Subgroups — C₁₆×C₂²⋊AGL(1,𝔽₅) #13025727

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂²⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅⋊(C₂²⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
5	(C₂×AGL(1,𝔽₅))⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₄×C₂²⋊AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
8	D₅⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
11	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂²×#672	C₄	1	1	yes	characteristic, complemented
13	C₂×D₅×C₁₆	—	2	4	no	—
14	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²⋊#674	C₄	1	2	yes	complemented
19	(C₂×AGL(1,𝔽₅))⋊C₈	C₄	1	2	yes	—
20	#64887	C₄	1	2	yes	complemented
21	C₁₆×AGL(1,𝔽₅)	—	2	4	no	—
22	C₁₆×C₂²⋊C₄	—	5	1	no	maximal, Sylow
23	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₂²⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
25	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
26	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅	—	2	2	no	—
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
35	C₂²⋊#170	C₈	1	2	yes	—
36	(C₂×AGL(1,𝔽₅))⋊C₄	C₈	1	2	yes	—
37	#4003	C₈	1	2	yes	—
38	C₂×#674	C₂×C₄	1	4	yes	—
39	C₈×AGL(1,𝔽₅)	—	2	2	no	—
40	C₈×AGL(1,𝔽₅)	—	2	2	no	—
41	D₅⋊C₁₆	—	2	2	no	—
42	D₅⋊C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#672	—	2	2	no	—
47	C₂×#672	C₂×C₄	1	1	yes	characteristic
48	C₂×#672	C₂×C₄	1	1	yes	characteristic
49	D₅×C₁₆	—	2	2	no	—
50	D₅×C₁₆	—	2	2	no	—
51	D₅×C₁₆	—	2	2	no	—
52	D₅×C₁₆	D₄	1	4	yes	—
53	C₂³×C₁₆	—	5	1	no	—
54	C₂×C₂²⋊C₁₆	—	5	1	no	—
55	C₂×C₄×C₁₆	—	5	2	no	—
56	C₈×C₂²⋊C₄	—	5	1	no	—
57	#1374	—	5	2	no	—
58	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
61	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₅	—	2	2	no	—
63	C₂×C₄×D₅	—	2	2	no	—
64	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
65	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²⋊AGL(1,𝔽₅)	C₁₆	1	8	yes	direct product factor
69	C₂×#170	C₂×C₈	1	4	yes	—
70	C₄×AGL(1,𝔽₅)	—	2	2	no	—
71	C₄×AGL(1,𝔽₅)	—	2	2	no	—
72	D₅⋊C₈	—	2	2	no	—
73	D₅⋊C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×#168	—	2	2	no	—
79	C₂×#168	C₄²	1	1	yes	characteristic
80	C₂×#168	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₅	—	2	2	no	—
82	C₈×D₅	—	2	1	no	—
83	C₈×D₅	—	2	1	no	—
84	C₈×D₅	—	2	2	no	—
85	C₈×D₅	C₂×D₄	1	2	yes	—
86	C₈×D₅	#41	1	2	yes	—
87	#674	—	2	4	no	—
88	C₅×C₁₆	—	2	1	no	—
89	C₅×C₁₆	C₂²⋊C₄	1	2	yes	complemented
90	#672	—	2	1	no	—
91	#672	C₂²⋊C₄	1	2	yes	complemented
92	C₂³×C₈	—	5	1	no	—
93	C₂²×C₁₆	—	5	1	no	—
94	C₂²×C₁₆	—	5	2	no	—
95	C₂²×C₁₆	—	5	1	no	—
96	C₂²×C₁₆	—	5	1	no	—
97	C₂²×C₁₆	—	5	1	no	—
98	C₂²×C₁₆	—	5	1	no	—
99	C₂²×C₁₆	—	10	4	no	—
