Subgroups — C₂×C₁₆×C₂²⋊D₅ #12252448

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64727	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×C₂²⋊D₅	C₂	1	8	yes	maximal, direct product factor
7	C₂×#64721	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	—	2	4	no	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	4	yes	complemented
16	C₂×C₂²⋊#168	C₄	1	1	yes	characteristic
17	C₈×C₂²⋊D₅	C₄	1	4	yes	—
18	C₈×C₂²⋊D₅	C₂²	1	4	yes	—
19	C₂×#3952	C₄	1	1	yes	characteristic
20	C₂×#3946	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₂²×#672	C₂²	1	1	yes	characteristic
23	C₂²×#672	C₂²	1	1	yes	characteristic
24	C₂×D₅×C₁₆	—	2	4	no	—
25	C₂×D₅×C₁₆	C₂²	1	4	yes	complemented
26	C₂²⋊#672	C₂²	1	4	yes	complemented
27	#64727	C₂²	1	4	yes	complemented
28	#64721	C₂²	1	4	yes	complemented
29	C₁₆×Dic₅	C₂²	1	4	yes	complemented
30	C₂×D₄×C₁₆	—	5	1	no	maximal, Sylow
31	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₂²⋊D₅	C₈	1	1	yes	characteristic
33	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊Dic₅	C₈	1	1	yes	characteristic
41	C₄×C₂²⋊D₅	C₈	1	4	yes	—
42	C₄×C₂²⋊D₅	C₂×C₄	1	4	yes	—
43	C₂×#685	C₈	1	1	yes	characteristic
44	C₂×#683	C₈	1	1	yes	characteristic
45	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₂²×#168	C₂×C₄	1	1	yes	characteristic
47	C₂²×#168	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅	—	2	2	no	—
49	C₂×C₈×D₅	—	2	2	no	—
50	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₅	C₂³	1	2	yes	—
52	C₂×C₅×C₁₆	—	2	2	no	—
53	C₂×C₅×C₁₆	—	2	8	no	—
54	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
55	C₂×C₅×C₁₆	C₂³	1	2	yes	—
56	C₂×C₅×C₁₆	C₂³	1	2	yes	—
57	C₂²⋊#168	C₂×C₄	1	4	yes	—
58	#3952	C₂×C₄	1	4	yes	—
59	#3946	C₂×C₄	1	4	yes	—
60	C₈×Dic₅	C₂×C₄	1	2	yes	—
61	C₈×Dic₅	C₂³	1	2	yes	—
62	C₂×#672	—	2	2	no	—
63	C₂×#672	—	2	2	no	—
64	C₂×#672	C₂³	1	2	yes	—
65	C₂×#672	C₂³	1	2	yes	—
66	D₅×C₁₆	—	2	8	no	—
67	C₂³×C₁₆	—	5	1	no	—
68	C₂³×C₁₆	D₅	1	1	yes	characteristic, complemented
69	C₂×C₈×D₄	—	5	1	no	—
70	D₄×C₁₆	—	5	8	no	—
71	C₂×C₄⋊C₁₆	—	5	1	no	—
72	C₂×C₂²⋊C₁₆	—	5	1	no	—
73	C₂×C₂²⋊C₁₆	—	5	1	no	—
74	C₂×C₄×C₁₆	—	5	1	no	—
75	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₅	C₁₆	1	8	yes	direct product factor
84	C₂×C₂²⋊D₅	C₂×C₈	1	4	yes	—
85	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
86	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
87	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	—	2	2	no	—
89	C₂×C₄×D₅	—	2	2	no	—
90	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
91	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	#41	1	2	yes	—
100	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₅	C₂×C₈	1	4	yes	—
102	#685	C₂×C₈	1	4	yes	—
103	#683	C₂×C₈	1	4	yes	—
104	C₄×Dic₅	C₂×C₈	1	2	yes	—
105	C₄×Dic₅	C₂²×C₄	1	2	yes	—
106	C₂×#168	—	2	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	C₂²×C₄	1	2	yes	—
109	C₂×#168	C₂²×C₄	1	2	yes	—
110	C₈×D₅	—	2	4	no	—
111	C₈×D₅	—	2	2	no	—
112	C₈×D₅	—	2	2	no	—
113	C₅×C₁₆	—	2	2	no	—
114	C₅×C₁₆	C₂×D₄	1	4	yes	complemented
115	#672	—	2	2	no	—
116	#672	—	2	2	no	—
117	C₂³×C₈	—	5	1	no	—
118	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
119	C₂×C₄×D₄	—	5	1	no	—
120	C₂²×C₁₆	—	2	4	no	—
121	C₂²×C₁₆	—	5	1	no	—
122	C₂²×C₁₆	—	5	1	no	—
123	C₂²×C₁₆	—	5	4	no	—
124	C₂²×C₁₆	—	10	4	no	—
125	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
