Subgroups — C₅×#20779 #11943499

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
2	C₅×#1840	C₂	1	2	yes	maximal, complemented
3	C₅×#1832	C₂	1	2	yes	maximal, complemented
4	C₅×#1817	C₂	1	2	yes	maximal
5	C₅×#1797	C₂	1	2	yes	maximal
6	C₅×#1768	C₂	1	2	yes	maximal, complemented
7	C₅×#1749	C₂	1	1	yes	maximal, characteristic
8	C₄×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₅×#1746	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#376	C₄	1	2	yes	—
19	C₅×#376	C₄	1	2	yes	—
20	C₅×#376	C₂²	1	1	yes	characteristic
21	C₅×#376	C₂²	1	1	yes	characteristic
22	C₅×#376	C₂²	1	1	yes	characteristic
23	C₅×#376	C₂²	1	1	yes	characteristic
24	C₂×C₅×#95	C₄	1	2	yes	—
25	C₂×C₅×#95	C₄	1	2	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	C₅×#336	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	#20779	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	4	yes	—
38	C₂²×C₅×C₈	C₂³	1	4	yes	—
39	C₂×C₄²×C₅	—	2	4	no	—
40	C₂×C₄²×C₅	—	2	4	no	—
41	C₂×C₄²×C₅	—	2	4	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	8	no	—
53	C₅×C₂²⋊C₈	—	2	4	no	—
54	C₅×C₂²⋊C₈	—	2	4	no	—
55	C₅×#97	—	2	4	no	—
56	C₄×C₅×C₈	—	2	4	no	—
57	C₅×#95	C₂×C₄	1	2	yes	—
58	C₅×#95	C₂×C₄	1	2	yes	—
59	C₅×#95	C₂×C₄	1	2	yes	—
60	C₅×#95	C₂×C₄	1	2	yes	—
61	C₅×#95	C₂×C₄	1	2	yes	—
62	C₅×#95	C₂×C₄	1	2	yes	—
63	C₅×#95	C₂×C₄	1	2	yes	—
64	C₅×#95	C₂×C₄	1	2	yes	—
65	C₂×#376	C₂×C₅	1	1	yes	characteristic
66	#1840	C₂×C₅	1	2	yes	complemented
67	#1832	C₂×C₅	1	2	yes	complemented
68	#1817	C₂×C₅	1	2	yes	—
69	#1797	C₂×C₅	1	2	yes	—
70	#1768	C₂×C₅	1	2	yes	complemented
71	#1749	C₂×C₅	1	1	yes	characteristic
72	C₄×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
73	#1746	C₂×C₅	1	2	yes	complemented
74	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	—	4	2	no	—
85	C₂²×C₄×C₅	—	4	2	no	—
86	C₂²×C₄×C₅	—	4	2	no	—
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅	#41	1	4	yes	—
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	2	4	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	4	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₂×C₅×C₈	—	2	4	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	—	2	4	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	#41	1	4	yes	—
113	C₄²×C₅	#41	1	4	yes	—
114	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
115	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
116	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
118	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
119	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
120	C₂×#97	C₂²×C₅	1	2	yes	—
121	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
122	#376	C₄×C₅	1	2	yes	—
123	#376	C₄×C₅	1	2	yes	—
124	#376	C₂²×C₅	1	1	yes	characteristic
125	#376	C₂²×C₅	1	1	yes	characteristic
126	#376	C₂²×C₅	1	1	yes	characteristic
127	#376	C₂²×C₅	1	1	yes	characteristic
128	C₂×#95	C₄×C₅	1	2	yes	—
129	C₂×#95	C₄×C₅	1	2	yes	—
130	#336	C₂²×C₅	1	4	yes	—
131	#336	C₂²×C₅	1	4	yes	—
132	#336	C₂²×C₅	1	4	yes	—
133	#336	C₂²×C₅	1	2	yes	—
134	#336	C₂²×C₅	1	2	yes	—
135	C₂³×C₅	—	2	1	no	—
136	C₂³×C₅	—	2	1	no	—
137	C₂³×C₅	—	2	2	no	—
138	C₂³×C₅	#117	1	2	yes	—
139	C₂³×C₅	#131	1	4	yes	—
140	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	8	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	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	4	8	no	—