100	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
101	C₂×C₂²⋊C₈	—	5	1	no	—
102	C₂×C₄×C₈	—	5	2	no	—
103	C₄×C₂²⋊C₄	—	5	1	no	—
104	C₂²⋊C₁₆	—	5	2	no	—
105	C₂²⋊C₁₆	—	5	2	no	—
106	C₄×C₁₆	—	10	4	no	—
107	#336	—	5	2	no	—
108	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
109	C₂²×D₅	—	2	2	no	—
110	C₂²×D₅	—	2	2	no	—
111	C₂²×D₅	C₄×C₈	1	1	yes	characteristic
112	C₂²×D₅	C₂×C₁₆	1	4	yes	—
113	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×AGL(1,𝔽₅)	—	2	4	no	—
115	C₂×AGL(1,𝔽₅)	—	2	4	no	—
116	C₂×AGL(1,𝔽₅)	C₂×C₁₆	1	8	yes	complemented
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
121	C₂×Dic₅	—	2	2	no	—
122	C₂×Dic₅	C₄×C₈	1	1	yes	characteristic
123	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
124	C₄×D₅	—	2	2	no	—
125	C₄×D₅	—	2	1	no	—
126	C₄×D₅	—	2	1	no	—
127	C₄×D₅	—	2	2	no	—
128	C₄×D₅	C₄×D₄	1	2	yes	—
129	C₄×D₅	#131	1	2	yes	—
130	#170	—	2	4	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₅×C₈	#117	1	1	yes	characteristic
134	#168	—	2	1	no	—
135	#168	C₄×D₄	1	2	yes	—
136	C₂³×C₄	—	5	1	no	—
137	C₂²×C₈	—	5	1	no	—
138	C₂²×C₈	—	5	2	no	—
139	C₂²×C₈	—	5	1	no	—
140	C₂²×C₈	—	5	1	no	—
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	5	1	no	—
143	C₂²×C₈	—	10	2	no	—
144	C₂²×C₈	—	10	2	no	—
145	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	—	5	2	no	—
147	C₂×C₄²	—	5	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	5	1	no	—
150	C₂×C₁₆	—	5	4	no	—
151	C₂×C₁₆	—	5	4	no	—
152	C₂×C₁₆	—	5	1	no	—
153	C₂×C₁₆	—	10	2	no	—
154	C₂×C₁₆	—	10	2	no	—
155	C₂×C₁₆	—	10	2	no	—
156	C₂×C₁₆	—	10	2	no	—
157	C₂×C₁₆	—	10	2	no	—
158	C₂×C₁₆	—	10	2	no	—
159	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂²⋊C₈	—	5	2	no	—
162	C₂²⋊C₈	—	5	2	no	—
163	C₄×C₈	—	10	2	no	—
164	C₄×C₈	—	10	2	no	—
165	#95	—	5	2	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	C₄×C₁₆	1	2	yes	complemented
169	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
170	C₂×D₅	—	2	1	no	—
171	C₂×D₅	—	2	1	no	—
172	C₂×D₅	—	2	8	no	—
173	C₂×D₅	C₄×C₁₆	1	2	yes	complemented
174	C₂×D₅	C₈×D₄	1	2	yes	—
175	C₂×D₅	C₂²×C₁₆	1	2	yes	—
176	C₂×D₅	#504	1	2	yes	—
177	AGL(1,𝔽₅)	—	2	8	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₅	#405	1	1	yes	characteristic
181	Dic₅	—	2	1	no	—
182	Dic₅	C₈×D₄	1	2	yes	—
183	C₂⁴	—	5	1	no	—
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	—	5	4	no	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	1	no	—
190	C₂²×C₄	—	10	2	no	—
191	C₂²×C₄	—	10	2	no	—
192	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	5	1	no	—
196	C₂×C₈	—	5	2	no	—
197	C₂×C₈	—	5	2	no	—
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	10	2	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	10	2	no	—
203	C₂×C₈	—	10	2	no	—
204	C₂×C₈	—	10	2	no	—
