Subgroups — C₅×#8604 #11931324

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1477	C₂	1	2	yes	maximal
5	C₅×#1283	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	C₄	1	1	yes	characteristic
8	C₄×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₅×#405	—	2	1	no	—
12	C₅×#405	—	2	1	no	—
13	C₅×#405	C₄	1	1	yes	characteristic
14	C₅×#405	C₄	1	1	yes	characteristic
15	C₄×C₅×M₁₆	—	2	1	no	—
16	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×Q₈⋊C₈	—	2	2	no	—
24	C₅×D₄⋊C₈	—	2	2	no	—
25	#8604	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
27	C₅×#131	—	2	2	no	—
28	C₅×#131	—	2	2	no	—
29	C₅×#131	—	2	2	no	—
30	C₅×#131	C₈	1	4	yes	complemented
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
41	C₄×C₅×Q₈	Q₈	1	1	yes	characteristic
42	C₄×C₅×D₄	—	2	1	no	—
43	C₄×C₅×D₄	—	2	1	no	—
44	C₄×C₅×D₄	D₄	1	1	yes	characteristic
45	C₄×C₅×D₄	Q₈	1	1	yes	characteristic
46	C₅×#117	—	2	1	no	—
47	C₅×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊C₈	—	2	2	no	—
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	2	no	—
54	C₅×#97	—	2	1	no	—
55	C₅×#97	—	2	1	no	—
56	C₅×#97	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	—	2	2	no	—
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	D₄	1	1	yes	characteristic
65	C₄×#131	C₂×C₅	1	1	yes	characteristic
66	#1564	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
68	#1477	C₂×C₅	1	2	yes	—
69	#1283	C₂×C₅	1	2	yes	complemented
70	C₅×#41	—	2	1	no	—
71	C₅×#41	—	2	1	no	—
72	C₅×#41	C₂×C₈	1	2	yes	—
73	C₅×#41	M₁₆	1	2	yes	—
74	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅	#41	1	1	yes	characteristic
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	1	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	C₂×C₈	1	2	yes	—
88	C₅×M₁₆	M₁₆	1	2	yes	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	4	2	no	—
96	C₂×C₅×C₈	—	4	2	no	—
97	C₂×C₅×C₈	—	8	2	no	—
98	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
103	C₂×C₅×C₈	M₁₆	1	2	yes	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
107	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
108	C₅×C₂²⋊C₄	—	2	1	no	—
109	C₅×C₂²⋊C₄	—	2	1	no	—
110	C₅×C₂²⋊C₄	—	2	1	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	—	2	1	no	—
113	C₄²×C₅	—	2	1	no	—
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
118	C₂×#131	C₄×C₅	1	1	yes	characteristic
119	C₂×#131	C₄×C₅	1	1	yes	characteristic
120	C₄×#41	C₂²×C₅	1	1	yes	characteristic
121	#431	C₂²×C₅	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
123	#405	—	2	1	no	—
124	#405	—	2	1	no	—
125	#405	C₄×C₅	1	1	yes	characteristic
126	#405	C₄×C₅	1	1	yes	characteristic
127	C₄×M₁₆	—	2	1	no	—
128	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	1	no	—
130	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
131	#335	—	2	1	no	—
132	#334	—	2	1	no	—
133	#334	—	2	1	no	—
134	#334	—	2	1	no	—
135	Q₈⋊C₈	—	2	2	no	—
136	D₄⋊C₈	—	2	2	no	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	#117	1	1	yes	characteristic
139	C₅×Q₈	—	2	1	no	—
140	C₅×Q₈	C₄⋊C₈	1	2	yes	—
141	C₅×D₄	—	2	2	no	—
142	C₅×D₄	—	2	2	no	—
143	C₅×D₄	—	2	1	no	—
144	C₅×D₄	C₄⋊C₈	1	2	yes	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	2	2	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	8	2	no	—
169	C₅×C₈	—	8	2	no	—
170	C₅×C₈	C₂²⋊C₈	1	2	yes	—
171	C₅×C₈	C₂²⋊C₈	1	2	yes	—
172	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	—	2	2	no	—
176	#131	C₅×C₈	1	4	yes	complemented
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
186	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
187	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	C₅×D₄	1	1	yes	characteristic
191	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
192	#117	—	2	1	no	—
193	#117	C₂×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄×C₈	C₅×D₄	1	1	yes	characteristic
209	C₄×C₈	C₅×D₄	1	1	yes	characteristic
210	C₄×C₈	C₅×D₄	1	1	yes	characteristic
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	4	1	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	#380	1	1	yes	characteristic
217	C₂²×C₅	#431	1	1	yes	characteristic
218	C₂²×C₅	#432	1	1	yes	characteristic
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	4	1	no	—
224	C₄×C₅	—	4	1	no	—
225	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
226	C₄×C₅	#407	1	1	yes	characteristic
227	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
228	C₄×C₅	#423	1	1	yes	characteristic
229	C₄×C₅	#471	1	2	yes	—
230	#41	—	2	1	no	—
231	#41	—	2	1	no	—
232	#41	C₂×C₅×C₈	1	2	yes	—
233	#41	C₅×M₁₆	1	2	yes	—
234	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₅×#41	1	1	yes	characteristic
242	C₂²×C₄	C₅×#41	1	1	yes	characteristic
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	C₂×C₅×C₈	1	2	yes	—
248	M₁₆	C₅×M₁₆	1	2	yes	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	8	2	no	—
258	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
263	C₂×C₈	C₅×M₁₆	1	2	yes	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
267	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
277	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	4	2	no	—
281	C₂×C₅	#1797	1	1	yes	characteristic
282	C₂×C₅	#1814	1	1	yes	characteristic
283	C₂×C₅	#1815	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	C₅×#117	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	C₅×C₄⋊C₈	1	2	yes	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	2	1	no	—
291	D₄	C₅×C₄⋊C₈	1	2	yes	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₅×#121	1	1	yes	characteristic
305	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
306	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	—	8	2	no	—
316	C₈	—	8	2	no	—
317	C₈	C₅×C₂²⋊C₈	1	2	yes	—
318	C₈	C₅×C₂²⋊C₈	1	2	yes	—
319	C₅	#8604	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₅×#380	1	1	yes	characteristic
326	C₂²	C₅×#431	1	1	yes	characteristic
327	C₂²	C₅×#432	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
335	C₄	C₅×#407	1	1	yes	characteristic
336	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
337	C₄	C₅×#423	1	1	yes	characteristic
338	C₄	C₅×#471	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	C₅×#1797	1	1	yes	characteristic
343	C₂	C₅×#1814	1	1	yes	characteristic
344	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1564	C₂	semidirect
3. C₂×C₅×#334	C₂	semidirect
5. C₅×#1283	C₂	semidirect
25. #8604	C₅	direct (N × H)
30. C₅×#131	C₈	semidirect
66. #1564	C₂×C₅	semidirect
67. C₂×#334	C₂×C₅	semidirect
69. #1283	C₂×C₅	semidirect
176. #131	C₅×C₈	semidirect
319. C₅	#8604	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×#131	2
C₅×#1564	2
C₂×C₅×#334	2
C₅×#1477	2
C₅×#1283	2
#8604	5

Subgroups — C5×#8604 #11931324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8604 #11931324