Subgroups — D₅×#1382 #12014191

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 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	C₂×D₅×#108	C₂	1	1	yes	maximal, characteristic
3	D₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₅×#1382	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598298	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598220	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅×M₁₆	—	2	1	no	—
9	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
10	D₅×#117	—	2	1	no	—
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#427	C₂²	1	1	yes	characteristic
14	C₂×#3954	—	2	1	no	—
15	C₂×#3951	C₂²	1	1	yes	characteristic
16	#65395	C₂²	1	1	yes	characteristic
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	#65287	—	2	1	no	—
19	C₂×D₅×C₁₆	—	2	1	no	—
20	C₂×D₅×C₁₆	—	2	1	no	—
21	C₂×D₅×C₁₆	—	2	2	no	—
22	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
23	D₅×#108	—	2	2	no	—
24	D₅×#108	C₄	1	4	yes	complemented
25	#65169	C₄	1	2	yes	—
26	D₅×#107	C₄	1	2	yes	—
27	#65149	—	2	1	no	—
28	D₅×#106	—	2	1	no	—
29	C₂×#1382	—	5	1	no	maximal, Sylow
30	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×#117	—	2	1	no	—
34	#4072	—	2	1	no	—
35	C₂×#681	—	2	1	no	—
36	C₂²×#168	D₄	1	1	yes	characteristic
37	D₅×M₁₆	—	2	4	no	—
38	D₅×M₁₆	—	4	2	no	—
39	C₂×#676	—	4	1	no	—
40	C₂×C₈×D₅	—	4	1	no	—
41	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅	D₄	1	1	yes	characteristic
47	C₂×C₈×D₅	D₄	1	2	yes	—
48	C₂×C₈×D₅	Q₈	1	2	yes	—
49	#4038	—	2	2	no	—
50	D₅×C₄⋊C₄	—	2	2	no	—
51	D₅×C₂²⋊C₄	—	4	1	no	—
52	#4023	—	4	1	no	—
53	#4018	—	4	1	no	—
54	C₄²×D₅	—	4	1	no	—
55	C₂×C₅×C₁₆	—	2	1	no	—
56	C₂×C₅×C₁₆	—	2	1	no	—
57	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
58	C₅×#108	—	2	1	no	—
59	C₅×#108	C₂×C₄	1	2	yes	complemented
60	C₅×#107	C₂×C₄	1	2	yes	—
61	C₅×#106	—	2	1	no	—
62	#3954	—	2	4	no	—
63	#3951	—	2	1	no	—
64	#3951	C₂×C₄	1	2	yes	complemented
65	#3950	C₂×C₄	1	2	yes	—
66	#3949	—	2	1	no	—
67	C₂×#672	—	2	1	no	—
68	C₂×#672	—	2	1	no	—
69	C₂×#672	C₂×C₄	1	2	yes	complemented
70	#3940	—	2	2	no	—
71	#3939	—	2	2	no	—
72	D₅×C₁₆	—	2	4	no	—
73	D₅×C₁₆	—	2	4	no	—
74	C₂³×C₁₆	—	5	1	no	—
75	C₂²×#108	—	5	1	no	—
76	C₂×#427	—	5	1	no	—
77	#1382	—	5	1	no	—
78	#1382	—	5	1	no	—
79	#1382	—	5	1	no	—
80	#1382	D₅	1	1	yes	characteristic, direct product factor
81	C₂³×D₅	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₅	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₅	—	4	2	no	—
85	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
89	C₂×C₄×D₅	D₈	1	1	yes	characteristic
90	C₂×C₄×D₅	D₈	1	1	yes	characteristic
91	C₂×C₄×D₅	SD₁₆	1	2	yes	—
92	C₂×C₄×D₅	Q₁₆	1	2	yes	—
93	C₅×M₁₆	—	2	2	no	—
94	C₅×M₁₆	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₅	—	4	1	no	—
105	C₂²⋊Dic₅	—	4	1	no	—
106	#685	—	4	2	no	—
107	C₄⋊Dic₅	—	2	2	no	—
108	#683	—	4	2	no	—
109	C₄×Dic₅	—	4	1	no	—
110	C₄×Dic₅	—	4	2	no	—
111	#681	—	2	2	no	—
112	#681	—	4	1	no	—
113	C₂×#168	—	4	1	no	—
114	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×#168	C₂²⋊C₄	1	2	yes	—
117	C₂×#168	C₄⋊C₄	1	2	yes	—
118	#676	—	4	4	no	—
119	C₈×D₅	—	4	4	no	—
120	C₈×D₅	C₄²	1	2	yes	—
121	C₈×D₅	C₄²	1	2	yes	—
122	C₈×D₅	C₂²⋊C₄	1	2	yes	—
123	C₈×D₅	C₂²⋊C₄	1	2	yes	—
124	C₈×D₅	C₄⋊C₄	1	2	yes	—
125	C₈×D₅	C₄⋊C₄	1	2	yes	—
126	C₈×D₅	C₄⋊C₄	1	2	yes	—
127	C₈×D₅	C₄⋊C₄	1	2	yes	—
128	C₅×C₁₆	—	2	2	no	—
129	#672	—	2	2	no	—
130	C₂²×M₁₆	—	10	1	no	—
131	C₂³×C₈	—	5	1	no	—
132	C₂×#117	—	10	1	no	—
133	C₂²×C₁₆	—	5	1	no	—
134	C₂²×C₁₆	—	5	4	no	—
