Subgroups — C₅×#8581 #11931301

Order 1280.

540 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#329	C₂	1	2	yes	maximal
5	C₅×#1295	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#124	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×#97	C₂²	1	2	yes	—
12	C₂×C₅×#97	C₂²	1	2	yes	—
13	C₅×#391	C₄	1	1	yes	characteristic
14	C₅×#390	C₄	1	1	yes	characteristic
15	C₅×#388	—	2	1	no	—
16	C₅×#388	—	2	1	no	—
17	C₅×#382	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#336	—	2	1	no	—
22	C₅×#336	—	2	1	no	—
23	C₅×#329	—	2	4	no	—
24	#8581	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	—	2	1	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₅×#124	—	2	1	no	—
31	C₅×#124	—	2	1	no	—
32	C₅×#124	D₄	1	2	yes	—
33	C₅×#124	Q₈	1	2	yes	—
34	C₄×C₅×Q₈	—	4	2	no	—
35	C₄×C₅×D₄	—	4	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
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	4	no	—
54	C₅×#97	—	2	2	no	—
55	C₅×#97	—	2	2	no	—
56	C₅×#97	—	2	2	no	—
57	C₅×#97	—	2	2	no	—
58	C₅×#95	—	2	1	no	—
59	C₅×#95	—	2	1	no	—
60	C₅×#95	—	2	1	no	—
61	C₅×#95	—	2	1	no	—
62	C₄×#124	C₂×C₅	1	1	yes	characteristic
63	#1767	C₂×C₅	1	1	yes	characteristic, complemented
64	#1767	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×#329	C₂×C₅	1	2	yes	—
66	#1295	C₂×C₅	1	2	yes	complemented
67	C₂⁴×C₅	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	1	no	—
69	C₂×C₅×Q₈	—	4	1	no	—
70	C₂×C₅×Q₈	—	4	1	no	—
71	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₅×D₄	—	2	1	no	—
73	C₂×C₅×D₄	—	4	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	—	4	2	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	4	no	—
91	C₂×C₅×C₈	—	4	2	no	—
92	C₂×C₅×C₈	—	4	2	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	2	2	no	—
99	C₅×C₂²⋊C₄	—	2	2	no	—
100	C₅×C₂²⋊C₄	—	2	2	no	—
101	C₅×C₂²⋊C₄	—	2	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	4	1	no	—
109	C₄²×C₅	—	4	1	no	—
110	C₄²×C₅	—	4	1	no	—
111	C₄²×C₅	—	4	2	no	—
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
114	C₄²×C₅	#41	1	1	yes	characteristic
115	C₄²×C₅	#41	1	1	yes	characteristic
116	C₂×#124	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
118	C₂×C₄×D₄	C₄×C₅	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×#97	C₂²×C₅	1	2	yes	—
122	C₂×#97	C₂²×C₅	1	2	yes	—
123	#391	C₄×C₅	1	1	yes	characteristic
124	#390	C₄×C₅	1	1	yes	characteristic
125	#388	—	2	1	no	—
126	#388	—	2	1	no	—
127	#382	C₂²×C₅	1	1	yes	characteristic
128	C₄×C₂²⋊C₄	—	2	1	no	—
129	C₄×C₂²⋊C₄	—	2	1	no	—
130	C₄³	C₂²×C₅	1	1	yes	characteristic
131	#336	—	2	1	no	—
132	#336	—	2	1	no	—
133	#329	—	2	4	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	—	4	1	no	—
136	C₂³×C₅	—	4	1	no	—
137	C₂³×C₅	—	4	1	no	—
138	C₂³×C₅	—	4	1	no	—
139	C₂³×C₅	—	4	1	no	—
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₅×Q₈	—	4	2	no	—
143	C₅×Q₈	—	4	2	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	4	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	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	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	—	4	1	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	8	4	no	—
171	C₂×C₄×C₅	#100	1	2	yes	—
172	C₂×C₄×C₅	#104	1	2	yes	—
173	C₂×C₄×C₅	#104	1	2	yes	—
174	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	#117	1	1	yes	characteristic
176	C₂×C₄×C₅	#121	1	1	yes	characteristic
177	C₂×C₄×C₅	#121	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
180	C₅×C₈	—	4	4	no	—
181	C₅×C₈	—	4	4	no	—
182	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
