Subgroups — C₅×#8626 #11931346

Order 1280.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2929	C₂	1	1	yes	maximal, characteristic
2	C₅×#1810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1807	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1513	C₂	1	2	yes	maximal
5	C₅×#1305	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#525	C₂²	1	1	yes	characteristic
9	C₅×#435	—	2	1	no	—
10	C₅×#435	—	2	1	no	—
11	C₅×C₈×D₄	—	2	1	no	—
12	C₅×C₈×D₄	—	2	1	no	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#106	—	2	1	no	—
15	C₅×#424	C₄	1	1	yes	characteristic, complemented
16	C₅×#408	C₄	1	1	yes	characteristic, complemented
17	C₅×#407	C₂²	1	2	yes	—
18	C₅×#407	C₂²	1	1	yes	characteristic
19	C₅×#405	C₄	1	1	yes	characteristic, complemented
20	C₅×#380	C₂²	1	2	yes	—
21	C₅×#341	—	2	1	no	—
22	C₅×#340	—	2	1	no	—
23	C₅×#328	—	2	2	no	—
24	C₅×#327	—	2	2	no	—
25	#8626	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₅×#131	—	4	2	no	—
29	C₂×C₅×M₁₆	—	2	1	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	—	4	2	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₅×#123	—	2	1	no	—
42	C₅×C₂²⋊Q₈	D₄	1	1	yes	characteristic
43	C₅×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
44	C₅×#121	D₄	1	1	yes	characteristic
45	C₅×#121	Q₈	1	1	yes	characteristic
46	C₅×#120	—	2	1	no	—
47	C₄×C₅×D₄	—	2	1	no	—
48	C₄×C₅×D₄	—	2	1	no	—
49	C₅×#117	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
52	C₅×#108	—	4	2	no	—
53	C₅×#106	—	4	2	no	—
54	C₅×C₄⋊C₈	—	2	1	no	—
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×C₂²⋊C₈	—	2	1	no	—
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₅×C₂²⋊C₈	—	2	1	no	—
59	C₅×C₂²⋊C₈	—	2	1	no	—
60	C₅×C₂²⋊C₈	—	2	2	no	—
61	C₅×C₂²⋊C₈	—	2	2	no	—
62	C₅×#97	—	2	1	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	C₅×#95	—	2	2	no	—
65	#2929	C₂×C₅	1	1	yes	characteristic
66	#1810	C₂×C₅	1	1	yes	characteristic, complemented
67	#1807	C₂×C₅	1	1	yes	characteristic, complemented
68	#1513	C₂×C₅	1	2	yes	—
69	#1305	C₂×C₅	1	2	yes	complemented
70	C₂⁴×C₅	#41	1	1	yes	characteristic
71	C₅×#41	—	4	1	no	—
72	C₅×#41	—	4	1	no	—
73	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₅×D₄	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	—	2	1	no	—
77	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	#41	1	1	yes	characteristic
88	C₅×M₁₆	—	4	2	no	—
89	C₅×M₁₆	—	4	1	no	—
90	C₅×M₁₆	—	4	1	no	—
91	C₅×M₁₆	—	4	2	no	—
92	C₅×M₁₆	—	4	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	4	2	no	—
102	C₂×C₅×C₈	—	4	2	no	—
103	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
113	C₅×C₂²⋊C₄	—	2	1	no	—
114	C₅×C₂²⋊C₄	—	2	1	no	—
115	C₅×C₂²⋊C₄	—	2	1	no	—
116	C₅×C₂²⋊C₄	—	2	1	no	—
117	C₅×C₂²⋊C₄	—	2	1	no	—
118	C₅×C₂²⋊C₄	—	8	2	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₂×#131	C₄×C₅	1	1	yes	characteristic, complemented
121	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
122	#525	C₂²×C₅	1	1	yes	characteristic
123	#435	—	2	1	no	—
124	#435	—	2	1	no	—
125	C₈×D₄	—	2	1	no	—
126	C₈×D₄	—	2	1	no	—
127	C₂×#108	—	2	1	no	—
128	C₂×#106	—	2	1	no	—
129	#424	C₄×C₅	1	1	yes	characteristic, complemented
130	#408	C₄×C₅	1	1	yes	characteristic, complemented
131	#407	C₂²×C₅	1	2	yes	—
132	#407	C₂²×C₅	1	1	yes	characteristic
133	#405	C₄×C₅	1	1	yes	characteristic, complemented
134	#380	C₂²×C₅	1	2	yes	—
135	#341	—	2	1	no	—
136	#340	—	2	1	no	—
137	#328	—	2	2	no	—
138	#327	—	2	2	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	1	no	—
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	#117	1	1	yes	characteristic
144	C₂³×C₅	#121	1	1	yes	characteristic
145	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
146	C₅×Q₈	—	4	1	no	—
147	C₅×D₄	—	4	1	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	4	2	no	—
