Subgroups — AGL(1,𝔽₅)×M₆₄ #13027521

Order 1280.

250 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊M₆₄	C₂	1	1	yes	maximal, characteristic
3	D₅×M₆₄	C₂	1	1	yes	maximal, characteristic
4	#10598486	C₂	1	2	yes	maximal, complemented
5	AGL(1,𝔽₅)×C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
7	C₂×D₅⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
9	#64877	C₄	1	2	yes	complemented
10	C₁₆×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
11	C₁₆×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
12	C₁₆×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
13	C₁₆×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
14	#64842	C₄	1	2	yes	complemented
15	D₅⋊C₃₂	C₂²	1	2	yes	—
16	C₅×M₆₄	C₄	1	1	yes	characteristic, complemented
17	#64719	C₄	1	1	yes	characteristic, complemented
18	#64667	C₄	1	2	yes	complemented
19	D₅×C₃₂	C₂²	1	2	yes	—
20	C₄×M₆₄	—	5	1	no	maximal, Sylow
21	C₂×C₄×AGL(1,𝔽₅)	C₈	1	1	yes	characteristic
22	C₂×D₅⋊C₈	C₈	1	1	yes	characteristic
23	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×#674	C₈	1	2	yes	—
25	C₈×AGL(1,𝔽₅)	C₈	1	1	yes	characteristic
26	C₈×AGL(1,𝔽₅)	C₈	1	1	yes	characteristic
27	C₈×AGL(1,𝔽₅)	C₂×C₄	1	1	yes	characteristic
28	C₈×AGL(1,𝔽₅)	C₂×C₄	1	1	yes	characteristic
29	D₅⋊C₁₆	C₈	1	2	yes	—
30	D₅⋊C₁₆	C₂×C₄	1	1	yes	characteristic
31	D₅⋊C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×#672	C₂×C₄	1	1	yes	characteristic
34	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
36	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
37	D₅×C₁₆	C₂³	1	1	yes	characteristic
38	#3928	C₂×C₄	1	4	yes	complemented
39	C₅×C₃₂	C₂×C₄	1	2	yes	complemented
40	#3926	C₂×C₄	1	2	yes	complemented
41	C₂×M₆₄	—	5	1	no	—
42	C₂×M₆₄	—	5	1	no	—
43	C₂×C₄×C₁₆	—	5	1	no	—
44	#1397	—	5	2	no	—
45	C₄×C₃₂	—	5	2	no	—
46	C₂²×AGL(1,𝔽₅)	C₁₆	1	2	yes	—
47	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
48	C₂×#170	C₁₆	1	2	yes	—
49	C₄×AGL(1,𝔽₅)	C₁₆	1	2	yes	—
50	C₄×AGL(1,𝔽₅)	C₂×C₈	1	1	yes	characteristic
51	C₄×AGL(1,𝔽₅)	C₂×C₈	1	1	yes	characteristic
52	D₅⋊C₈	C₁₆	1	2	yes	—
53	D₅⋊C₈	C₂×C₈	1	1	yes	characteristic
54	D₅⋊C₈	C₂×C₈	1	1	yes	characteristic
55	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
56	C₂×#168	C₂×C₈	1	1	yes	characteristic
57	C₈×D₅	C₄²	1	1	yes	characteristic
58	C₈×D₅	C₂×C₈	1	1	yes	characteristic
59	C₈×D₅	C₂×C₈	1	1	yes	characteristic
60	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
61	#674	C₂×C₈	1	2	yes	—
62	#674	C₂×C₈	1	2	yes	—
63	C₅×C₁₆	C₄²	1	1	yes	characteristic
64	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	#672	C₄²	1	1	yes	characteristic
66	#672	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₁₆	—	5	1	no	—
68	C₂²×C₁₆	—	5	1	no	—
69	C₂×C₄×C₈	—	5	1	no	—
70	M₆₄	—	5	2	no	—
71	M₆₄	—	5	2	no	—
72	M₆₄	—	5	1	no	—
73	M₆₄	—	5	2	no	—
74	M₆₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
75	C₂×C₃₂	—	5	2	no	—
76	C₂×C₃₂	—	5	2	no	—
77	C₄×C₁₆	—	5	1	no	—
78	C₄×C₁₆	—	5	1	no	—
79	C₄×C₁₆	—	5	1	no	—
80	C₄×C₁₆	—	5	1	no	—
81	C₂²×D₅	C₂×C₁₆	1	1	yes	characteristic
82	C₂×AGL(1,𝔽₅)	—	2	4	no	—
83	C₂×AGL(1,𝔽₅)	C₂×C₁₆	1	2	yes	—
84	C₂×AGL(1,𝔽₅)	C₂×C₁₆	1	2	yes	—
85	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
86	C₂×Dic₅	C₂×C₁₆	1	1	yes	characteristic
87	C₄×D₅	C₄×C₈	1	1	yes	characteristic
88	C₄×D₅	C₂×C₁₆	1	1	yes	characteristic
89	C₄×D₅	C₂×C₁₆	1	1	yes	characteristic
90	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
