Subgroups — C₅×#8569 #11931289

Order 1280.

548 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2293	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₅×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₅×#390	C₄	1	1	yes	characteristic
12	C₅×#390	C₄	1	1	yes	characteristic
13	C₅×#389	C₄	1	1	yes	characteristic
14	C₅×#389	C₄	1	1	yes	characteristic
15	C₅×#388	—	2	1	no	—
16	C₅×#388	—	2	1	no	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₅×#336	—	2	1	no	—
21	C₅×#336	—	2	1	no	—
22	C₅×#336	—	2	1	no	—
23	C₅×D₄⋊C₈	—	2	4	no	—
24	#8569	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	D₄	1	2	yes	—
32	C₄×C₅×D₄	Q₈	1	2	yes	—
33	C₅×#117	—	4	4	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
43	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₂×C₄²×C₅	—	2	1	no	—
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	C₅×#95	—	2	1	no	—
58	C₅×#95	—	2	1	no	—
59	C₅×#95	—	2	1	no	—
60	#2293	C₂×C₅	1	1	yes	characteristic
61	#1831	C₂×C₅	1	1	yes	characteristic, complemented
62	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×D₄⋊C₈	C₂×C₅	1	2	yes	—
64	#1279	C₂×C₅	1	2	yes	complemented
65	C₂⁴×C₅	—	2	1	no	—
66	C₂×C₅×D₄	—	2	1	no	—
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	—	4	2	no	—
77	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₅×C₄⋊C₄	—	2	1	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
96	C₅×C₂²⋊C₄	—	2	2	no	—
97	C₅×C₂²⋊C₄	—	2	2	no	—
98	C₅×C₂²⋊C₄	—	4	2	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₅×C₂²⋊C₄	—	4	2	no	—
101	C₅×C₂²⋊C₄	—	4	2	no	—
102	C₄²×C₅	—	2	2	no	—
103	C₄²×C₅	—	2	2	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅	#41	1	1	yes	characteristic
112	C₄²×C₅	#41	1	1	yes	characteristic
113	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
114	C₂×#117	—	2	1	no	—
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
118	#390	C₄×C₅	1	1	yes	characteristic
119	#390	C₄×C₅	1	1	yes	characteristic
120	#389	C₄×C₅	1	1	yes	characteristic
121	#389	C₄×C₅	1	1	yes	characteristic
122	#388	—	2	1	no	—
123	#388	—	2	1	no	—
124	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	C₄×C₂²⋊C₄	—	2	1	no	—
126	C₄³	C₂²×C₅	1	1	yes	characteristic
127	#336	—	2	1	no	—
128	#336	—	2	1	no	—
129	#336	—	2	1	no	—
130	D₄⋊C₈	—	2	4	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	4	1	no	—
133	C₂³×C₅	—	4	1	no	—
134	C₂³×C₅	—	4	1	no	—
135	C₂³×C₅	—	4	1	no	—
136	C₂³×C₅	—	4	1	no	—
137	C₂³×C₅	—	4	1	no	—
138	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₅×D₄	—	2	4	no	—
140	C₅×D₄	—	4	2	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	—	4	4	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	#104	1	2	yes	—
166	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	#117	1	1	yes	characteristic
168	C₂×C₄×C₅	#121	1	1	yes	characteristic
169	C₂×C₄×C₅	#121	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄×C₅	#131	1	2	yes	—
173	C₂×C₄×C₅	#131	1	2	yes	—
174	C₂×C₄×C₅	#135	1	2	yes	—
175	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
176	C₅×C₈	—	4	4	no	—
177	C₅×C₈	—	8	2	no	—
178	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂³×C₄	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₄×D₄	—	2	1	no	—
183	C₄×D₄	—	2	1	no	—
184	C₄×D₄	C₅×D₄	1	2	yes	—
185	C₄×D₄	C₅×Q₈	1	2	yes	—
186	#117	—	4	4	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