150	C₅×D₄	—	4	2	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	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	8	2	no	—
166	C₂×C₄×C₅	—	8	2	no	—
167	C₂×C₄×C₅	—	8	2	no	—
168	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	4	1	no	—
176	C₅×C₈	—	4	1	no	—
177	C₅×C₈	—	8	2	no	—
178	C₅×C₈	#99	1	2	yes	—
179	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
181	#131	—	4	2	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	4	2	no	—
188	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
192	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
193	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
194	#123	—	2	1	no	—
195	C₂²⋊Q₈	C₅×D₄	1	1	yes	characteristic
196	C₂²⋊Q₈	C₅×Q₈	1	1	yes	characteristic
197	#121	C₅×D₄	1	1	yes	characteristic
198	#121	C₅×Q₈	1	1	yes	characteristic
199	#120	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	#117	C₂×C₄×C₅	1	1	yes	characteristic
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	#108	—	4	2	no	—
206	#106	—	4	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	C₂²⋊C₈	—	2	1	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₂²⋊C₈	—	2	2	no	—
215	#97	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	#95	—	2	2	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	4	4	no	—
227	C₂²×C₅	—	4	1	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	1	no	—
230	C₂²×C₅	#380	1	1	yes	characteristic
231	C₂²×C₅	#471	1	2	yes	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	4	1	no	—
236	C₄×C₅	—	4	1	no	—
237	C₄×C₅	—	4	1	no	—
238	C₄×C₅	—	16	2	no	—
239	C₄×C₅	C₂×#99	1	1	yes	characteristic
240	C₄×C₅	#410	1	1	yes	characteristic
241	C₂⁴	C₅×#41	1	1	yes	characteristic
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
258	C₂²×C₄	C₅×#41	1	1	yes	characteristic
259	M₁₆	—	4	2	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
283	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₄²	—	2	1	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	4	1	no	—
293	C₂×C₅	—	4	1	no	—
294	C₂×C₅	—	8	1	no	—
295	C₂×C₅	#1799	1	1	yes	characteristic
296	C₂×C₅	#1814	1	1	yes	characteristic
297	C₂×C₅	#1815	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	4	1	no	—
302	C₂³	C₅×#117	1	1	yes	characteristic
303	C₂³	C₅×#121	1	1	yes	characteristic
304	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
305	Q₈	—	4	1	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	C₂×C₄	—	2	1	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	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₅×#121	1	1	yes	characteristic
330	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	8	2	no	—
337	C₈	C₅×#99	1	2	yes	—
338	C₅	#8626	1	1	yes	characteristic, direct product factor, Sylow
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	C₅×#380	1	1	yes	characteristic
352	C₂²	C₅×#471	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	16	2	no	—
360	C₄	C₂×C₅×#99	1	1	yes	characteristic
361	C₄	C₅×#410	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	—	4	1	no	—
365	C₂	—	8	1	no	—
366	C₂	C₅×#1799	1	1	yes	characteristic
367	C₂	C₅×#1814	1	1	yes	characteristic
368	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₅×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1810	C₂	semidirect
3. C₅×#1807	C₂	semidirect
5. C₅×#1305	C₂	semidirect
6. C₂×C₅×#131	C₄	semidirect
15. C₅×#424	C₄	semidirect
16. C₅×#408	C₄	semidirect
19. C₅×#405	C₄	semidirect
25. #8626	C₅	direct (N × H)
66. #1810	C₂×C₅	semidirect
67. #1807	C₂×C₅	semidirect
69. #1305	C₂×C₅	semidirect
120. C₂×#131	C₄×C₅	semidirect
129. #424	C₄×C₅	semidirect
130. #408	C₄×C₅	semidirect
133. #405	C₄×C₅	semidirect
338. C₅	#8626	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×#2929	2
C₅×#1810	2
C₅×#1807	2
C₅×#1513	2
C₅×#1305	2
#8626	5

Subgroups — C5×#8626 #11931346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8626 #11931346