91	#170	C₂×C₁₆	1	2	yes	—
92	#170	C₂×C₁₆	1	2	yes	—
93	C₅×C₈	C₄×C₈	1	1	yes	characteristic
94	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
95	#168	C₄×C₈	1	1	yes	characteristic
96	#168	C₂²×C₈	1	1	yes	characteristic
97	C₂²×C₈	—	5	1	no	—
98	C₂²×C₈	—	5	1	no	—
99	C₂×C₄²	—	5	1	no	—
100	C₂×C₁₆	—	5	1	no	—
101	C₂×C₁₆	—	5	1	no	—
102	C₂×C₁₆	—	5	2	no	—
103	C₂×C₁₆	—	5	2	no	—
104	C₂×C₁₆	—	5	1	no	—
105	C₂×C₁₆	—	5	1	no	—
106	C₂×C₁₆	—	5	1	no	—
107	C₂×C₁₆	—	5	1	no	—
108	C₂×C₁₆	—	5	1	no	—
109	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₄×C₈	—	5	1	no	—
111	C₄×C₈	—	5	1	no	—
112	C₄×C₈	—	5	1	no	—
113	C₄×C₈	—	5	1	no	—
114	C₃₂	—	5	4	no	—
115	C₃₂	—	5	2	no	—
116	C₃₂	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
117	C₂²×C₅	C₄×C₁₆	1	1	yes	characteristic
118	C₂×D₅	—	2	2	no	—
119	C₂×D₅	C₄×C₁₆	1	1	yes	characteristic
120	C₂×D₅	C₂²×C₁₆	1	1	yes	characteristic
121	AGL(1,𝔽₅)	—	2	2	no	—
122	AGL(1,𝔽₅)	M₆₄	1	4	yes	direct product factor
123	C₄×C₅	C₄×C₁₆	1	1	yes	characteristic
124	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
125	Dic₅	C₄×C₁₆	1	1	yes	characteristic
126	Dic₅	C₂²×C₁₆	1	1	yes	characteristic
127	C₂²×C₄	—	5	1	no	—
128	C₂²×C₄	—	5	2	no	—
129	C₂×C₈	—	5	1	no	—
130	C₂×C₈	—	5	1	no	—
131	C₂×C₈	—	5	2	no	—
132	C₂×C₈	—	5	2	no	—
133	C₂×C₈	—	5	1	no	—
134	C₂×C₈	—	5	1	no	—
135	C₂×C₈	—	5	1	no	—
136	C₂×C₈	—	5	1	no	—
137	C₂×C₈	—	5	1	no	—
138	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₄²	—	5	1	no	—
140	C₄²	—	5	1	no	—
141	C₄²	—	5	2	no	—
142	C₁₆	—	5	2	no	—
143	C₁₆	—	5	2	no	—
144	C₁₆	—	5	1	no	—
145	C₁₆	—	5	1	no	—
146	C₁₆	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₁₆	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₅	—	2	1	no	—
149	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
150	D₅	—	2	1	no	—
151	D₅	C₂×M₆₄	1	2	yes	—
152	C₂³	—	5	1	no	—
153	C₂×C₄	—	5	2	no	—
154	C₂×C₄	—	5	1	no	—
155	C₂×C₄	—	5	1	no	—
156	C₂×C₄	—	5	1	no	—
157	C₂×C₄	—	5	1	no	—
158	C₂×C₄	—	5	1	no	—
159	C₂×C₄	—	5	2	no	—
160	C₂×C₄	—	10	4	no	—
161	C₂×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₈	—	5	2	no	—
163	C₈	—	5	2	no	—
164	C₈	—	5	1	no	—
165	C₈	—	5	1	no	—
166	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₅	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	5	1	no	—
170	C₂²	—	5	1	no	—
171	C₂²	—	10	2	no	—
172	C₂²	C₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₄	—	5	4	no	—
174	C₄	—	5	1	no	—
175	C₄	—	5	1	no	—
176	C₄	—	10	2	no	—
177	C₄	C₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	5	2	no	—
181	C₂	—	10	1	no	—
182	C₂	C₂×C₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂×C₄×C₁₆
Fitting	C₅×M₆₄	C₄
prime radical	C₂²×D₅	C₂×C₁₆
cosocle	D₅×C₁₆	C₂³
hypercenter	M₆₄	AGL(1,𝔽₅)
hypocenter	C₅	C₄×M₆₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
4. #10598486	C₂	semidirect
5. AGL(1,𝔽₅)×C₃₂	C₂	semidirect
9. #64877	C₄	semidirect
14. #64842	C₄	semidirect
16. C₅×M₆₄	C₄	semidirect
17. #64719	C₄	semidirect
18. #64667	C₄	semidirect
38. #3928	C₂×C₄	semidirect
39. C₅×C₃₂	C₂×C₄	semidirect
40. #3926	C₂×C₄	semidirect
74. M₆₄	AGL(1,𝔽₅)	direct (N × H)
116. C₃₂	C₂×AGL(1,𝔽₅)	semidirect
122. AGL(1,𝔽₅)	M₆₄	direct (N × H)
168. C₅	C₄×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×AGL(1,𝔽₅)	2
D₅⋊M₆₄	2
D₅×M₆₄	2
#10598486	2
AGL(1,𝔽₅)×C₃₂	2
C₄×M₆₄	5

Subgroups — AGL(1,𝔽5)×M64 #13027521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₅)×M₆₄ #13027521