Subgroups — C₅×#8628 #11931348

Order 1280.

456 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	C₅×#2929	C₂	1	1	yes	maximal, characteristic
2	C₅×#1810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1808	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×#107	—	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	#8628	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×#107	—	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	#1808	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×#107	—	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	1	no	—
140	C₂³×C₅	—	2	1	no	—
141	C₂³×C₅	—	2	1	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	#117	1	1	yes	characteristic
145	C₂³×C₅	#121	1	1	yes	characteristic
146	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
147	C₅×Q₈	—	4	1	no	—
148	C₅×D₄	—	4	1	no	—
149	C₅×D₄	—	4	2	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	4	1	no	—
152	C₅×D₄	—	4	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	—	2	1	no	—
161	C₂×C₄×C₅	—	2	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	—	4	1	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	8	2	no	—
168	C₂×C₄×C₅	—	8	2	no	—
169	C₂×C₄×C₅	—	8	2	no	—
170	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	#121	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	4	1	no	—
179	C₅×C₈	—	8	2	no	—
180	C₅×C₈	#99	1	2	yes	—
181	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
183	#131	—	4	2	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
193	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
194	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
195	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
196	#123	—	2	1	no	—
197	C₂²⋊Q₈	C₅×D₄	1	1	yes	characteristic
198	C₂²⋊Q₈	C₅×Q₈	1	1	yes	characteristic
199	#121	C₅×D₄	1	1	yes	characteristic
200	#121	C₅×Q₈	1	1	yes	characteristic
201	#120	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	#117	C₂×C₄×C₅	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	#108	—	4	2	no	—
208	#107	—	4	2	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	C₂²⋊C₈	—	2	1	no	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	C₂²⋊C₈	—	2	2	no	—
217	#97	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	#95	—	2	2	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	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	4	2	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	—	4	1	no	—
231	C₂²×C₅	—	4	1	no	—
232	C₂²×C₅	—	4	1	no	—
233	C₂²×C₅	#380	1	1	yes	characteristic
234	C₂²×C₅	#472	1	1	yes	characteristic
235	C₂²×C₅	#473	1	1	yes	characteristic
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	4	1	no	—
240	C₄×C₅	—	4	1	no	—
241	C₄×C₅	—	4	1	no	—
242	C₄×C₅	—	16	2	no	—
243	C₄×C₅	C₂×#99	1	1	yes	characteristic
244	C₄×C₅	#410	1	1	yes	characteristic
245	C₂⁴	C₅×#41	1	1	yes	characteristic
246	#41	—	4	1	no	—
247	#41	—	4	1	no	—
248	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
262	C₂²×C₄	C₅×#41	1	1	yes	characteristic
263	M₁₆	—	4	2	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	4	2	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
287	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	2	1	no	—
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	8	2	no	—
294	C₄²	—	2	1	no	—
295	C₂×C₅	—	2	1	no	—
296	C₂×C₅	—	2	1	no	—
297	C₂×C₅	—	4	1	no	—
298	C₂×C₅	—	4	1	no	—
299	C₂×C₅	—	8	1	no	—
300	C₂×C₅	#1799	1	1	yes	characteristic
301	C₂×C₅	#1814	1	1	yes	characteristic
302	C₂×C₅	#1816	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	C₅×#117	1	1	yes	characteristic
309	C₂³	C₅×#121	1	1	yes	characteristic
310	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
311	Q₈	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	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	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
335	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
336	C₂×C₄	C₅×#121	1	1	yes	characteristic
337	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	8	2	no	—
344	C₈	C₅×#99	1	2	yes	—
345	C₅	#8628	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	C₅×#380	1	1	yes	characteristic
360	C₂²	C₅×#472	1	1	yes	characteristic
361	C₂²	C₅×#473	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	16	2	no	—
369	C₄	C₂×C₅×#99	1	1	yes	characteristic
370	C₄	C₅×#410	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	4	1	no	—
374	C₂	—	4	1	no	—
375	C₂	—	8	1	no	—
376	C₂	C₅×#1799	1	1	yes	characteristic
377	C₂	C₅×#1814	1	1	yes	characteristic
378	C₂	C₅×#1816	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×#1808	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. #8628	C₅	direct (N × H)
66. #1810	C₂×C₅	semidirect
67. #1808	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
345. C₅	#8628	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×#1808	2
C₅×#1513	2
C₅×#1305	2
#8628	5

Subgroups — C5×#8628 #11931348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8628 #11931348