Subgroups — C₅×#17890 #11940610

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
3	C₅×#1594	C₂	1	2	yes	maximal, complemented
4	C₅×#1593	C₂	1	2	yes	maximal, complemented
5	C₅×#1592	C₂	1	4	yes	maximal, complemented
6	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1476	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₄	1	1	yes	characteristic
10	C₄×C₅×#41	C₂²	1	2	yes	—
11	C₅×#446	C₂²	1	4	yes	—
12	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₅×SD₁₆	C₄	1	2	yes	—
15	C₄×C₅×SD₁₆	C₂²	1	2	yes	—
16	C₄×C₅×D₈	C₄	1	1	yes	characteristic
17	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
18	C₅×#436	C₂²	1	4	yes	—
19	C₅×#435	—	2	4	no	—
20	C₅×#431	—	2	2	no	—
21	C₅×#431	C₂²	1	2	yes	—
22	C₅×#427	C₄	1	1	yes	characteristic
23	C₅×#416	C₄	1	2	yes	—
24	C₄×C₅×M₁₆	—	2	2	no	—
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#335	C₂²	1	2	yes	—
29	C₅×#334	C₂²	1	2	yes	—
30	C₅×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₅×D₄⋊C₈	C₂²	1	4	yes	—
32	C₅×#322	C₂²	1	2	yes	—
33	C₅×#322	C₂²	1	2	yes	—
34	#17890	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅×#41	C₂×C₄	1	2	yes	—
36	C₅×#135	—	2	4	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	—	2	4	no	—
41	C₂²×C₅×C₈	—	2	1	no	—
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	D₄	1	2	yes	—
45	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×Q₈	C₂³	1	2	yes	—
47	C₄×C₅×D₄	—	2	2	no	—
48	C₄×C₅×D₄	—	2	2	no	—
49	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
50	C₄×C₅×D₄	C₂³	1	2	yes	—
51	C₅×#117	—	2	2	no	—
52	C₅×#117	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅	—	2	2	no	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#107	C₂×C₄	1	1	yes	characteristic
56	C₅×#107	C₂×C₄	1	1	yes	characteristic
57	C₅×#106	C₂×C₄	1	2	yes	—
58	C₅×C₄⋊C₈	—	2	4	no	—
59	C₅×C₄⋊C₈	C₂³	1	4	yes	—
60	C₅×#103	C₂×C₄	1	2	yes	—
61	C₅×#103	C₂×C₄	1	2	yes	—
62	C₅×#102	C₂×C₄	1	2	yes	—
63	C₅×#102	C₂×C₄	1	2	yes	—
64	C₅×C₂²⋊C₈	—	2	2	no	—
65	C₅×C₂²⋊C₈	—	2	4	no	—
66	C₅×#97	—	2	4	no	—
67	C₄×C₅×C₈	—	2	2	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₅×C₈	D₄	1	2	yes	—
72	C₄×C₅×C₈	C₂³	1	2	yes	—
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	2	yes	—
76	#2965	C₂×C₅	1	2	yes	complemented
77	C₄×#135	C₂×C₅	1	1	yes	characteristic
78	#1594	C₂×C₅	1	2	yes	complemented
79	#1593	C₂×C₅	1	2	yes	complemented
80	#1592	C₂×C₅	1	4	yes	complemented
81	#1564	C₂×C₅	1	1	yes	characteristic, complemented
82	#1476	C₂×C₅	1	2	yes	complemented
83	C₂×#322	C₂×C₅	1	1	yes	characteristic, complemented
84	C₅×#41	—	2	4	no	—
85	C₅×#41	—	2	4	no	—
86	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	#41	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×SD₁₆	—	2	4	no	—
97	C₅×D₈	—	2	2	no	—
98	C₅×M₁₆	—	4	8	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	4	2	no	—
103	C₂×C₅×C₈	—	4	2	no	—
104	C₂×C₅×C₈	M₁₆	1	4	yes	complemented
105	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	2	2	no	—
120	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅	#41	1	1	yes	characteristic
123	C₄²×C₅	C₂⁴	1	1	yes	characteristic
124	C₂×#135	C₄×C₅	1	1	yes	characteristic
125	C₄×#41	C₂²×C₅	1	2	yes	—
126	#446	C₂²×C₅	1	4	yes	—
127	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
128	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
129	C₄×SD₁₆	C₄×C₅	1	2	yes	—
130	C₄×SD₁₆	C₂²×C₅	1	2	yes	—
131	C₄×D₈	C₄×C₅	1	1	yes	characteristic
132	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
133	#436	C₂²×C₅	1	4	yes	—
134	#435	—	2	4	no	—
135	#431	—	2	2	no	—
136	#431	C₂²×C₅	1	2	yes	—
137	#427	C₄×C₅	1	1	yes	characteristic
138	#416	C₄×C₅	1	2	yes	—
139	C₄×M₁₆	—	2	2	no	—
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
143	#335	C₂²×C₅	1	2	yes	—
144	#334	C₂²×C₅	1	2	yes	—
145	Q₈⋊C₈	C₂²×C₅	1	4	yes	—
146	D₄⋊C₈	C₂²×C₅	1	4	yes	—
147	#322	C₂²×C₅	1	2	yes	—
148	#322	C₂²×C₅	1	2	yes	—
149	C₂³×C₅	—	2	2	no	—
150	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	2	no	—
154	C₅×D₄	—	2	2	no	—
155	C₅×D₄	—	4	4	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	—	4	4	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	—	4	4	no	—
