Subgroups — C₅×#9783 #11932503

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₅×#1964	C₂	1	2	yes	maximal, complemented
3	C₅×#1808	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1519	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#121	C₂²	1	2	yes	—
9	C₅×#466	—	2	1	no	—
10	C₅×#466	—	2	1	no	—
11	C₅×#466	—	2	1	no	—
12	C₅×#466	—	2	1	no	—
13	C₂×C₅×#107	—	2	1	no	—
14	C₂×C₅×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅×#102	C₄	1	2	yes	complemented
16	C₂×C₅×#99	—	2	4	no	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	—	2	1	no	—
19	C₅×#380	C₂²	1	2	yes	—
20	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₅×#340	—	2	1	no	—
22	C₅×#336	—	2	1	no	—
23	C₅×#327	—	2	4	no	—
24	#9783	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	2	no	—
26	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	—	4	2	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	4	2	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₅×#121	—	2	2	no	—
40	C₅×#121	—	2	2	no	—
41	C₅×#121	—	2	2	no	—
42	C₅×#121	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₅×#107	—	2	1	no	—
50	C₅×#107	—	2	1	no	—
51	C₅×#107	—	4	2	no	—
52	C₅×#102	—	2	2	no	—
53	C₅×#102	—	2	2	no	—
54	C₅×#99	—	4	8	no	—
55	C₅×C₂²⋊C₈	—	2	1	no	—
56	C₅×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₅×C₈	—	4	2	no	—
58	C₅×#95	—	2	1	no	—
59	C₅×#95	—	2	2	no	—
60	C₂×#466	C₂×C₅	1	1	yes	characteristic
61	#1964	C₂×C₅	1	2	yes	complemented
62	#1808	C₂×C₅	1	1	yes	characteristic, complemented
63	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
64	#1519	C₂×C₅	1	2	yes	complemented
65	C₂⁴×C₅	—	2	2	no	—
66	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	—	4	2	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂×C₅×D₄	—	4	2	no	—
73	C₂×C₅×D₄	—	4	4	no	—
74	C₂×C₅×D₄	—	4	4	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	2	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	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	#41	1	1	yes	characteristic
85	C₂²×C₄×C₅	#41	1	1	yes	characteristic
86	C₅×D₈	—	4	2	no	—
87	C₅×D₈	—	4	2	no	—
88	C₂×C₅×C₈	—	2	2	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	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₂²⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₅×C₂²⋊C₄	—	4	2	no	—
108	C₅×C₂²⋊C₄	—	4	4	no	—
109	C₅×C₂²⋊C₄	D₈	1	4	yes	complemented
110	C₄²×C₅	—	4	1	no	—
111	C₄²×C₅	—	4	1	no	—
112	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
113	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×#121	C₂²×C₅	1	2	yes	—
115	#466	—	2	1	no	—
116	#466	—	2	1	no	—
117	#466	—	2	1	no	—
118	#466	—	2	1	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₄×C₅	1	1	yes	characteristic, complemented
121	C₂×#102	C₄×C₅	1	2	yes	complemented
122	C₂×#99	—	2	4	no	—
123	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×C₈	—	2	1	no	—
125	#380	C₂²×C₅	1	2	yes	—
126	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	#340	—	2	1	no	—
128	#336	—	2	1	no	—
129	#327	—	2	4	no	—
130	C₂³×C₅	—	4	1	no	—
131	C₂³×C₅	—	4	1	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	4	4	no	—
134	C₂³×C₅	—	8	2	no	—
135	C₂³×C₅	—	8	2	no	—
136	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂³×C₅	#121	1	1	yes	characteristic
138	C₂³×C₅	#127	1	1	yes	characteristic
139	C₂³×C₅	C₂×D₈	1	2	yes	—
140	C₂³×C₅	#135	1	2	yes	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	8	2	no	—
144	C₅×D₄	—	8	4	no	—
145	C₅×D₄	—	8	2	no	—
146	C₅×D₄	—	8	2	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	4	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	—	8	2	no	—
162	C₂×C₄×C₅	—	8	2	no	—
163	C₂×C₄×C₅	—	8	2	no	—
164	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	#121	1	1	yes	characteristic
167	C₂×C₄×C₅	#124	1	1	yes	characteristic
