Subgroups — C₅×#8605 #11931325

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#335	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×#335	—	2	1	no	—
21	C₅×#335	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×Q₈⋊C₈	—	2	2	no	—
24	C₅×D₄⋊C₈	—	2	2	no	—
25	#8605	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×#335	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	#335	—	2	1	no	—
133	#335	—	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	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	—	4	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	#121	1	1	yes	characteristic
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
166	C₅×C₈	—	2	2	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	2	2	no	—
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	8	2	no	—
171	C₅×C₈	—	8	2	no	—
172	C₅×C₈	C₂²⋊C₈	1	2	yes	—
173	C₅×C₈	C₂²⋊C₈	1	2	yes	—
174	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
175	#131	—	2	2	no	—
176	#131	—	2	2	no	—
177	#131	—	2	2	no	—
178	#131	C₅×C₈	1	4	yes	complemented
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
188	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
189	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	C₅×D₄	1	1	yes	characteristic
193	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	#97	—	2	1	no	—
203	#97	—	2	1	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	C₅×D₄	1	1	yes	characteristic
210	C₄×C₈	C₅×D₄	1	1	yes	characteristic
211	C₄×C₈	C₅×D₄	1	1	yes	characteristic
212	C₄×C₈	C₅×D₄	1	1	yes	characteristic
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	4	1	no	—
217	C₂²×C₅	—	4	1	no	—
218	C₂²×C₅	#380	1	1	yes	characteristic
219	C₂²×C₅	#431	1	1	yes	characteristic
220	C₂²×C₅	#432	1	1	yes	characteristic
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	4	1	no	—
226	C₄×C₅	—	4	1	no	—
227	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
228	C₄×C₅	#407	1	1	yes	characteristic
229	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
230	C₄×C₅	#423	1	1	yes	characteristic
231	C₄×C₅	#472	1	1	yes	characteristic
232	C₄×C₅	#473	1	1	yes	characteristic
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	C₂×C₅×C₈	1	2	yes	—
236	#41	C₅×M₁₆	1	2	yes	—
237	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₅×#41	1	1	yes	characteristic
245	C₂²×C₄	C₅×#41	1	1	yes	characteristic
246	M₁₆	—	2	4	no	—
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	C₂×C₅×C₈	1	2	yes	—
251	M₁₆	C₅×M₁₆	1	2	yes	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	8	2	no	—
261	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
266	C₂×C₈	C₅×M₁₆	1	2	yes	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
270	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
278	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
279	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
280	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	4	2	no	—
284	C₂×C₅	#1797	1	1	yes	characteristic
285	C₂×C₅	#1814	1	1	yes	characteristic
286	C₂×C₅	#1816	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	C₅×#117	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	C₅×C₄⋊C₈	1	2	yes	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	1	no	—
294	D₄	C₅×C₄⋊C₈	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	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	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₅×#121	1	1	yes	characteristic
309	C₂×C₄	C₅×#121	1	1	yes	characteristic
310	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
311	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
314	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	—	8	2	no	—
321	C₈	—	8	2	no	—
322	C₈	C₅×C₂²⋊C₈	1	2	yes	—
323	C₈	C₅×C₂²⋊C₈	1	2	yes	—
324	C₅	#8605	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₅×#380	1	1	yes	characteristic
331	C₂²	C₅×#431	1	1	yes	characteristic
332	C₂²	C₅×#432	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
340	C₄	C₅×#407	1	1	yes	characteristic
341	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
342	C₄	C₅×#423	1	1	yes	characteristic
343	C₄	C₅×#472	1	1	yes	characteristic
344	C₄	C₅×#473	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	2	no	—
348	C₂	C₅×#1797	1	1	yes	characteristic
349	C₂	C₅×#1814	1	1	yes	characteristic
350	C₂	C₅×#1816	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×#335	C₂	semidirect
5. C₅×#1283	C₂	semidirect
25. #8605	C₅	direct (N × H)
30. C₅×#131	C₈	semidirect
66. #1564	C₂×C₅	semidirect
67. C₂×#335	C₂×C₅	semidirect
69. #1283	C₂×C₅	semidirect
178. #131	C₅×C₈	semidirect
324. C₅	#8605	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×#335	2
C₅×#1477	2
C₅×#1283	2
#8605	5

Subgroups — C5×#8605 #11931325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8605 #11931325