127	C₂²×C₁₆	C₂×D₅	1	4	yes	complemented
128	C₈×D₄	—	5	4	no	—
129	C₈×D₄	—	5	4	no	—
130	C₂×C₄⋊C₈	—	5	1	no	—
131	C₂×C₂²⋊C₈	—	5	1	no	—
132	C₂×C₂²⋊C₈	—	5	1	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	C₄⋊C₁₆	—	5	4	no	—
135	C₂²⋊C₁₆	—	5	4	no	—
136	C₂²⋊C₁₆	—	5	4	no	—
137	C₄×C₁₆	—	5	4	no	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	C₂×C₁₆	1	4	yes	complemented
141	C₂³×C₅	C₂²×C₈	1	2	yes	—
142	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
143	C₂²×D₅	—	2	2	no	—
144	C₂²×D₅	—	2	2	no	—
145	C₂²×D₅	C₂×C₁₆	1	4	yes	complemented
146	C₂²×D₅	C₂²×C₈	1	2	yes	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₅	#131	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂²⋊D₅	C₂×C₁₆	1	16	yes	direct product factor
157	C₂×Dic₅	—	2	2	no	—
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	C₂×C₁₆	1	4	yes	complemented
160	C₂×Dic₅	C₂²×C₈	1	2	yes	—
161	C₂×Dic₅	C₂²×C₈	1	2	yes	—
162	C₂×Dic₅	C₂²×C₈	1	2	yes	—
163	C₄×D₅	—	2	2	no	—
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	2	4	no	—
166	C₅×C₈	—	2	2	no	—
167	C₅×C₈	C₄×D₄	1	2	yes	—
168	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
169	C₅×C₈	C₂×#41	1	1	yes	characteristic
170	#168	—	2	2	no	—
171	#168	—	2	2	no	—
172	C₂²×D₄	—	5	1	no	—
173	C₂³×C₄	—	5	1	no	—
174	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	5	1	no	—
178	C₂²×C₈	—	5	1	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	—	5	2	no	—
181	C₂²×C₈	—	10	2	no	—
182	C₂²×C₈	—	10	2	no	—
183	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₅	1	2	yes	—
185	C₂²×C₈	C₂²×D₅	1	2	yes	—
186	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
187	C₄×D₄	—	5	4	no	—
188	C₄×D₄	—	5	4	no	—
189	C₂×C₄⋊C₄	—	5	1	no	—
190	C₂×C₂²⋊C₄	—	5	1	no	—
191	C₂×C₂²⋊C₄	—	5	1	no	—
192	C₂×C₄²	—	5	1	no	—
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	8	no	—
195	C₂×C₁₆	—	5	2	no	—
196	C₂×C₁₆	—	5	2	no	—
197	C₂×C₁₆	—	10	2	no	—
198	C₂×C₁₆	—	10	2	no	—
199	C₂×C₁₆	—	10	8	no	—
200	C₂×C₁₆	C₂²⋊D₅	1	4	yes	direct product factor
201	C₂×C₁₆	C₂²×D₅	1	2	yes	—
202	C₂×C₁₆	C₂²×D₅	1	2	yes	—
203	C₄⋊C₈	—	5	4	no	—
204	C₂²⋊C₈	—	5	4	no	—
205	C₂²⋊C₈	—	5	4	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	2	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	8	no	—
211	C₂²×C₅	C₈×D₄	1	2	yes	—
212	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
213	C₂²×C₅	C₂²×C₁₆	1	4	yes	complemented
214	C₂²×C₅	#504	1	2	yes	—
215	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	2	8	no	—
219	C₂×D₅	C₂²×C₁₆	1	4	yes	complemented
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	C₈×D₄	1	2	yes	—
222	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	C₄×C₅	C₂×#131	1	1	yes	characteristic
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	C₂²×C₁₆	1	4	yes	complemented
227	C₂⁴	—	5	1	no	—
228	C₂⁴	C₈×D₅	1	1	yes	characteristic
229	C₂×D₄	—	5	8	no	—
230	C₂×D₄	—	5	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	10	2	no	—
239	C₂²×C₄	—	10	2	no	—
240	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
241	C₂²×C₄	C₈×D₅	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
243	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	5	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	10	2	no	—
254	C₂×C₈	—	10	4	no	—
255	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
257	C₂×C₈	#709	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
