Subgroups — C₅×#8204 #11930924

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×C₈	—	2	1	no	—
9	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₅×C₈×D₄	—	2	2	no	—
11	C₅×C₈×D₄	C₄	1	4	yes	complemented
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#102	C₄	1	2	yes	—
15	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₅×#97	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅×#340	C₄	1	2	yes	—
22	C₅×#336	—	2	1	no	—
23	C₅×D₄⋊C₈	C₄	1	4	yes	complemented
24	C₅×#322	—	2	2	no	—
25	#8204	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	—	2	1	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	4	no	—
32	C₂²×C₅×C₈	—	4	2	no	—
33	C₂²×C₅×C₈	—	4	2	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
40	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₂×C₄²×C₅	—	2	1	no	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₅×#102	—	2	4	no	—
51	C₅×C₂²⋊C₈	—	2	4	no	—
52	C₅×#97	—	4	2	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
57	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₅×#95	—	2	1	no	—
60	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
61	#1770	C₂×C₅	1	1	yes	characteristic, complemented
62	C₄×#102	C₂×C₅	1	1	yes	characteristic
63	C₂×D₄⋊C₈	C₂×C₅	1	1	yes	characteristic
64	C₂×#322	C₂×C₅	1	1	yes	characteristic, complemented
65	#1279	C₂×C₅	1	1	yes	characteristic, complemented
66	#1277	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂⁴×C₅	—	2	1	no	—
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	C₄²	1	2	yes	—
71	C₂×C₅×D₄	M₁₆	1	4	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	#41	1	1	yes	characteristic
83	C₂×C₅×C₈	—	2	4	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	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	4	8	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	—	4	1	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	1	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	C₄²	1	2	yes	—
105	C₅×C₂²⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₄²×C₅	—	4	1	no	—
109	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	#41	1	1	yes	characteristic
113	C₂³×C₈	—	2	1	no	—
114	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
115	C₈×D₄	—	2	2	no	—
116	C₈×D₄	C₄×C₅	1	4	yes	complemented
117	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
119	C₂×#102	C₄×C₅	1	2	yes	—
120	C₂×C₂²⋊C₈	—	2	1	no	—
121	C₂×#97	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
126	#340	C₄×C₅	1	2	yes	—
127	#336	—	2	1	no	—
128	D₄⋊C₈	C₄×C₅	1	4	yes	complemented
129	#322	—	2	2	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	4	no	—
132	C₂³×C₅	—	4	1	no	—
133	C₂³×C₅	—	4	1	no	—
134	C₂³×C₅	—	4	2	no	—
135	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	4	no	—
138	C₅×D₄	#97	1	4	yes	complemented
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	4	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	#117	1	1	yes	characteristic
153	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
157	C₂×C₄×C₅	#131	1	2	yes	—
158	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×C₈	—	2	4	no	—
163	C₅×C₈	—	2	2	no	—
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	—	4	2	no	—
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	—	4	2	no	—
168	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂³×C₄	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	4	2	no	—
175	C₂²×C₈	—	4	2	no	—
176	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
177	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
182	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
191	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
192	#102	—	2	4	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	#97	—	4	2	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
199	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
200	C₄×C₈	C₅×D₄	1	2	yes	—
201	#95	—	2	1	no	—
202	C₂²×C₅	—	2	4	no	—
203	C₂²×C₅	—	4	4	no	—
204	C₂²×C₅	—	4	4	no	—
205	C₂²×C₅	—	4	2	no	—
206	C₂²×C₅	—	4	2	no	—
207	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₅	#435	1	2	yes	—
209	C₂²×C₅	#440	1	1	yes	characteristic
210	C₂²×C₅	#442	1	1	yes	characteristic
211	C₂²×C₅	#443	1	2	yes	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	—	8	2	no	—
216	C₄×C₅	C₂×#97	1	1	yes	characteristic
217	C₄×C₅	#405	1	1	yes	characteristic
218	C₄×C₅	#417	1	1	yes	characteristic
219	C₄×C₅	#418	1	1	yes	characteristic
220	C₄×C₅	#435	1	2	yes	—
221	C₄×C₅	#440	1	1	yes	characteristic
222	C₄×C₅	#442	1	1	yes	characteristic
223	C₂⁴	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	C₄²×C₅	1	2	yes	—
227	C₂×D₄	C₅×M₁₆	1	4	yes	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂²×C₄	C₅×#41	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	8	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
256	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₄²×C₅	1	2	yes	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
266	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
267	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
268	C₄²	C₅×#41	1	1	yes	characteristic
269	C₂×C₅	—	4	4	no	—
270	C₂×C₅	#1585	1	2	yes	—
271	C₂×C₅	#1586	1	2	yes	—
272	C₂×C₅	#1752	1	1	yes	characteristic
273	C₂×C₅	#1762	1	1	yes	characteristic
274	C₂×C₅	#1764	1	1	yes	characteristic
275	C₂³	—	2	2	no	—
276	C₂³	—	2	4	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
281	D₄	—	2	2	no	—
282	D₄	—	2	4	no	—
283	D₄	C₅×#97	1	4	yes	complemented
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
296	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₅×#117	1	1	yes	characteristic
298	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
302	C₂×C₄	C₅×#131	1	2	yes	—
303	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₄	C₅×#137	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₅	#8204	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	4	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
320	C₂²	C₅×#435	1	2	yes	—
321	C₂²	C₅×#440	1	1	yes	characteristic
322	C₂²	C₅×#442	1	1	yes	characteristic
323	C₂²	C₅×#443	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₂×C₅×#97	1	1	yes	characteristic
329	C₄	C₅×#405	1	1	yes	characteristic
330	C₄	C₅×#417	1	1	yes	characteristic
331	C₄	C₅×#418	1	1	yes	characteristic
332	C₄	C₅×#435	1	2	yes	—
333	C₄	C₅×#440	1	1	yes	characteristic
334	C₄	C₅×#442	1	1	yes	characteristic
335	C₂	—	4	4	no	—
336	C₂	C₅×#1585	1	2	yes	—
337	C₂	C₅×#1586	1	2	yes	—
338	C₂	C₅×#1752	1	1	yes	characteristic
339	C₂	C₅×#1762	1	1	yes	characteristic
340	C₂	C₅×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1770	C₂	semidirect
5. C₂×C₅×#322	C₂	semidirect
6. C₅×#1279	C₂	semidirect
7. C₅×#1277	C₂	semidirect
11. C₅×C₈×D₄	C₄	semidirect
23. C₅×D₄⋊C₈	C₄	semidirect
25. #8204	C₅	direct (N × H)
61. #1770	C₂×C₅	semidirect
64. C₂×#322	C₂×C₅	semidirect
65. #1279	C₂×C₅	semidirect
66. #1277	C₂×C₅	semidirect
116. C₈×D₄	C₄×C₅	semidirect
128. D₄⋊C₈	C₄×C₅	semidirect
138. C₅×D₄	#97	semidirect
283. D₄	C₅×#97	semidirect
313. C₅	#8204	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×C₈×D₄	2
C₅×#1770	2
C₄×C₅×#102	2
C₂×C₅×D₄⋊C₈	2
C₂×C₅×#322	2
C₅×#1279	2
C₅×#1277	2
#8204	5

Subgroups — C5×#8204 #11930924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8204 #11930924