Subgroups — C₅×#8667 #11931387

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#1806	C₂	1	2	yes	maximal
3	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1303	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×C₈×Q₈	C₄	1	1	yes	characteristic, complemented
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic, complemented
9	C₅×C₈×D₄	—	2	1	no	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	1	yes	characteristic, complemented
12	C₅×C₈×D₄	C₄	1	1	yes	characteristic, complemented
13	C₅×#431	—	2	1	no	—
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₅×#107	—	2	1	no	—
16	C₅×#423	C₂²	1	2	yes	—
17	C₂×C₅×#104	—	2	2	no	—
18	C₂×C₄×C₅×C₈	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₅×#340	—	2	1	no	—
22	C₅×#339	—	2	2	no	—
23	C₅×#335	—	2	1	no	—
24	C₅×#335	—	2	1	no	—
25	#8667	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	D₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×Q₈	Q₈	1	1	yes	characteristic
34	C₄×C₅×D₄	—	2	1	no	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×D₄	Q₈	1	1	yes	characteristic
38	C₅×#117	—	2	1	no	—
39	C₅×#117	D₄	1	1	yes	characteristic
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	—	2	2	no	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#107	—	2	4	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₅×#104	—	4	4	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₅×#95	—	2	2	no	—
63	C₈×#41	C₂×C₅	1	1	yes	characteristic
64	#1806	C₂×C₅	1	2	yes	—
65	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
66	#1567	C₂×C₅	1	1	yes	characteristic, complemented
67	#1303	C₂×C₅	1	2	yes	complemented
68	C₅×#41	—	2	1	no	—
69	C₅×#41	—	2	1	no	—
70	C₅×#41	D₈	1	2	yes	—
71	C₅×#41	Q₁₆	1	2	yes	—
72	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	4	2	no	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₅×M₁₆	—	4	2	no	—
83	C₅×M₁₆	—	4	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₂×C₅×C₈	—	4	2	no	—
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	—	2	1	no	—
97	C₅×C₄⋊C₄	—	2	4	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₅×C₂²⋊C₄	—	2	1	no	—
103	C₅×C₂²⋊C₄	—	2	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₄²×C₅	—	2	1	no	—
106	C₄²×C₅	—	2	1	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₅	#41	1	1	yes	characteristic
113	C₄×#41	C₂²×C₅	1	1	yes	characteristic
114	C₈×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
115	C₈×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
116	C₈×D₄	—	2	1	no	—
117	C₈×D₄	—	2	1	no	—
118	C₈×D₄	C₄×C₅	1	1	yes	characteristic, complemented
119	C₈×D₄	C₄×C₅	1	1	yes	characteristic, complemented
120	#431	—	2	1	no	—
121	#431	C₂²×C₅	1	1	yes	characteristic
122	C₂×#107	—	2	1	no	—
123	#423	C₂²×C₅	1	2	yes	—
124	C₂×#104	—	2	2	no	—
125	C₂×C₄×C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
127	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
128	#340	—	2	1	no	—
129	#339	—	2	2	no	—
130	#335	—	2	1	no	—
131	#335	—	2	1	no	—
132	C₂³×C₅	—	2	1	no	—
133	C₂³×C₅	#121	1	1	yes	characteristic
134	C₅×Q₈	—	2	1	no	—
135	C₅×Q₈	#107	1	2	yes	complemented
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	2	1	no	—
139	C₅×D₄	#107	1	2	yes	complemented
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	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	—	4	4	no	—
152	C₂×C₄×C₅	—	8	2	no	—
153	C₂×C₄×C₅	—	8	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	#117	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	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
162	C₂×C₄×C₅	#135	1	1	yes	characteristic
163	C₂×C₄×C₅	#135	1	1	yes	characteristic
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	4	1	no	—
168	C₅×C₈	—	4	1	no	—
169	C₅×C₈	—	8	2	no	—
170	C₅×C₈	#104	1	2	yes	complemented
171	C₂×#41	C₅×D₄	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