135	C₂²×C₁₆	—	10	1	no	—
136	C₂²×C₁₆	—	10	1	no	—
137	C₂²×C₁₆	—	10	2	no	—
138	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
139	C₂×#108	—	5	1	no	—
140	C₂×#108	—	5	4	no	—
141	C₂×#108	—	10	2	no	—
142	C₂×#108	—	10	1	no	—
143	C₂×#108	C₂×D₅	1	1	yes	characteristic
144	#427	—	5	1	no	—
145	#427	—	5	2	no	—
146	#427	—	10	1	no	—
147	#427	—	10	1	no	—
148	#427	C₂×D₅	1	1	yes	characteristic
149	C₂×#107	—	5	2	no	—
150	C₂×#106	—	10	1	no	—
151	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
152	C₂²×D₅	—	2	4	no	—
153	C₂²×D₅	#102	1	1	yes	characteristic
154	C₂²×D₅	#102	1	1	yes	characteristic
155	C₂²×D₅	#103	1	2	yes	—
156	C₂²×D₅	#106	1	2	yes	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
164	C₂×Dic₅	—	4	2	no	—
165	C₂×Dic₅	#102	1	1	yes	characteristic
166	C₂×Dic₅	#102	1	1	yes	characteristic
167	C₂×Dic₅	#103	1	2	yes	—
168	C₂×Dic₅	#106	1	2	yes	—
169	C₄×D₅	—	8	4	no	—
170	C₄×D₅	#95	1	2	yes	—
171	C₄×D₅	#95	1	2	yes	—
172	C₄×D₅	#102	1	2	yes	—
173	C₄×D₅	#102	1	2	yes	—
174	C₄×D₅	#103	1	2	yes	—
175	C₄×D₅	#103	1	2	yes	—
176	C₄×D₅	#107	1	2	yes	—
177	C₄×D₅	#107	1	2	yes	—
178	C₅×C₈	—	4	2	no	—
179	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
180	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	#168	—	4	2	no	—
184	#168	#95	1	2	yes	—
185	#168	#95	1	2	yes	—
186	C₂³×C₄	—	5	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	10	4	no	—
189	C₂×M₁₆	—	10	1	no	—
190	C₂×M₁₆	—	20	1	no	—
191	C₂×M₁₆	—	20	2	no	—
192	C₂²×C₈	—	5	1	no	—
193	C₂²×C₈	—	5	1	no	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	—	5	2	no	—
196	C₂²×C₈	—	5	2	no	—
197	C₂²×C₈	—	5	1	no	—
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	—	5	2	no	—
200	C₂²×C₈	—	5	2	no	—
201	C₂²×C₈	—	20	1	no	—
202	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
203	#117	—	2	1	no	—
204	#117	—	10	2	no	—
205	#117	—	10	1	no	—
206	#117	—	20	1	no	—
207	#117	—	20	1	no	—
208	C₂×C₄⋊C₄	—	10	2	no	—
209	C₂×C₂²⋊C₄	—	20	1	no	—
210	C₂×C₄²	—	20	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	5	2	no	—
214	C₂×C₁₆	—	10	4	no	—
215	C₂×C₁₆	—	10	4	no	—
216	C₂×C₁₆	—	10	1	no	—
217	C₂×C₁₆	—	10	1	no	—
218	C₂×C₁₆	C₄×D₅	1	2	yes	complemented
219	#108	—	2	1	no	—
220	#108	—	5	2	no	—
221	#108	—	10	4	no	—
222	#108	—	10	1	no	—
223	#108	C₄×D₅	1	2	yes	complemented
224	#107	—	5	2	no	—
225	#107	—	10	2	no	—
226	#107	C₄×D₅	1	2	yes	—
227	#106	—	2	1	no	—
228	#106	—	10	2	no	—
229	#106	—	10	1	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	C₂×#102	1	1	yes	characteristic
232	C₂²×C₅	C₂×#103	1	1	yes	characteristic
233	C₂²×C₅	C₂×#106	1	1	yes	characteristic
234	C₂×D₅	—	2	2	no	—
235	C₂×D₅	—	2	4	no	—
236	C₂×D₅	—	2	4	no	—
237	C₂×D₅	#340	1	2	yes	—
238	C₂×D₅	#340	1	2	yes	—
239	C₂×D₅	#359	1	4	yes	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	C₂×#95	1	1	yes	characteristic
242	C₄×C₅	C₂×#102	1	1	yes	characteristic
243	C₄×C₅	C₂×#103	1	1	yes	characteristic
244	C₄×C₅	C₂×#107	1	1	yes	characteristic
245	Dic₅	—	8	2	no	—
246	Dic₅	#340	1	2	yes	—
247	Dic₅	#340	1	2	yes	—
248	C₂⁴	—	5	1	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	5	2	no	—
253	C₂²×C₄	—	5	2	no	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	5	1	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	—	5	2	no	—
258	C₂²×C₄	—	20	2	no	—
259	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	10	2	no	—
263	M₁₆	—	20	4	no	—
264	M₁₆	—	20	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	5	1	no	—
267	C₂×C₈	—	5	1	no	—
268	C₂×C₈	—	5	2	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	2	no	—