202	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
203	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	2	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	8	1	no	—
217	C₂²×C₅	—	8	1	no	—
218	C₂²×C₅	—	8	1	no	—
219	C₂²×C₅	—	8	1	no	—
220	C₂²×C₅	#380	1	1	yes	characteristic
221	C₂²×C₅	#408	1	1	yes	characteristic
222	C₂²×C₅	#408	1	1	yes	characteristic
223	C₂²×C₅	#416	1	1	yes	characteristic
224	C₂²×C₅	#418	1	1	yes	characteristic
225	C₂²×C₅	C₂×#104	1	1	yes	characteristic
226	C₂²×C₅	#421	1	1	yes	characteristic
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	1	no	—
232	C₄×C₅	—	4	1	no	—
233	C₄×C₅	—	4	1	no	—
234	C₄×C₅	—	8	1	no	—
235	C₄×C₅	—	8	1	no	—
236	C₄×C₅	#424	1	2	yes	—
237	C₄×C₅	#433	1	2	yes	—
238	C₄×C₅	#463	1	2	yes	—
239	C₄×C₅	#478	1	2	yes	—
240	C₂⁴	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
285	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
286	C₄²	C₅×#41	1	1	yes	characteristic
287	C₄²	C₅×#41	1	1	yes	characteristic
288	C₂×C₅	—	8	2	no	—
289	C₂×C₅	#1482	1	1	yes	characteristic
290	C₂×C₅	#1484	1	1	yes	characteristic
291	C₂×C₅	#1494	1	1	yes	characteristic
292	C₂×C₅	#1496	1	1	yes	characteristic
293	C₂×C₅	#1798	1	1	yes	characteristic
294	C₂×C₅	#1804	1	1	yes	characteristic
295	C₂×C₅	#1806	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	4	1	no	—
302	C₂³	—	4	1	no	—
303	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
304	D₄	—	2	4	no	—
305	D₄	—	4	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	C₅×#104	1	2	yes	—
331	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₅×#117	1	1	yes	characteristic
333	C₂×C₄	C₅×#121	1	1	yes	characteristic
334	C₂×C₄	C₅×#121	1	1	yes	characteristic
335	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
336	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
337	C₂×C₄	C₅×#131	1	2	yes	—
338	C₂×C₄	C₅×#131	1	2	yes	—
339	C₂×C₄	C₅×#135	1	2	yes	—
340	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
341	C₈	—	4	4	no	—
342	C₈	—	8	2	no	—
343	C₅	#8569	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	8	1	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	C₅×#380	1	1	yes	characteristic
352	C₂²	C₅×#408	1	1	yes	characteristic
353	C₂²	C₅×#408	1	1	yes	characteristic
354	C₂²	C₅×#416	1	1	yes	characteristic
355	C₂²	C₅×#418	1	1	yes	characteristic
356	C₂²	C₂×C₅×#104	1	1	yes	characteristic
357	C₂²	C₅×#421	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	C₅×#424	1	2	yes	—
368	C₄	C₅×#433	1	2	yes	—
369	C₄	C₅×#463	1	2	yes	—
370	C₄	C₅×#478	1	2	yes	—
371	C₂	—	8	2	no	—
372	C₂	C₅×#1482	1	1	yes	characteristic
373	C₂	C₅×#1484	1	1	yes	characteristic
374	C₂	C₅×#1494	1	1	yes	characteristic
375	C₂	C₅×#1496	1	1	yes	characteristic
376	C₂	C₅×#1798	1	1	yes	characteristic
377	C₂	C₅×#1804	1	1	yes	characteristic
378	C₂	C₅×#1806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1831	C₂	semidirect
3. C₄×C₅×C₄⋊C₈	C₂	semidirect
5. C₅×#1279	C₂	semidirect
24. #8569	C₅	direct (N × H)
61. #1831	C₂×C₅	semidirect
62. C₄×C₄⋊C₈	C₂×C₅	semidirect
64. #1279	C₂×C₅	semidirect
343. C₅	#8569	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2293	2
C₅×#1831	2
C₄×C₅×C₄⋊C₈	2
C₂×C₅×D₄⋊C₈	2
C₅×#1279	2
#8569	5

Subgroups — C5×#8569 #11931289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8569 #11931289