes	—
24	C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
25	C₅×D₄	C₂×C₄	1	3	yes	complemented
26	C₂×C₄×C₅	C₂×C₄	1	3	yes	complemented
27	C₂²⋊D₅	C₂×C₄	1	6	yes	complemented
28	C₂×Dic₅	C₂×C₄	1	3	yes	complemented
29	D₂₀	C₂×C₄	1	3	yes	complemented
30	C₄×D₅	C₂×C₄	1	3	yes	complemented
31	C₄×D₅	C₂×C₄	1	3	yes	complemented
32	C₄×D₅	C₂³	1	1	yes	characteristic
33	C₄×D₅	C₂³	1	3	yes	—
34	Dic₁₀	C₂×C₄	1	3	yes	complemented
35	C₂×#41	—	5	1	no	—
36	C₄×Q₈	—	5	2	no	—
37	C₄×D₄	—	5	6	no	—
38	#117	—	5	3	no	—
39	C₂×C₄²	—	5	3	no	—
40	C₂²×C₅	C₂²×C₄	1	3	yes	—
41	C₂×D₅	—	2	6	no	—
42	C₂×D₅	C₂²×C₄	1	3	yes	—
43	C₂×D₅	C₂⁴	1	1	yes	characteristic
44	AGL(1,𝔽₅)	—	2	6	no	—
45	AGL(1,𝔽₅)	#41	1	4	yes	direct product factor
46	C₄×C₅	C₂²×C₄	1	3	yes	—
47	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	Dic₅	C₂²×C₄	1	3	yes	—
49	Dic₅	C₂²×C₄	1	1	yes	characteristic
50	#41	—	5	1	no	—
51	#41	—	5	3	no	—
52	#41	—	5	3	no	—
53	#41	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
54	C₂×Q₈	—	5	1	no	—
55	C₂×D₄	—	5	3	no	—
56	C₂²×C₄	—	5	3	no	—
57	C₂²×C₄	—	5	6	no	—
58	C₄⋊C₄	—	5	6	no	—
59	C₂²⋊C₄	—	5	6	no	—
60	C₄²	—	5	1	no	—
61	C₄²	—	5	3	no	—
62	C₄²	—	5	6	no	—
63	C₂×C₅	—	2	3	no	—
64	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
65	D₅	—	2	3	no	—
66	D₅	C₂×#41	1	2	yes	—
67	C₂³	—	5	3	no	—
68	Q₈	—	5	3	no	—
69	Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
70	D₄	—	5	6	no	—
71	D₄	—	5	3	no	—
72	D₄	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
73	C₂×C₄	—	5	2	no	—
74	C₂×C₄	—	5	3	no	—
75	C₂×C₄	—	5	3	no	—
76	C₂×C₄	—	5	3	no	—
77	C₂×C₄	—	5	1	no	—
78	C₂×C₄	—	5	3	no	—
79	C₂×C₄	—	5	6	no	—
80	C₂×C₄	—	10	12	no	—
81	C₂×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
82	C₅	C₄×#41	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	5	1	no	—
84	C₂²	—	5	3	no	—
85	C₂²	—	10	6	no	—
86	C₂²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
87	C₄	—	5	4	no	—
88	C₄	—	5	1	no	—
89	C₄	—	5	3	no	—
90	C₄	—	10	6	no	—
91	C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
92	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	—	5	2	no	—
95	C₂	—	10	3	no	—
96	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#41	C₄
prime radical	D₅×#41	C₂
cosocle	C₂×D₅	C₂⁴
hypercenter	#41	AGL(1,𝔽₅)
hypocenter	C₅	C₄×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. Q₈×AGL(1,𝔽₅)	C₂	semidirect
3. D₄×AGL(1,𝔽₅)	C₂	semidirect
4. #4130	C₂	semidirect
5. C₂×C₄×AGL(1,𝔽₅)	C₂	semidirect
7. C₅×#41	C₄	semidirect
8. #713	C₄	semidirect
10. #711	C₄	semidirect
12. #709	C₄	semidirect
24. C₅×Q₈	C₂×C₄	semidirect
25. C₅×D₄	C₂×C₄	semidirect
26. C₂×C₄×C₅	C₂×C₄	semidirect
27. C₂²⋊D₅	C₂×C₄	semidirect
28. C₂×Dic₅	C₂×C₄	semidirect
29. D₂₀	C₂×C₄	semidirect
30. C₄×D₅	C₂×C₄	semidirect
31. C₄×D₅	C₂×C₄	semidirect
34. Dic₁₀	C₂×C₄	semidirect
45. AGL(1,𝔽₅)	#41	direct (N × H)
53. #41	AGL(1,𝔽₅)	direct (N × H)
69. Q₈	C₂×AGL(1,𝔽₅)	semidirect
72. D₄	C₂×AGL(1,𝔽₅)	semidirect
81. C₂×C₄	C₂×AGL(1,𝔽₅)	semidirect
82. C₅	C₄×#41	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
Q₈×AGL(1,𝔽₅)	2
D₄×AGL(1,𝔽₅)	2
#4130	2
C₂×C₄×AGL(1,𝔽₅)	2
C₄×#41	5

Subgroups — #41×AGL(1,𝔽5) #66265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×AGL(1,𝔽₅) #66265