164	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
168	C₂×C₄×C₅	#131	1	2	yes	—
169	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	4	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	4	4	no	—
177	C₅×C₈	C₂×M₁₆	1	2	yes	—
178	C₅×C₈	C₂×M₁₆	1	2	yes	—
179	C₂×#41	C₂×C₄×C₅	1	2	yes	—
180	#135	—	2	4	no	—
181	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×SD₁₆	C₂×C₄×C₅	1	2	yes	—
183	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
188	C₂²×C₈	C₅×D₄	1	2	yes	—
189	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
190	C₄×Q₈	C₂³×C₅	1	2	yes	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
194	C₄×D₄	C₂³×C₅	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	#107	C₂×C₄×C₅	1	1	yes	characteristic
200	#107	C₂×C₄×C₅	1	1	yes	characteristic
201	#106	C₂×C₄×C₅	1	2	yes	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	C₂³×C₅	1	4	yes	—
204	#103	C₂×C₄×C₅	1	2	yes	—
205	#103	C₂×C₄×C₅	1	2	yes	—
206	#102	C₂×C₄×C₅	1	2	yes	—
207	#102	C₂×C₄×C₅	1	2	yes	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	#97	—	2	4	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
214	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
215	C₄×C₈	C₅×D₄	1	2	yes	—
216	C₄×C₈	C₂³×C₅	1	2	yes	—
217	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅	1	2	yes	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	4	2	no	—
223	C₂²×C₅	—	4	2	no	—
224	C₂²×C₅	—	4	2	no	—
225	C₂²×C₅	#436	1	2	yes	—
226	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	#436	1	2	yes	—
233	C₄×C₅	#444	1	2	yes	—
234	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
235	C₄×C₅	C₂×#131	1	1	yes	characteristic
236	C₄×C₅	#575	1	2	yes	—
237	#41	—	2	4	no	—
238	#41	—	2	4	no	—
239	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₅×#41	1	1	yes	characteristic
248	Q₁₆	—	2	2	no	—
249	SD₁₆	—	2	4	no	—
250	D₈	—	2	2	no	—
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	C₅×M₁₆	1	4	yes	complemented
258	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
259	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
261	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
266	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
274	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄²	C₅×#41	1	1	yes	characteristic
276	C₄²	C₂⁴×C₅	1	1	yes	characteristic
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	8	2	no	—
279	C₂×C₅	C₂×#436	1	1	yes	characteristic
280	C₂×C₅	#2943	1	1	yes	characteristic
281	C₂×C₅	C₂×#444	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	4	no	—
289	C₂×C₄	—	2	2	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	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
301	C₂×C₄	C₅×#131	1	2	yes	—
302	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	4	no	—
310	C₈	C₂×C₅×M₁₆	1	2	yes	—
311	C₈	C₂×C₅×M₁₆	1	2	yes	—
312	C₅	#17890	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₅×#436	1	2	yes	—
319	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₅×#436	1	2	yes	—
326	C₄	C₅×#444	1	2	yes	—
327	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
328	C₄	C₂×C₅×#131	1	1	yes	characteristic
329	C₄	C₅×#575	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×C₅×#436	1	1	yes	characteristic
333	C₂	C₅×#2943	1	1	yes	characteristic
334	C₂	C₂×C₅×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×C₅×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2965	C₂	semidirect
3. C₅×#1594	C₂	semidirect
4. C₅×#1593	C₂	semidirect
5. C₅×#1592	C₂	semidirect
6. C₅×#1564	C₂	semidirect
7. C₅×#1476	C₂	semidirect
8. C₂×C₅×#322	C₂	semidirect
34. #17890	C₅	direct (N × H)
76. #2965	C₂×C₅	semidirect
78. #1594	C₂×C₅	semidirect
79. #1593	C₂×C₅	semidirect
80. #1592	C₂×C₅	semidirect
81. #1564	C₂×C₅	semidirect
82. #1476	C₂×C₅	semidirect
83. C₂×#322	C₂×C₅	semidirect
104. C₂×C₅×C₈	M₁₆	semidirect
257. C₂×C₈	C₅×M₁₆	semidirect
312. C₅	#17890	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×#2965	2
C₄×C₅×#135	2
C₅×#1594	2
C₅×#1593	2
C₅×#1592	2
C₅×#1564	2
C₅×#1476	2
C₂×C₅×#322	2
#17890	5

Subgroups — C5×#17890 #11940610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17890 #11940610