Subgroups — C₅×#8666 #11931386

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1566	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×#106	—	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×#334	—	2	1	no	—
24	C₅×#334	—	2	1	no	—
25	#8666	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×#106	—	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×#106	C₂×C₅	1	1	yes	characteristic, complemented
66	#1566	C₂×C₅	1	1	yes	characteristic, complemented
67	#1303	C₂×C₅	1	2	yes	complemented
68	C₅×#41	—	2	2	no	—
69	C₅×#41	SD₁₆	1	4	yes	—
70	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	4	2	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	1	yes	characteristic
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	4	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
98	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₅×C₂²⋊C₄	—	2	1	no	—
100	C₅×C₂²⋊C₄	—	2	1	no	—
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₄²×C₅	—	4	2	no	—
106	C₄²×C₅	—	4	2	no	—
107	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₅	#41	1	1	yes	characteristic
111	C₄×#41	C₂²×C₅	1	1	yes	characteristic
112	C₈×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
113	C₈×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
114	C₈×D₄	—	2	1	no	—
115	C₈×D₄	—	2	1	no	—
116	C₈×D₄	C₄×C₅	1	1	yes	characteristic, complemented
117	C₈×D₄	C₄×C₅	1	1	yes	characteristic, complemented
118	#431	—	2	1	no	—
119	#431	C₂²×C₅	1	1	yes	characteristic
120	C₂×#106	—	2	1	no	—
121	#423	C₂²×C₅	1	2	yes	—
122	C₂×#104	—	2	2	no	—
123	C₂×C₄×C₈	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
126	#340	—	2	1	no	—
127	#339	—	2	2	no	—
128	#334	—	2	1	no	—
129	#334	—	2	1	no	—
130	C₂³×C₅	—	2	1	no	—
131	C₂³×C₅	#121	1	1	yes	characteristic
132	C₅×Q₈	—	2	1	no	—
133	C₅×Q₈	#106	1	2	yes	complemented
134	C₅×D₄	—	2	4	no	—
135	C₅×D₄	—	2	1	no	—
136	C₅×D₄	#106	1	2	yes	complemented
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	4	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	—	8	2	no	—
149	C₂×C₄×C₅	—	8	2	no	—
150	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	#117	1	1	yes	characteristic
153	C₂×C₄×C₅	#121	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₅	#135	1	2	yes	—
158	C₅×C₈	—	2	1	no	—
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	—	2	1	no	—
161	C₅×C₈	—	4	1	no	—
162	C₅×C₈	—	4	1	no	—
163	C₅×C₈	—	8	2	no	—
164	C₅×C₈	#104	1	2	yes	complemented
165	C₂×#41	C₅×D₄	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
171	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
176	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
177	#117	—	2	1	no	—
178	#117	C₅×D₄	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	1	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
183	#106	—	2	4	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
189	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
190	#104	—	4	4	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	1	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
198	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
199	C₄×C₈	C₅×D₄	1	1	yes	characteristic
200	C₄×C₈	C₅×D₄	1	1	yes	characteristic
201	#95	—	2	2	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	4	1	no	—
206	C₂²×C₅	—	4	1	no	—
207	C₂²×C₅	#380	1	1	yes	characteristic
208	C₂²×C₅	#465	1	2	yes	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	4	1	no	—
215	C₄×C₅	—	8	4	no	—
216	C₄×C₅	C₂×#104	1	1	yes	characteristic
217	C₄×C₅	#421	1	1	yes	characteristic
218	C₄×C₅	C₂×#106	1	1	yes	characteristic
219	C₄×C₅	#427	1	1	yes	characteristic
220	C₄×C₅	#465	1	2	yes	—
221	#41	—	2	2	no	—
222	#41	C₅×SD₁₆	1	4	yes	—
223	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₅×#41	1	1	yes	characteristic
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
251	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
261	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
262	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
263	C₄²	C₅×#41	1	1	yes	characteristic
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#1806	1	1	yes	characteristic
267	C₂×C₅	#1807	1	1	yes	characteristic
268	C₂×C₅	#1814	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	C₂³	C₅×#121	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	C₅×#106	1	2	yes	complemented
273	D₄	—	2	4	no	—
274	D₄	—	2	1	no	—
275	D₄	C₅×#106	1	2	yes	complemented
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₅×#117	1	1	yes	characteristic
292	C₂×C₄	C₅×#121	1	1	yes	characteristic
293	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
296	C₂×C₄	C₅×#135	1	2	yes	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	8	2	no	—
303	C₈	C₅×#104	1	2	yes	complemented
304	C₅	#8666	1	1	yes	characteristic, direct product factor, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	C₅×#380	1	1	yes	characteristic
311	C₂²	C₅×#465	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	4	no	—
319	C₄	C₂×C₅×#104	1	1	yes	characteristic
320	C₄	C₅×#421	1	1	yes	characteristic
321	C₄	C₂×C₅×#106	1	1	yes	characteristic
322	C₄	C₅×#427	1	1	yes	characteristic
323	C₄	C₅×#465	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₅×#1806	1	1	yes	characteristic
327	C₂	C₅×#1807	1	1	yes	characteristic
328	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×#106	C₂	semidirect
4. C₅×#1566	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. #8666	C₅	direct (N × H)
65. C₄×#106	C₂×C₅	semidirect
66. #1566	C₂×C₅	semidirect
67. #1303	C₂×C₅	semidirect
112. C₈×Q₈	C₄×C₅	semidirect
113. C₈×Q₈	C₄×C₅	semidirect
116. C₈×D₄	C₄×C₅	semidirect
117. C₈×D₄	C₄×C₅	semidirect
133. C₅×Q₈	#106	semidirect
136. C₅×D₄	#106	semidirect
164. C₅×C₈	#104	semidirect
272. Q₈	C₅×#106	semidirect
275. D₄	C₅×#106	semidirect
303. C₈	C₅×#104	semidirect
304. C₅	#8666	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×#106	2
C₅×#1566	2
C₅×#1303	2
#8666	5

Subgroups — C5×#8666 #11931386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8666 #11931386