168	C₅×C₈	—	2	1	no	—
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	4	2	no	—
172	C₅×C₈	#99	1	2	yes	complemented
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
175	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
185	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
186	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
187	#121	—	2	2	no	—
188	#121	—	2	2	no	—
189	#121	—	2	2	no	—
190	#121	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	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	C₅×D₄	1	2	yes	—
196	C₂×C₄²	—	2	1	no	—
197	#107	—	2	1	no	—
198	#107	—	2	1	no	—
199	#107	—	4	2	no	—
200	#102	—	2	2	no	—
201	#102	—	2	2	no	—
202	#99	—	4	8	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	C₅×D₄	1	2	yes	—
205	C₄×C₈	—	4	2	no	—
206	#95	—	2	1	no	—
207	#95	—	2	2	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	8	2	no	—
217	C₂²×C₅	—	8	2	no	—
218	C₂²×C₅	—	8	2	no	—
219	C₂²×C₅	—	16	4	no	—
220	C₂²×C₅	#390	1	1	yes	characteristic
221	C₂²×C₅	C₄×D₈	1	2	yes	—
222	C₂²×C₅	#466	1	2	yes	—
223	C₂²×C₅	#493	1	1	yes	characteristic
224	C₂²×C₅	#495	1	1	yes	characteristic
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	4	4	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	—	16	2	no	—
231	C₄×C₅	C₂×#99	1	1	yes	characteristic
232	C₄×C₅	#410	1	1	yes	characteristic
233	C₂⁴	—	2	2	no	—
234	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₅×#41	1	1	yes	characteristic
253	C₂²×C₄	C₅×#41	1	1	yes	characteristic
254	D₈	—	4	2	no	—
255	D₈	—	4	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	C₅×D₈	1	4	yes	complemented
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	2	no	—
283	C₂×C₅	—	2	1	no	—
284	C₂×C₅	—	16	2	no	—
285	C₂×C₅	#1964	1	1	yes	characteristic
286	C₂×C₅	#1970	1	1	yes	characteristic
287	C₂×C₅	#1976	1	1	yes	characteristic
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	4	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
295	C₂³	C₅×#121	1	1	yes	characteristic
296	C₂³	C₅×#127	1	1	yes	characteristic
297	C₂³	C₂×C₅×D₈	1	2	yes	—
298	C₂³	C₅×#135	1	2	yes	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
324	C₂×C₄	C₅×#121	1	1	yes	characteristic
325	C₂×C₄	C₅×#124	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	C₅×#99	1	2	yes	complemented
331	C₅	#9783	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	C₅×#390	1	1	yes	characteristic
345	C₂²	C₄×C₅×D₈	1	2	yes	—
346	C₂²	C₅×#466	1	2	yes	—
347	C₂²	C₅×#493	1	1	yes	characteristic
348	C₂²	C₅×#495	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	16	2	no	—
355	C₄	C₂×C₅×#99	1	1	yes	characteristic
356	C₄	C₅×#410	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	2	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	C₅×#1964	1	1	yes	characteristic
363	C₂	C₅×#1970	1	1	yes	characteristic
364	C₂	C₅×#1976	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#390
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂×C₅×#466	C₂
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×#1964	C₂	semidirect
3. C₅×#1808	C₂	semidirect
4. C₅×C₈×C₂²⋊C₄	C₂	semidirect
5. C₅×#1519	C₂	semidirect
6. C₂²×C₅×D₈	C₄	semidirect
14. C₂×C₅×#107	C₄	semidirect
15. C₂×C₅×#102	C₄	semidirect
24. #9783	C₅	direct (N × H)
61. #1964	C₂×C₅	semidirect
62. #1808	C₂×C₅	semidirect
63. C₈×C₂²⋊C₄	C₂×C₅	semidirect
64. #1519	C₂×C₅	semidirect
109. C₅×C₂²⋊C₄	D₈	semidirect
112. C₂²×D₈	C₄×C₅	semidirect
120. C₂×#107	C₄×C₅	semidirect
121. C₂×#102	C₄×C₅	semidirect
172. C₅×C₈	#99	semidirect
277. C₂²⋊C₄	C₅×D₈	semidirect
330. C₈	C₅×#99	semidirect
331. C₅	#9783	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×#466	2
C₅×#1964	2
C₅×#1808	2
C₅×C₈×C₂²⋊C₄	2
C₅×#1519	2
#9783	5

Subgroups — C5×#9783 #11932503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9783 #11932503