259	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
260	C₄⋊C₄	—	5	4	no	—
261	C₂²⋊C₄	—	5	4	no	—
262	C₂²⋊C₄	—	5	4	no	—
263	C₄²	—	5	2	no	—
264	C₄²	—	5	2	no	—
265	C₁₆	—	2	2	no	—
266	C₁₆	—	10	2	no	—
267	C₁₆	—	10	2	no	—
268	C₁₆	C₂×C₂²⋊D₅	1	4	yes	direct product factor
269	C₂×C₅	—	2	4	no	—
270	C₂×C₅	D₄×C₁₆	1	4	yes	complemented
271	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
272	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
273	C₂×C₅	C₂×#504	1	1	yes	characteristic
274	D₅	—	2	4	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	5	2	no	—
278	C₂³	—	5	4	no	—
279	C₂³	—	10	2	no	—
280	C₂³	—	10	2	no	—
281	C₂³	D₅×C₁₆	1	4	yes	complemented
282	C₂³	C₂×C₈×D₅	1	2	yes	—
283	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
284	D₄	—	5	16	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	4	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	10	2	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
300	C₂×C₄	#4047	1	2	yes	—
301	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
302	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	10	2	no	—
305	C₈	—	10	2	no	—
306	C₈	C₄×C₂²⋊D₅	1	2	yes	—
307	C₈	C₂×#709	1	1	yes	characteristic
308	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
309	C₅	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	2	no	—
311	C₂²	—	2	8	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	5	4	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	10	8	no	—
316	C₂²	—	10	2	no	—
317	C₂²	C₂×D₅×C₁₆	1	2	yes	—
318	C₂²	C₂×D₅×C₁₆	1	4	yes	complemented
319	C₂²	#65190	1	2	yes	—
320	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
321	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	5	4	no	—
324	C₄	—	10	2	no	—
325	C₄	—	10	2	no	—
326	C₄	C₈×C₂²⋊D₅	1	2	yes	—
327	C₄	C₂×#4047	1	1	yes	characteristic
328	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	10	4	no	—
331	C₂	C₁₆×C₂²⋊D₅	1	4	yes	direct product factor
332	C₂	C₂²×D₅×C₁₆	1	1	yes	characteristic
333	C₂	C₂×#65190	1	1	yes	characteristic
334	C₂	C₂×C₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×C₂²⋊D₅	C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×C₁₆	C₂	semidirect
3. C₂²×D₅×C₁₆	C₂	semidirect
4. C₂×C₂²⋊#672	C₂	semidirect
5. C₂×#64727	C₂	semidirect
6. C₁₆×C₂²⋊D₅	C₂	direct (N × H)
7. C₂×#64721	C₂	semidirect
8. C₂×C₁₆×Dic₅	C₂	semidirect
15. C₂²×C₅×C₁₆	C₂²	semidirect
25. C₂×D₅×C₁₆	C₂²	semidirect
26. C₂²⋊#672	C₂²	semidirect
27. #64727	C₂²	semidirect
28. #64721	C₂²	semidirect
29. C₁₆×Dic₅	C₂²	semidirect
54. C₂×C₅×C₁₆	D₄	semidirect
68. C₂³×C₁₆	D₅	semidirect
83. C₂×C₂²⋊D₅	C₁₆	direct (N × H)
114. C₅×C₁₆	C₂×D₄	semidirect
127. C₂²×C₁₆	C₂×D₅	semidirect
140. C₂³×C₅	C₂×C₁₆	semidirect
145. C₂²×D₅	C₂×C₁₆	semidirect
156. C₂²⋊D₅	C₂×C₁₆	direct (N × H)
159. C₂×Dic₅	C₂×C₁₆	semidirect
200. C₂×C₁₆	C₂²⋊D₅	direct (N × H)
213. C₂²×C₅	C₂²×C₁₆	semidirect
219. C₂×D₅	C₂²×C₁₆	semidirect
226. Dic₅	C₂²×C₁₆	semidirect
268. C₁₆	C₂×C₂²⋊D₅	direct (N × H)
270. C₂×C₅	D₄×C₁₆	semidirect
281. C₂³	D₅×C₁₆	semidirect
309. C₅	C₂×D₄×C₁₆	semidirect
318. C₂²	C₂×D₅×C₁₆	semidirect
331. C₂	C₁₆×C₂²⋊D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂×C₈×C₂²⋊D₅	2
C₂²×D₅×C₁₆	2
C₂×C₂²⋊#672	2
C₂×#64727	2
C₁₆×C₂²⋊D₅	2
C₂×#64721	2
C₂×C₁₆×Dic₅	2
C₂×D₄×C₁₆	5

Subgroups — C2×C16×C22⋊D5 #12252448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×C₂²⋊D₅ #12252448