Subgroups — C₅×#8588 #11931308

Order 1280.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1831	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#331	C₂	1	2	yes	maximal
5	C₅×#1296	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#127	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	—	2	1	no	—
8	C₂×C₄×C₅×D₄	—	2	1	no	—
9	C₂×C₄×C₅×D₄	C₄	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₅×#390	—	2	1	no	—
15	C₅×#390	—	2	1	no	—
16	C₅×#390	C₄	1	2	yes	—
17	C₅×#384	C₂²	1	1	yes	characteristic
18	C₄³×C₅	C₂²	1	1	yes	characteristic
19	C₅×#336	—	2	1	no	—
20	C₅×#336	—	2	1	no	—
21	C₅×#331	—	2	4	no	—
22	#8588	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	—	2	1	no	—
24	C₂²×C₅×D₄	—	2	1	no	—
25	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₅×#127	—	2	1	no	—
30	C₅×#127	—	2	1	no	—
31	C₅×#127	D₄	1	2	yes	—
32	C₅×#127	Q₈	1	2	yes	—
33	C₄×C₅×D₄	—	4	4	no	—
34	C₄×C₅×D₄	—	4	4	no	—
35	C₄×C₅×D₄	—	4	2	no	—
36	C₄×C₅×D₄	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
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×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	—	4	2	no	—
55	C₅×C₄⋊C₈	—	4	2	no	—
56	C₄×#127	C₂×C₅	1	1	yes	characteristic
57	#1842	C₂×C₅	1	1	yes	characteristic, complemented
58	#1831	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#331	C₂×C₅	1	2	yes	—
60	#1296	C₂×C₅	1	2	yes	complemented
61	C₂⁴×C₅	—	2	2	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	4	2	no	—
68	C₂×C₅×D₄	—	4	2	no	—
69	C₂×C₅×D₄	—	4	1	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂×C₅×D₄	—	4	2	no	—
73	C₂×C₅×D₄	—	4	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂×C₅×D₄	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	4	4	no	—
78	C₂²×C₄×C₅	—	4	4	no	—
79	C₂²×C₄×C₅	—	4	4	no	—
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	—	4	2	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	—	4	2	no	—
92	C₂×C₅×C₈	—	4	2	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₅×C₄⋊C₄	—	4	1	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₂²⋊C₄	—	4	4	no	—
98	C₅×C₂²⋊C₄	—	4	4	no	—
99	C₅×C₂²⋊C₄	—	4	4	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	2	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₄²×C₅	—	4	1	no	—
106	C₄²×C₅	D₈	1	2	yes	—
107	C₄²×C₅	SD₁₆	1	2	yes	—
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	C₄²×C₅	#41	1	1	yes	characteristic
112	C₂×#127	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×D₄	—	2	1	no	—
114	C₂×C₄×D₄	—	2	1	no	—
115	C₂×C₄×D₄	C₄×C₅	1	2	yes	—
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
120	#390	—	2	1	no	—
121	#390	—	2	1	no	—
122	#390	C₄×C₅	1	2	yes	—
123	#384	C₂²×C₅	1	1	yes	characteristic
124	C₄³	C₂²×C₅	1	1	yes	characteristic
125	#336	—	2	1	no	—
126	#336	—	2	1	no	—
127	#331	—	2	4	no	—
128	C₂³×C₅	—	2	4	no	—
129	C₂³×C₅	—	4	2	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	—	4	2	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	4	2	no	—
134	C₂³×C₅	—	4	2	no	—
135	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₅×D₄	—	4	4	no	—
137	C₅×D₄	—	4	4	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	4	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	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	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	8	8	no	—
159	C₂×C₄×C₅	#100	1	2	yes	—
160	C₂×C₄×C₅	#102	1	2	yes	—
161	C₂×C₄×C₅	#102	1	2	yes	—
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	#117	1	1	yes	characteristic
164	C₂×C₄×C₅	#121	1	1	yes	characteristic
165	C₂×C₄×C₅	#121	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
170	C₂×C₄×C₅	#135	1	2	yes	—
171	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₅	#137	1	1	yes	characteristic
174	C₅×C₈	—	8	2	no	—
175	C₅×C₈	—	8	2	no	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
179	C₂³×C₄	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	#127	—	2	1	no	—
183	#127	—	2	1	no	—
184	#127	C₅×D₄	1	2	yes	—
185	#127	C₅×Q₈	1	2	yes	—
186	C₄×D₄	—	4	4	no	—
