Subgroups — C₅×#21679 #11944399

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	1	yes	maximal, characteristic
3	C₅×#1940	C₂	1	1	yes	maximal, characteristic
4	C₅×#1916	C₂	1	2	yes	maximal
5	C₅×#1916	C₂	1	2	yes	maximal
6	C₅×#1832	C₂	1	1	yes	maximal, characteristic
7	C₅×#1771	C₂	1	2	yes	maximal
8	C₅×#1770	C₂	1	2	yes	maximal
9	C₅×#1770	C₂	1	1	yes	maximal, characteristic
10	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₅×#1726	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#400	C₄	1	2	yes	—
25	C₅×#400	C₄	1	2	yes	—
26	C₅×#398	C₄	1	2	yes	—
27	C₅×#398	C₄	1	2	yes	—
28	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
32	C₅×#376	C₂²	1	1	yes	characteristic
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	C₅×#336	C₂²	1	2	yes	—
36	C₅×#336	C₂²	1	2	yes	—
37	C₅×#336	C₂²	1	2	yes	—
38	C₅×#336	C₂²	1	2	yes	—
39	#21679	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	4	no	—
58	C₅×C₄⋊C₈	—	2	4	no	—
59	C₅×C₄⋊C₈	—	2	2	no	—
60	C₅×C₄⋊C₈	—	2	2	no	—
61	C₅×#97	—	2	2	no	—
62	C₅×#97	—	2	2	no	—
63	C₅×#97	—	2	4	no	—
64	C₅×#97	—	2	2	no	—
65	C₄×C₅×C₈	—	2	2	no	—
66	C₄×C₅×C₈	—	2	2	no	—
67	C₄×C₅×C₈	—	2	4	no	—
68	C₄×C₅×C₈	Q₈	1	4	yes	—
69	C₅×#95	C₂×C₄	1	2	yes	—
70	C₅×#95	C₂×C₄	1	2	yes	—
71	C₅×#95	C₂×C₄	1	1	yes	characteristic
72	C₅×#95	C₂×C₄	1	1	yes	characteristic
73	C₅×#95	C₂×C₄	1	2	yes	—
74	C₅×#95	C₂×C₄	1	2	yes	—
75	C₅×#95	C₂×C₄	1	2	yes	—
76	C₅×#95	C₂×C₄	1	2	yes	—
77	C₅×#95	C₂×C₄	1	1	yes	characteristic
78	C₅×#95	C₂×C₄	1	1	yes	characteristic
79	#2402	C₂×C₅	1	1	yes	characteristic
80	#1940	C₂×C₅	1	1	yes	characteristic
81	#1940	C₂×C₅	1	1	yes	characteristic
82	#1916	C₂×C₅	1	2	yes	—
83	#1916	C₂×C₅	1	2	yes	—
84	#1832	C₂×C₅	1	1	yes	characteristic
85	#1771	C₂×C₅	1	2	yes	—
86	#1770	C₂×C₅	1	2	yes	—
87	#1770	C₂×C₅	1	1	yes	characteristic
88	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
89	#1726	C₂×C₅	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
112	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
113	C₂×C₅×C₈	#41	1	2	yes	—
114	C₂×C₅×C₈	#41	1	2	yes	—
115	C₂×C₅×C₈	#41	1	2	yes	—
116	C₂×C₅×C₈	#41	1	2	yes	—
117	C₅×C₄⋊C₄	—	2	4	no	—
118	C₅×C₄⋊C₄	—	2	4	no	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	1	no	—
125	C₄²×C₅	—	2	1	no	—
126	C₄²×C₅	—	2	1	no	—
127	C₄²×C₅	—	2	1	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	—	2	2	no	—
130	C₄²×C₅	—	2	2	no	—
131	C₄²×C₅	C₂×Q₈	1	2	yes	—
132	C₄²×C₅	#41	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
137	C₂×#97	C₂²×C₅	1	1	yes	characteristic
138	C₂×#97	C₂²×C₅	1	1	yes	characteristic
139	C₂×#97	C₂²×C₅	1	1	yes	characteristic
140	C₂×#97	C₂²×C₅	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
145	#400	C₄×C₅	1	2	yes	—
146	#400	C₄×C₅	1	2	yes	—
147	#398	C₄×C₅	1	2	yes	—
148	#398	C₄×C₅	1	2	yes	—
149	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
150	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
151	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
153	#376	C₂²×C₅	1	1	yes	characteristic
154	#336	C₂²×C₅	1	2	yes	—
155	#336	C₂²×C₅	1	2	yes	—
156	#336	C₂²×C₅	1	2	yes	—
157	#336	C₂²×C₅	1	2	yes	—
158	#336	C₂²×C₅	1	2	yes	—
159	#336	C₂²×C₅	1	2	yes	—
160	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
182	C₂×C₄×C₅	#131	1	4	yes	—
183	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
189	C₅×C₈	—	2	2	no	—
190	C₅×C₈	—	2	2	no	—
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	4	2	no	—