205	C₂×C₈	—	10	1	no	—
206	C₂×C₈	—	10	1	no	—
207	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
208	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₂²⋊C₄	—	5	8	no	—
210	C₄²	—	10	2	no	—
211	C₄²	—	10	2	no	—
212	C₁₆	—	2	1	no	—
213	C₁₆	—	5	2	no	—
214	C₁₆	—	10	4	no	—
215	C₁₆	—	10	1	no	—
216	C₁₆	C₂²⋊AGL(1,𝔽₅)	1	2	yes	direct product factor
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
220	C₂×C₅	#2108	1	1	yes	characteristic
221	D₅	—	2	2	no	—
222	D₅	D₄×C₁₆	1	4	yes	—
223	C₂³	—	5	1	no	—
224	C₂³	—	5	4	no	—
225	C₂³	—	5	1	no	—
226	C₂³	—	10	2	no	—
227	C₂³	—	10	2	no	—
228	C₂³	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	2	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	5	8	no	—
235	C₂×C₄	—	5	1	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	—	10	2	no	—
239	C₂×C₄	—	10	4	no	—
240	C₂×C₄	—	10	4	no	—
241	C₂×C₄	—	10	1	no	—
242	C₂×C₄	—	10	1	no	—
243	C₂×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	5	2	no	—
247	C₈	—	10	4	no	—
248	C₈	—	10	1	no	—
249	C₈	#4130	1	1	yes	characteristic
250	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
251	C₅	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	5	2	no	—
255	C₂²	—	5	2	no	—
256	C₂²	—	5	2	no	—
257	C₂²	—	5	2	no	—
258	C₂²	—	10	8	no	—
259	C₂²	—	10	1	no	—
260	C₂²	—	10	1	no	—
261	C₂²	C₁₆×AGL(1,𝔽₅)	1	2	yes	complemented
262	C₂²	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	5	2	no	—
265	C₄	—	10	8	no	—
266	C₄	—	10	1	no	—
267	C₄	#65718	1	1	yes	characteristic
268	C₄	C₄×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	5	4	no	—
271	C₂	—	10	2	no	—
272	C₂	C₂×C₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
273	C₂	#10604098	1	1	yes	characteristic
274	C₂	C₈×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂²×C₅	C₂×C₄×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₈×D₅	C₂³
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₆×AGL(1,𝔽₅)	C₂	semidirect
5. (C₂×AGL(1,𝔽₅))⋊C₁₆	C₂	semidirect
11. C₂²×C₅×C₁₆	C₄	semidirect
12. C₂²×#672	C₄	semidirect
14. C₂×D₅×C₁₆	C₄	semidirect
15. C₂×D₅×C₁₆	C₄	semidirect
18. C₂²⋊#674	C₄	semidirect
20. #64887	C₄	semidirect
68. C₂²⋊AGL(1,𝔽₅)	C₁₆	direct (N × H)
89. C₅×C₁₆	C₂²⋊C₄	semidirect
91. #672	C₂²⋊C₄	semidirect
100. C₂²×C₁₆	AGL(1,𝔽₅)	semidirect
116. C₂×AGL(1,𝔽₅)	C₂×C₁₆	semidirect
168. C₂²×C₅	C₄×C₁₆	semidirect
173. C₂×D₅	C₄×C₁₆	semidirect
216. C₁₆	C₂²⋊AGL(1,𝔽₅)	direct (N × H)
251. C₅	C₁₆×C₂²⋊C₄	semidirect
261. C₂²	C₁₆×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×C₁₆	2
C₈×C₂²⋊AGL(1,𝔽₅)	2
D₅⋊(C₂²⋊C₁₆)	2
C₂×C₁₆×AGL(1,𝔽₅)	2
(C₂×AGL(1,𝔽₅))⋊C₁₆	2
C₁₆×C₂²⋊C₄	5

Subgroups — C16×C22⋊AGL(1,𝔽5) #13025727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊AGL(1,𝔽₅) #13025727