Subgroups — #11990337

Order 1280.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2966	C₂	1	1	yes	maximal, characteristic
2	#10602547	C₂	1	1	yes	maximal, characteristic
3	#10602539	C₂	1	1	yes	maximal, characteristic
4	#10600542	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598242	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598239	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#529	C₂²	1	1	yes	characteristic
9	C₅×#446	C₄	1	1	yes	characteristic, complemented
10	C₅×C₈×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₅×#436	C₄	1	1	yes	characteristic, complemented
12	C₅×#435	—	2	1	no	—
13	C₅×C₈×D₄	—	2	1	no	—
14	C₅×C₈×D₄	C₄	1	1	yes	characteristic, complemented
15	C₅×#433	—	2	1	no	—
16	C₅×#432	C₂²	1	1	yes	characteristic
17	C₅×#405	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×#3947	—	2	1	no	—
20	C₂×C₈×Dic₅	—	2	1	no	—
21	#65221	C₂²	1	1	yes	characteristic
22	#65214	C₂²	1	1	yes	characteristic
23	C₄×#681	C₂²	1	1	yes	characteristic
24	C₂×#3935	C₂²	1	1	yes	characteristic
25	#64763	C₄	1	2	yes	complemented
26	#64760	C₄	1	2	yes	complemented
27	#64758	C₄	1	2	yes	—
28	#64757	C₄	1	2	yes	—
29	#64674	—	2	1	no	—
30	C₈×#168	—	2	1	no	—
31	#8247	—	5	1	no	maximal, Sylow
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₅×#126	—	2	1	no	—
37	C₅×#125	C₂×C₄	1	1	yes	characteristic
38	C₅×#124	C₂×C₄	1	1	yes	characteristic
39	C₅×#123	—	2	1	no	—
40	C₅×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
41	C₅×#121	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×D₄	—	2	1	no	—
44	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
45	C₅×#117	—	2	1	no	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₅	—	2	1	no	—
48	C₂×#681	—	2	1	no	—
49	C₂²×#168	—	2	1	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₅×C₂²⋊C₈	—	2	1	no	—
56	C₅×C₂²⋊C₈	—	2	1	no	—
57	C₅×#97	—	2	1	no	—
58	C₅×#97	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	#3963	—	2	1	no	—
65	#3963	C₂×C₄	1	2	yes	—
66	C₂²⋊#168	C₂×C₄	1	2	yes	—
67	#3947	—	2	4	no	—
68	C₈×Dic₅	—	2	4	no	—
69	C₄⋊#168	—	2	1	no	—
70	#3935	—	2	1	no	—
71	#3935	C₂×C₄	1	2	yes	—
72	#3935	C₂×C₄	1	2	yes	—
73	C₄×#168	—	2	1	no	—
74	C₄×#168	—	2	1	no	—
75	#2966	D₅	1	1	yes	characteristic
76	#1726	—	5	1	no	—
77	#1524	—	5	1	no	—
78	#1500	—	5	1	no	—
79	C₈×M₁₆	—	5	1	no	—
80	#1302	—	5	1	no	—
81	#1300	—	5	1	no	—
82	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
83	C₂×C₅×D₄	—	2	1	no	—
84	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	#41	1	1	yes	characteristic
89	C₂²×Dic₅	—	2	2	no	—
90	C₅×M₁₆	—	4	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	4	2	no	—
96	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₅×C₄⋊C₄	—	2	1	no	—
101	C₅×C₄⋊C₄	—	2	1	no	—
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
105	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
106	C₅×C₂²⋊C₄	—	2	1	no	—
107	C₅×C₂²⋊C₄	—	2	1	no	—
108	C₅×C₂²⋊C₄	—	2	1	no	—
109	C₅×C₂²⋊C₄	—	2	1	no	—
110	C₅×C₂²⋊C₄	—	2	1	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅	#41	1	1	yes	characteristic
116	C₄×Dic₅	—	2	2	no	—
117	C₄×Dic₅	—	2	2	no	—
118	#681	—	2	4	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#168	—	2	2	no	—
124	#529	C₂×D₅	1	1	yes	characteristic
125	#446	Dic₅	1	1	yes	characteristic, complemented
126	C₈×Q₈	Dic₅	1	1	yes	characteristic, complemented
127	#436	Dic₅	1	1	yes	characteristic, complemented
128	#435	—	2	1	no	—
129	C₈×D₄	—	2	1	no	—
130	C₈×D₄	Dic₅	1	1	yes	characteristic, complemented
131	#433	—	2	1	no	—
132	#432	C₂×D₅	1	1	yes	characteristic
133	#431	—	5	1	no	—
134	#405	—	2	1	no	—
135	C₄×M₁₆	—	5	1	no	—
136	C₂×#97	—	5	1	no	—
137	C₂×#97	—	10	1	no	—
138	C₂×C₄×C₈	—	10	1	no	—
139	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
140	#376	—	5	1	no	—
141	#330	—	5	2	no	—
142	#329	—	5	2	no	—
143	#328	—	5	2	no	—
144	#327	—	5	2	no	—
145	#322	—	10	1	no	—
146	C₈²	—	10	1	no	—
147	C₂³×C₅	—	2	1	no	—
148	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
149	C₅×Q₈	—	4	1	no	—
150	C₅×D₄	—	4	1	no	—
151	C₅×D₄	—	4	2	no	—
152	C₂×C₄×C₅	—	2	2	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	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	#117	1	1	yes	characteristic