176	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
177	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
182	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	C₅×D₄	1	1	yes	characteristic
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
189	#107	—	2	4	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
195	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
196	#104	—	4	4	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
205	C₄×C₈	C₅×D₄	1	1	yes	characteristic
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	#95	—	2	2	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	4	1	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	#380	1	1	yes	characteristic
214	C₂²×C₅	#466	1	1	yes	characteristic
215	C₂²×C₅	#467	1	1	yes	characteristic
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	4	1	no	—
222	C₄×C₅	—	8	4	no	—
223	C₄×C₅	C₂×#104	1	1	yes	characteristic
224	C₄×C₅	#421	1	1	yes	characteristic
225	C₄×C₅	C₂×#107	1	1	yes	characteristic
226	C₄×C₅	#427	1	1	yes	characteristic
227	C₄×C₅	#466	1	1	yes	characteristic
228	C₄×C₅	#467	1	1	yes	characteristic
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	C₅×D₈	1	2	yes	—
232	#41	C₅×Q₁₆	1	2	yes	—
233	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂²×C₄	C₅×#41	1	1	yes	characteristic
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
271	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
272	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
273	C₄²	C₅×#41	1	1	yes	characteristic
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	—	4	2	no	—
277	C₂×C₅	#1806	1	1	yes	characteristic
278	C₂×C₅	#1808	1	1	yes	characteristic
279	C₂×C₅	#1814	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	C₅×#121	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	C₅×#107	1	2	yes	complemented
284	D₄	—	2	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	1	no	—
287	D₄	C₅×#107	1	2	yes	complemented
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₅×#117	1	1	yes	characteristic
305	C₂×C₄	C₅×#121	1	1	yes	characteristic
306	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
310	C₂×C₄	C₅×#135	1	1	yes	characteristic
311	C₂×C₄	C₅×#135	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	8	2	no	—
318	C₈	C₅×#104	1	2	yes	complemented
319	C₅	#8667	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×#466	1	1	yes	characteristic
327	C₂²	C₅×#467	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	8	4	no	—
335	C₄	C₂×C₅×#104	1	1	yes	characteristic
336	C₄	C₅×#421	1	1	yes	characteristic
337	C₄	C₂×C₅×#107	1	1	yes	characteristic
338	C₄	C₅×#427	1	1	yes	characteristic
339	C₄	C₅×#466	1	1	yes	characteristic
340	C₄	C₅×#467	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	C₅×#1806	1	1	yes	characteristic
345	C₂	C₅×#1808	1	1	yes	characteristic
346	C₂	C₅×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ 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
3. C₄×C₅×#107	C₂	semidirect
4. C₅×#1567	C₂	semidirect
5. C₅×#1303	C₂	semidirect
7. C₅×C₈×Q₈	C₄	semidirect
8. C₅×C₈×Q₈	C₄	semidirect
11. C₅×C₈×D₄	C₄	semidirect
12. C₅×C₈×D₄	C₄	semidirect
25. #8667	C₅	direct (N × H)
65. C₄×#107	C₂×C₅	semidirect
66. #1567	C₂×C₅	semidirect
67. #1303	C₂×C₅	semidirect
114. C₈×Q₈	C₄×C₅	semidirect
115. C₈×Q₈	C₄×C₅	semidirect
118. C₈×D₄	C₄×C₅	semidirect
119. C₈×D₄	C₄×C₅	semidirect
135. C₅×Q₈	#107	semidirect
139. C₅×D₄	#107	semidirect
170. C₅×C₈	#104	semidirect
283. Q₈	C₅×#107	semidirect
287. D₄	C₅×#107	semidirect
318. C₈	C₅×#104	semidirect
319. C₅	#8667	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×C₈×#41	2
C₅×#1806	2
C₄×C₅×#107	2
C₅×#1567	2
C₅×#1303	2
#8667	5

Subgroups — C5×#8667 #11931387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8667 #11931387