Subgroups — C₅×#7560 #11930280

Order 1280.

538 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	C₅×#2293	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1271	C₂	1	2	yes	maximal, complemented
5	C₅×#1270	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₅×C₈	—	2	1	no	—
10	C₂×C₄×C₅×C₈	—	2	1	no	—
11	C₅×#390	C₄	1	1	yes	characteristic
12	C₅×#390	C₂²	1	1	yes	characteristic
13	C₅×#389	C₄	1	1	yes	characteristic
14	C₅×#389	C₂²	1	1	yes	characteristic
15	C₅×#388	—	2	1	no	—
16	C₅×#388	—	2	1	no	—
17	C₄×C₅×C₄⋊C₄	C₄	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	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	2	yes	—
23	#7560	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	—	2	1	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₄×C₅×D₄	—	2	2	no	—
29	C₄×C₅×D₄	C₈	1	4	yes	complemented
30	C₅×#117	—	4	2	no	—
31	C₅×#117	—	4	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
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×C₄²×C₅	—	2	1	no	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	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	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₄×C₅×C₈	—	2	4	no	—
52	C₄×C₅×C₈	—	4	2	no	—
53	C₅×#95	—	2	1	no	—
54	C₅×#95	—	2	1	no	—
55	C₅×#95	—	2	1	no	—
56	#2293	C₂×C₅	1	1	yes	characteristic
57	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	#1744	C₂×C₅	1	1	yes	characteristic, complemented
59	#1271	C₂×C₅	1	2	yes	complemented
60	#1270	C₂×C₅	1	2	yes	—
61	C₂⁴×C₅	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₄⋊C₄	—	4	1	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₅×C₂²⋊C₄	—	4	2	no	—
95	C₅×C₂²⋊C₄	—	4	1	no	—
96	C₅×C₂²⋊C₄	—	4	1	no	—
97	C₅×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₄²×C₅	—	4	2	no	—
104	C₄²×C₅	C₂×C₈	1	2	yes	—
105	C₄²×C₅	M₁₆	1	2	yes	—
106	C₄²×C₅	D₈	1	2	yes	—
107	C₄²×C₅	SD₁₆	1	2	yes	—
108	C₂×C₄×D₄	C₄×C₅	1	1	yes	characteristic
109	C₂×#117	—	2	1	no	—
110	C₂×C₄⋊C₈	—	2	1	no	—
111	C₂×C₄×C₈	—	2	1	no	—
112	C₂×C₄×C₈	—	2	1	no	—
113	#390	C₄×C₅	1	1	yes	characteristic
114	#390	C₂²×C₅	1	1	yes	characteristic
115	#389	C₄×C₅	1	1	yes	characteristic
116	#389	C₂²×C₅	1	1	yes	characteristic
117	#388	—	2	1	no	—
118	#388	—	2	1	no	—
119	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
120	C₄×C₂²⋊C₄	—	2	1	no	—
121	C₄³	C₂²×C₅	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#336	C₂²×C₅	1	2	yes	—
124	#336	C₂²×C₅	1	2	yes	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	—	4	1	no	—
127	C₂³×C₅	—	4	1	no	—
128	C₂³×C₅	—	4	1	no	—
129	C₂³×C₅	—	4	1	no	—
130	C₂³×C₅	—	4	1	no	—
131	C₂³×C₅	—	4	1	no	—
132	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	2	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	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	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	8	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
158	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
159	C₂×C₄×C₅	#102	1	2	yes	—
160	C₂×C₄×C₅	#102	1	2	yes	—
161	C₂×C₄×C₅	#104	1	2	yes	—
162	C₂×C₄×C₅	#104	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₅	#135	1	2	yes	—
168	C₅×C₈	—	4	4	no	—
169	C₅×C₈	—	4	4	no	—
170	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