193	C₅×C₈	—	4	2	no	—
194	C₅×C₈	#125	1	4	yes	—
195	C₂²×C₈	C₂³×C₅	1	2	yes	—
196	C₂²×C₈	C₂³×C₅	1	2	yes	—
197	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
204	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
207	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₅	1	2	yes	—
210	C₂×C₄²	C₂³×C₅	1	2	yes	—
211	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
212	C₄⋊C₈	—	2	4	no	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	2	no	—
216	#97	—	2	2	no	—
217	#97	—	2	2	no	—
218	#97	—	2	4	no	—
219	#97	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	2	4	no	—
223	C₄×C₈	C₅×Q₈	1	4	yes	—
224	#95	C₂×C₄×C₅	1	2	yes	—
225	#95	C₂×C₄×C₅	1	2	yes	—
226	#95	C₂×C₄×C₅	1	1	yes	characteristic
227	#95	C₂×C₄×C₅	1	1	yes	characteristic
228	#95	C₂×C₄×C₅	1	2	yes	—
229	#95	C₂×C₄×C₅	1	2	yes	—
230	#95	C₂×C₄×C₅	1	2	yes	—
231	#95	C₂×C₄×C₅	1	2	yes	—
232	#95	C₂×C₄×C₅	1	1	yes	characteristic
233	#95	C₂×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
235	C₂²×C₅	C₄×#41	1	1	yes	characteristic
236	C₂²×C₅	C₄×#41	1	1	yes	characteristic
237	C₂²×C₅	C₂×#131	1	2	yes	—
238	C₂²×C₅	#568	1	2	yes	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	4	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	4	2	no	—
244	C₄×C₅	—	4	2	no	—
245	C₄×C₅	—	4	2	no	—
246	C₄×C₅	—	4	2	no	—
247	C₄×C₅	C₂×#125	1	1	yes	characteristic
248	C₄×C₅	#529	1	1	yes	characteristic
249	C₄×C₅	#529	1	1	yes	characteristic
250	C₄×C₅	#533	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
259	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
273	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
274	C₂×C₈	C₅×#41	1	2	yes	—
275	C₂×C₈	C₅×#41	1	2	yes	—
276	C₂×C₈	C₅×#41	1	2	yes	—
277	C₂×C₈	C₅×#41	1	2	yes	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂×C₅×Q₈	1	2	yes	—
293	C₄²	C₅×#41	1	2	yes	—
294	C₂×C₅	C₄×#125	1	1	yes	characteristic
295	C₂×C₅	#2960	1	1	yes	characteristic
296	C₂×C₅	#2961	1	1	yes	characteristic
297	C₂×C₅	#2968	1	2	yes	—
298	C₂×C₅	#2969	1	2	yes	—
299	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
320	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
321	C₂×C₄	C₅×#131	1	4	yes	—
322	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
325	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
326	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
327	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	C₅×#125	1	4	yes	—
334	C₅	#21679	1	1	yes	characteristic, direct product factor, Sylow
335	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
336	C₂²	C₄×C₅×#41	1	1	yes	characteristic
337	C₂²	C₄×C₅×#41	1	1	yes	characteristic
338	C₂²	C₂×C₅×#131	1	2	yes	—
339	C₂²	C₅×#568	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	C₂×C₅×#125	1	1	yes	characteristic
349	C₄	C₅×#529	1	1	yes	characteristic
350	C₄	C₅×#529	1	1	yes	characteristic
351	C₄	C₅×#533	1	1	yes	characteristic
352	C₂	C₄×C₅×#125	1	1	yes	characteristic
353	C₂	C₅×#2960	1	1	yes	characteristic
354	C₂	C₅×#2961	1	1	yes	characteristic
355	C₂	C₅×#2968	1	2	yes	—
356	C₂	C₅×#2969	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
39. #21679	C₅	direct (N × H)
334. C₅	#21679	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×#2402	2
C₅×#1940	2
C₅×#1916	2
C₅×#1832	2
C₅×#1771	2
C₅×#1770	2
C₅×C₈×C₄⋊C₄	2
C₅×#1726	2
#21679	5

Subgroups — C5×#21679 #11944399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21679 #11944399