163	C₂×C₄×C₅	—	4	8	no	—
164	C₂×C₄×C₅	—	4	4	no	—
165	C₂×C₄×C₅	—	4	4	no	—
166	C₂×C₄×C₅	#117	1	2	yes	—
167	C₂×C₄×C₅	#117	1	4	yes	—
168	C₂×C₄×C₅	#117	1	4	yes	—
169	C₂×C₄×C₅	C₂×#41	1	2	yes	—
170	C₂×C₄×C₅	C₂×#41	1	2	yes	—
171	C₂×C₄×C₅	C₂×#41	1	2	yes	—
172	C₅×C₈	—	4	4	no	—
173	C₅×C₈	—	4	4	no	—
174	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
177	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	4	yes	—
180	C₂²×C₈	C₂³×C₅	1	4	yes	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	—	2	4	no	—
184	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	C₄⋊C₈	—	2	8	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	#97	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	#95	C₂×C₄×C₅	1	2	yes	—
200	#95	C₂×C₄×C₅	1	2	yes	—
201	#95	C₂×C₄×C₅	1	2	yes	—
202	#95	C₂×C₄×C₅	1	2	yes	—
203	#95	C₂×C₄×C₅	1	2	yes	—
204	#95	C₂×C₄×C₅	1	2	yes	—
205	#95	C₂×C₄×C₅	1	2	yes	—
206	#95	C₂×C₄×C₅	1	2	yes	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	4	no	—
209	C₂²×C₅	—	2	4	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	#433	1	4	yes	—
212	C₂²×C₅	C₂×#117	1	1	yes	characteristic
213	C₂²×C₅	C₂×#117	1	1	yes	characteristic
214	C₂²×C₅	C₂×#117	1	1	yes	characteristic
215	C₂²×C₅	C₂×#131	1	2	yes	—
216	C₂²×C₅	#568	1	2	yes	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	2	4	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	4	1	no	—
222	C₄×C₅	—	4	1	no	—
223	C₄×C₅	—	4	1	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	#529	1	4	yes	—
227	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₅×#41	1	4	yes	—
250	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	C₅×#41	1	4	yes	—
266	C₄²	C₅×#41	1	4	yes	—
267	C₂×C₅	—	2	4	no	—
268	C₂×C₅	#2283	1	1	yes	characteristic
269	C₂×C₅	C₂×#433	1	1	yes	characteristic
270	C₂×C₅	#2921	1	1	yes	characteristic
271	C₂×C₅	#2924	1	2	yes	—
272	C₂×C₅	#2924	1	2	yes	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₅×#117	1	2	yes	—
277	C₂³	C₅×#131	1	4	yes	—
278	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	C₅×#117	1	2	yes	—
305	C₂×C₄	C₅×#117	1	4	yes	—
306	C₂×C₄	C₅×#117	1	4	yes	—
307	C₂×C₄	C₂×C₅×#41	1	2	yes	—
308	C₂×C₄	C₂×C₅×#41	1	2	yes	—
309	C₂×C₄	C₂×C₅×#41	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₈	—	4	4	no	—
312	C₅	#20779	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	C₅×#433	1	4	yes	—
318	C₂²	C₂×C₅×#117	1	1	yes	characteristic
319	C₂²	C₂×C₅×#117	1	1	yes	characteristic
320	C₂²	C₂×C₅×#117	1	1	yes	characteristic
321	C₂²	C₂×C₅×#131	1	2	yes	—
322	C₂²	C₅×#568	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	C₅×#529	1	4	yes	—
333	C₂	—	2	4	no	—
334	C₂	C₅×#2283	1	1	yes	characteristic
335	C₂	C₂×C₅×#433	1	1	yes	characteristic
336	C₂	C₅×#2921	1	1	yes	characteristic
337	C₂	C₅×#2924	1	2	yes	—
338	C₂	C₅×#2924	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1840	C₂	semidirect
3. C₅×#1832	C₂	semidirect
6. C₅×#1768	C₂	semidirect
9. C₅×#1746	C₂	semidirect
31. #20779	C₅	direct (N × H)
66. #1840	C₂×C₅	semidirect
67. #1832	C₂×C₅	semidirect
70. #1768	C₂×C₅	semidirect
73. #1746	C₂×C₅	semidirect
312. C₅	#20779	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×#376	2
C₅×#1840	2
C₅×#1832	2
C₅×#1817	2
C₅×#1797	2
C₅×#1768	2
C₅×#1749	2
C₄×C₅×C₂²⋊C₈	2
C₅×#1746	2
#20779	5

Subgroups — C5×#20779 #11943499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20779 #11943499