271	C₂×C₈	—	5	2	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	5	2	no	—
275	C₂×C₈	—	5	2	no	—
276	C₂×C₈	—	5	2	no	—
277	C₂×C₈	—	5	2	no	—
278	C₂×C₈	—	20	4	no	—
279	C₂×C₈	—	20	1	no	—
280	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
282	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
283	C₂×C₈	D₄×D₅	1	1	yes	characteristic
284	C₂×C₈	D₄×D₅	1	1	yes	characteristic
285	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	10	2	no	—
288	C₄⋊C₄	—	20	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	20	2	no	—
291	C₂²⋊C₄	—	20	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	20	2	no	—
294	C₄²	—	20	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	10	2	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	C₂×#340	1	1	yes	characteristic
300	C₂×C₅	C₂×#359	1	2	yes	—
301	D₅	—	2	2	no	—
302	D₅	#1382	1	4	yes	direct product factor
303	C₂³	—	5	1	no	—
304	C₂³	—	5	1	no	—
305	C₂³	—	5	2	no	—
306	C₂³	—	5	2	no	—
307	C₂³	—	10	4	no	—
308	C₂³	D₅×SD₁₆	1	1	yes	characteristic
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	5	1	no	—
311	C₂×C₄	—	5	1	no	—
312	C₂×C₄	—	5	2	no	—
313	C₂×C₄	—	5	2	no	—
314	C₂×C₄	—	5	2	no	—
315	C₂×C₄	—	5	2	no	—
316	C₂×C₄	—	5	2	no	—
317	C₂×C₄	—	5	2	no	—
318	C₂×C₄	—	5	2	no	—
319	C₂×C₄	—	5	2	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	5	2	no	—
322	C₂×C₄	—	20	2	no	—
323	C₂×C₄	—	40	4	no	—
324	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
325	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	D₅×D₈	1	1	yes	characteristic
328	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
329	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	5	2	no	—
332	C₈	—	5	2	no	—
333	C₈	—	20	2	no	—
334	C₈	C₄²×D₅	1	1	yes	characteristic
335	C₈	D₅×C₂²⋊C₄	1	1	yes	characteristic
336	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
337	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
338	C₅	C₂×#1382	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	2	2	no	—
340	C₂²	—	5	4	no	—
341	C₂²	—	5	2	no	—
342	C₂²	—	5	2	no	—
343	C₂²	—	10	4	no	—
344	C₂²	—	10	4	no	—
345	C₂²	—	10	2	no	—
346	C₂²	D₅×#102	1	1	yes	characteristic
347	C₂²	D₅×#103	1	1	yes	characteristic
348	C₂²	D₅×#106	1	1	yes	characteristic
349	C₄	—	5	2	no	—
350	C₄	—	5	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	40	2	no	—
353	C₄	D₅×#95	1	1	yes	characteristic
354	C₄	D₅×#102	1	1	yes	characteristic
355	C₄	D₅×#103	1	1	yes	characteristic
356	C₄	D₅×#107	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	5	4	no	—
360	C₂	—	10	2	no	—
361	C₂	D₅×#340	1	1	yes	characteristic
362	C₂	D₅×#359	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1382	C₂
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×D₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1382	D₅
hypocenter	C₅	C₂×#1382
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₅×#1382	C₂	semidirect
5. #10598298	C₂	semidirect
6. #10598220	C₂	semidirect
7. #10598215	C₂	semidirect
22. C₂×D₅×C₁₆	C₄	semidirect
24. D₅×#108	C₄	semidirect
57. C₂×C₅×C₁₆	C₂×C₄	semidirect
59. C₅×#108	C₂×C₄	semidirect
64. #3951	C₂×C₄	semidirect
69. C₂×#672	C₂×C₄	semidirect
80. #1382	D₅	direct (N × H)
218. C₂×C₁₆	C₄×D₅	semidirect
223. #108	C₄×D₅	semidirect
302. D₅	#1382	direct (N × H)
338. C₅	C₂×#1382	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
C₂×D₅×#108	2
D₅×#427	2
C₅×#1382	2
#10598298	2
#10598220	2
#10598215	2
C₂×#1382	5

Subgroups — D5×#1382 #12014191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1382 #12014191