187	C₄×D₄	—	4	4	no	—
188	C₄×D₄	—	4	2	no	—
189	C₄×D₄	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
199	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	4	2	no	—
208	C₄⋊C₈	—	4	2	no	—
209	C₂²×C₅	—	4	4	no	—
210	C₂²×C₅	—	4	4	no	—
211	C₂²×C₅	—	4	4	no	—
212	C₂²×C₅	—	8	2	no	—
213	C₂²×C₅	—	8	2	no	—
214	C₂²×C₅	—	8	2	no	—
215	C₂²×C₅	—	8	2	no	—
216	C₂²×C₅	#380	1	1	yes	characteristic
217	C₂²×C₅	C₂×#100	1	1	yes	characteristic
218	C₂²×C₅	#413	1	1	yes	characteristic
219	C₂²×C₅	C₂×#102	1	1	yes	characteristic
220	C₂²×C₅	#416	1	1	yes	characteristic
221	C₂²×C₅	#417	1	1	yes	characteristic
222	C₂²×C₅	#418	1	1	yes	characteristic
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	#459	1	1	yes	characteristic
231	C₄×C₅	#461	1	1	yes	characteristic
232	C₄×C₅	#463	1	2	yes	—
233	C₄×C₅	#474	1	1	yes	characteristic
234	C₄×C₅	#476	1	1	yes	characteristic
235	C₄×C₅	#478	1	2	yes	—
236	C₂⁴	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	C₅×C₄⋊C₄	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	4	no	—
253	C₂²×C₄	—	4	4	no	—
254	C₂²×C₄	—	4	4	no	—
255	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₂²⋊C₄	—	4	4	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	C₅×D₈	1	2	yes	—
282	C₄²	C₅×SD₁₆	1	2	yes	—
283	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
284	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
285	C₄²	C₅×#41	1	1	yes	characteristic
286	C₄²	C₅×#41	1	1	yes	characteristic
287	C₂×C₅	—	8	4	no	—
288	C₂×C₅	#1540	1	1	yes	characteristic
289	C₂×C₅	#1545	1	1	yes	characteristic
290	C₂×C₅	#1551	1	1	yes	characteristic
291	C₂×C₅	#1557	1	1	yes	characteristic
292	C₂×C₅	#1800	1	1	yes	characteristic
293	C₂×C₅	#1802	1	1	yes	characteristic
294	C₂×C₅	#1804	1	1	yes	characteristic
295	C₂³	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	8	no	—
326	C₂×C₄	C₅×#100	1	2	yes	—
327	C₂×C₄	C₅×#102	1	2	yes	—
328	C₂×C₄	C₅×#102	1	2	yes	—
329	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₅×#117	1	1	yes	characteristic
331	C₂×C₄	C₅×#121	1	1	yes	characteristic
332	C₂×C₄	C₅×#121	1	1	yes	characteristic
333	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
334	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
335	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
336	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
337	C₂×C₄	C₅×#135	1	2	yes	—
338	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
339	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
340	C₂×C₄	C₅×#137	1	1	yes	characteristic
341	C₈	—	8	2	no	—
342	C₈	—	8	2	no	—
343	C₅	#8588	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	4	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	C₅×#380	1	1	yes	characteristic
352	C₂²	C₂×C₅×#100	1	1	yes	characteristic
353	C₂²	C₅×#413	1	1	yes	characteristic
354	C₂²	C₂×C₅×#102	1	1	yes	characteristic
355	C₂²	C₅×#416	1	1	yes	characteristic
356	C₂²	C₅×#417	1	1	yes	characteristic
357	C₂²	C₅×#418	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	8	2	no	—
365	C₄	C₅×#459	1	1	yes	characteristic
366	C₄	C₅×#461	1	1	yes	characteristic
367	C₄	C₅×#463	1	2	yes	—
368	C₄	C₅×#474	1	1	yes	characteristic
369	C₄	C₅×#476	1	1	yes	characteristic
370	C₄	C₅×#478	1	2	yes	—
371	C₂	—	8	4	no	—
372	C₂	C₅×#1540	1	1	yes	characteristic
373	C₂	C₅×#1545	1	1	yes	characteristic
374	C₂	C₅×#1551	1	1	yes	characteristic
375	C₂	C₅×#1557	1	1	yes	characteristic
376	C₂	C₅×#1800	1	1	yes	characteristic
377	C₂	C₅×#1802	1	1	yes	characteristic
378	C₂	C₅×#1804	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1842	C₂	semidirect
3. C₅×#1831	C₂	semidirect
5. C₅×#1296	C₂	semidirect
22. #8588	C₅	direct (N × H)
57. #1842	C₂×C₅	semidirect
58. #1831	C₂×C₅	semidirect
60. #1296	C₂×C₅	semidirect
343. C₅	#8588	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×#127	2
C₅×#1842	2
C₅×#1831	2
C₂×C₅×#331	2
C₅×#1296	2
#8588	5

Subgroups — C5×#8588 #11931308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8588 #11931308