171	C₂³×C₄	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	C₅×C₈	1	4	yes	complemented
176	#117	—	4	2	no	—
177	#117	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₂²⋊C₄	—	2	1	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
192	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
193	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
194	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	4	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	4	2	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	C₂²×C₅	—	4	2	no	—
203	C₂²×C₅	—	4	2	no	—
204	C₂²×C₅	—	4	2	no	—
205	C₂²×C₅	—	8	1	no	—
206	C₂²×C₅	—	8	1	no	—
207	C₂²×C₅	—	8	1	no	—
208	C₂²×C₅	—	8	1	no	—
209	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
210	C₂²×C₅	#410	1	1	yes	characteristic
211	C₂²×C₅	#413	1	1	yes	characteristic
212	C₂²×C₅	C₂×#102	1	1	yes	characteristic
213	C₂²×C₅	#416	1	1	yes	characteristic
214	C₂²×C₅	C₂×#104	1	1	yes	characteristic
215	C₂²×C₅	C₂×#104	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	8	1	no	—
226	C₄×C₅	—	8	1	no	—
227	C₄×C₅	D₄⋊C₈	1	4	yes	—
228	C₂⁴	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₂×C₅×C₈	1	2	yes	—
272	C₄²	C₅×M₁₆	1	2	yes	—
273	C₄²	C₅×D₈	1	2	yes	—
274	C₄²	C₅×SD₁₆	1	2	yes	—
275	C₂×C₅	—	8	2	no	—
276	C₂×C₅	#1458	1	1	yes	characteristic
277	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
278	C₂×C₅	#1476	1	1	yes	characteristic
279	C₂×C₅	#1500	1	1	yes	characteristic
280	C₂×C₅	#1502	1	1	yes	characteristic
281	C₂×C₅	#1524	1	1	yes	characteristic
282	C₂×C₅	#1540	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	8	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
316	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
317	C₂×C₄	C₅×#102	1	2	yes	—
318	C₂×C₄	C₅×#102	1	2	yes	—
319	C₂×C₄	C₅×#104	1	2	yes	—
320	C₂×C₄	C₅×#104	1	2	yes	—
321	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
325	C₂×C₄	C₅×#135	1	2	yes	—
326	C₈	—	4	4	no	—
327	C₈	—	4	4	no	—
328	C₅	#7560	1	1	yes	characteristic, direct product factor, Sylow
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
337	C₂²	C₅×#410	1	1	yes	characteristic
338	C₂²	C₅×#413	1	1	yes	characteristic
339	C₂²	C₂×C₅×#102	1	1	yes	characteristic
340	C₂²	C₅×#416	1	1	yes	characteristic
341	C₂²	C₂×C₅×#104	1	1	yes	characteristic
342	C₂²	C₂×C₅×#104	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	1	no	—
354	C₄	C₅×D₄⋊C₈	1	4	yes	—
355	C₂	—	8	2	no	—
356	C₂	C₅×#1458	1	1	yes	characteristic
357	C₂	C₂×C₅×D₄⋊C₈	1	1	yes	characteristic
358	C₂	C₅×#1476	1	1	yes	characteristic
359	C₂	C₅×#1500	1	1	yes	characteristic
360	C₂	C₅×#1502	1	1	yes	characteristic
361	C₂	C₅×#1524	1	1	yes	characteristic
362	C₂	C₅×#1540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. C₄×C₅×C₄⋊C₈	C₂	semidirect
3. C₅×#1744	C₂	semidirect
4. C₅×#1271	C₂	semidirect
23. #7560	C₅	direct (N × H)
29. C₄×C₅×D₄	C₈	semidirect
57. C₄×C₄⋊C₈	C₂×C₅	semidirect
58. #1744	C₂×C₅	semidirect
59. #1271	C₂×C₅	semidirect
175. C₄×D₄	C₅×C₈	semidirect
328. C₅	#7560	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
C₄×C₅×C₄⋊C₈	2
C₅×#1744	2
C₅×#1271	2
C₅×#1270	2
#7560	5

Subgroups — C5×#7560 #11930280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7560 #11930280