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×Dic₅	—	2	4	no	—
168	C₂×Dic₅	—	4	2	no	—
169	C₂×Dic₅	—	4	2	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	4	1	no	—
174	C₅×C₈	#104	1	2	yes	—
175	C₅×C₈	#104	1	2	yes	—
176	#168	—	2	4	no	—
177	#168	—	2	4	no	—
178	#168	—	4	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	10	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	10	1	no	—
183	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
185	#126	—	2	1	no	—
186	#125	C₄×D₅	1	1	yes	characteristic
187	#124	C₄×D₅	1	1	yes	characteristic
188	#123	—	2	1	no	—
189	C₂²⋊Q₈	C₄×D₅	1	1	yes	characteristic
190	#121	C₄×D₅	1	1	yes	characteristic
191	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	C₂×Dic₅	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	C₂×C₄²	—	10	1	no	—
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	10	1	no	—
200	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	5	2	no	—
206	#97	—	2	1	no	—
207	#97	—	2	1	no	—
208	#97	—	5	2	no	—
209	#97	—	5	2	no	—
210	#97	—	10	4	no	—
211	#97	—	10	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	10	4	no	—
214	C₄×C₈	—	10	1	no	—
215	C₄×C₈	—	10	1	no	—
216	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
217	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
219	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
220	#95	—	5	2	no	—
221	#95	—	10	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	—	4	1	no	—
226	C₂²×C₅	—	4	1	no	—
227	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₅	#443	1	2	yes	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	4	1	no	—
235	C₄×C₅	—	4	1	no	—
236	C₄×C₅	—	4	1	no	—
237	C₄×C₅	C₂×#104	1	1	yes	characteristic
238	C₄×C₅	C₂×#104	1	1	yes	characteristic
239	Dic₅	—	4	4	no	—
240	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄×Dic₅	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂²×C₄	#709	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
248	M₁₆	—	4	2	no	—
249	M₁₆	—	10	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
269	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	10	2	no	—
277	C₄²	—	10	2	no	—
278	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
279	C₄²	#709	1	1	yes	characteristic
280	C₄²	C₂²×Dic₅	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
282	C₂×C₅	—	2	2	no	—
283	C₂×C₅	—	8	1	no	—
284	C₂×C₅	#1757	1	1	yes	characteristic
285	C₂×C₅	C₄×#104	1	1	yes	characteristic
286	C₂×C₅	#1764	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	C₄×C₂²⋊D₅	1	1	yes	characteristic
289	Q₈	—	4	1	no	—
290	D₄	—	4	1	no	—
291	D₄	—	4	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	10	4	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	2	no	—
304	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
305	C₂×C₄	#4072	1	1	yes	characteristic
306	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
307	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	4	1	no	—
314	C₈	—	10	4	no	—
315	C₈	—	10	4	no	—
316	C₈	—	20	2	no	—
317	C₈	#3969	1	2	yes	—
318	C₈	#3969	1	2	yes	—
319	C₅	#8247	1	1	yes	characteristic, complemented, Sylow
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#65433	1	2	yes	—
326	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	20	4	no	—
336	C₄	C₂×#3969	1	1	yes	characteristic
337	C₄	C₂×#3969	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	C₄×#3969	1	1	yes	characteristic
341	C₂	#10602801	1	1	yes	characteristic
342	C₂	#10602804	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2966	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₅×#121	C₂×C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2966	D₅
hypocenter	C₅	#8247
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
4. #10600542	C₂	semidirect
5. C₈×#681	C₂	semidirect
6. #10598242	C₂	semidirect
7. #10598239	C₂	semidirect
9. C₅×#446	C₄	semidirect
10. C₅×C₈×Q₈	C₄	semidirect
11. C₅×#436	C₄	semidirect
14. C₅×C₈×D₄	C₄	semidirect
25. #64763	C₄	semidirect
26. #64760	C₄	semidirect
125. #446	Dic₅	semidirect
126. C₈×Q₈	Dic₅	semidirect
127. #436	Dic₅	semidirect
130. C₈×D₄	Dic₅	semidirect
319. C₅	#8247	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2966	2
#10602547	2
#10602539	2
#10600542	2
C₈×#681	2
#10598242	2
#10598239	2
#8247	5