184	C₂³×C₄	—	2	1	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	#124	—	2	1	no	—
188	#124	—	2	1	no	—
189	#124	C₅×D₄	1	2	yes	—
190	#124	C₅×Q₈	1	2	yes	—
191	C₄×Q₈	—	4	2	no	—
192	C₄×D₄	—	4	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
205	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
206	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
207	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	2	4	no	—
211	#97	—	2	2	no	—
212	#97	—	2	2	no	—
213	#97	—	2	2	no	—
214	#97	—	2	2	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	1	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	—	4	2	no	—
221	C₂²×C₅	—	4	2	no	—
222	C₂²×C₅	—	8	1	no	—
223	C₂²×C₅	—	8	1	no	—
224	C₂²×C₅	—	8	1	no	—
225	C₂²×C₅	—	8	1	no	—
226	C₂²×C₅	#380	1	1	yes	characteristic
227	C₂²×C₅	C₂×#100	1	1	yes	characteristic
228	C₂²×C₅	#413	1	1	yes	characteristic
229	C₂²×C₅	C₂×#104	1	1	yes	characteristic
230	C₂²×C₅	C₂×#104	1	1	yes	characteristic
231	C₂²×C₅	#421	1	1	yes	characteristic
232	C₂²×C₅	#421	1	1	yes	characteristic
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	4	1	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	—	8	1	no	—
246	C₂⁴	—	2	1	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
292	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
293	C₄²	C₅×#41	1	1	yes	characteristic
294	C₄²	C₅×#41	1	1	yes	characteristic
295	C₂×C₅	—	8	2	no	—
296	C₂×C₅	#1524	1	1	yes	characteristic
297	C₂×C₅	#1527	1	1	yes	characteristic
298	C₂×C₅	#1530	1	1	yes	characteristic
299	C₂×C₅	#1535	1	1	yes	characteristic
300	C₂×C₅	#1800	1	1	yes	characteristic
301	C₂×C₅	#1806	1	1	yes	characteristic
302	C₂×C₅	#1806	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	4	1	no	—
310	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	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	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	C₅×#100	1	2	yes	—
341	C₂×C₄	C₅×#104	1	2	yes	—
342	C₂×C₄	C₅×#104	1	2	yes	—
343	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
344	C₂×C₄	C₅×#117	1	1	yes	characteristic
345	C₂×C₄	C₅×#121	1	1	yes	characteristic
346	C₂×C₄	C₅×#121	1	1	yes	characteristic
347	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
348	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
349	C₈	—	4	4	no	—
350	C₈	—	4	4	no	—
351	C₅	#8581	1	1	yes	characteristic, direct product factor, Sylow
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	C₅×#380	1	1	yes	characteristic
360	C₂²	C₂×C₅×#100	1	1	yes	characteristic
361	C₂²	C₅×#413	1	1	yes	characteristic
362	C₂²	C₂×C₅×#104	1	1	yes	characteristic
363	C₂²	C₂×C₅×#104	1	1	yes	characteristic
364	C₂²	C₅×#421	1	1	yes	characteristic
365	C₂²	C₅×#421	1	1	yes	characteristic
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₂	—	8	2	no	—
380	C₂	C₅×#1524	1	1	yes	characteristic
381	C₂	C₅×#1527	1	1	yes	characteristic
382	C₂	C₅×#1530	1	1	yes	characteristic
383	C₂	C₅×#1535	1	1	yes	characteristic
384	C₂	C₅×#1800	1	1	yes	characteristic
385	C₂	C₅×#1806	1	1	yes	characteristic
386	C₂	C₅×#1806	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅×#1767	C₂	semidirect
3. C₅×#1767	C₂	semidirect
5. C₅×#1295	C₂	semidirect
24. #8581	C₅	direct (N × H)
63. #1767	C₂×C₅	semidirect
64. #1767	C₂×C₅	semidirect
66. #1295	C₂×C₅	semidirect
351. C₅	#8581	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#124	2
C₅×#1767	2
C₂×C₅×#329	2
C₅×#1295	2
#8581	5

Subgroups — C5×#8581 #11931301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8581 #11931301