Subgroups — #11989470

Order 1280.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611013	C₂	1	1	yes	maximal, characteristic
2	#10610769	C₂	1	1	yes	maximal, characteristic
3	#10610248	C₂	1	1	yes	maximal, characteristic
4	C₅×#1299	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598234	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598201	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₅×#423	C₂²	1	1	yes	characteristic
11	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	#65415	—	2	1	no	—
13	#65415	C₄	1	2	yes	—
14	#65407	—	2	1	no	—
15	M₁₆×Dic₅	C₄	1	2	yes	—
16	#65214	C₂²	1	1	yes	characteristic
17	#65213	C₂²	1	1	yes	characteristic
18	C₄×#681	C₂²	1	1	yes	characteristic
19	#65126	C₄	1	2	yes	—
20	#65124	—	2	1	no	—
21	#65124	—	2	1	no	—
22	#65124	C₄	1	2	yes	—
23	#65122	—	2	1	no	—
24	#65122	—	2	1	no	—
25	#65015	—	2	1	no	—
26	#65008	—	2	1	no	—
27	#64994	C₄	1	2	yes	—
28	#64994	C₄	1	2	yes	—
29	C₄×#676	—	2	1	no	—
30	#64974	—	2	1	no	—
31	#64974	—	2	1	no	—
32	#64974	—	2	1	no	—
33	#8062	—	5	1	no	maximal, Sylow
34	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₅	Q₈	1	1	yes	characteristic
37	C₂×C₅×M₁₆	—	2	1	no	—
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₅	Q₈	1	1	yes	characteristic
43	C₂×#681	—	2	1	no	—
44	C₂×#681	C₂×C₄	1	2	yes	—
45	C₂×#676	—	2	2	no	—
46	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅	D₄	1	1	yes	characteristic
51	C₄²×D₅	D₄	1	1	yes	characteristic
52	C₄²×D₅	D₄	1	1	yes	characteristic
53	C₄²×D₅	D₄	1	1	yes	characteristic
54	C₅×C₄⋊C₈	—	2	1	no	—
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₅×#97	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	—	2	1	no	—
61	C₂²⋊#168	—	2	1	no	—
62	#3952	—	2	2	no	—
63	#3952	—	2	2	no	—
64	#3952	—	2	2	no	—
65	#3952	—	2	2	no	—
66	#3952	—	2	2	no	—
67	#3947	—	2	2	no	—
68	#3947	—	2	2	no	—
69	#3946	—	2	2	no	—
70	#3946	—	2	2	no	—
71	#3946	—	2	2	no	—
72	C₈×Dic₅	—	2	2	no	—
73	C₄⋊#168	—	2	1	no	—
74	C₄⋊#168	—	2	1	no	—
75	C₄⋊#168	C₂×C₄	1	2	yes	—
76	#3935	C₂×C₄	1	2	yes	—
77	C₄×#168	—	2	1	no	—
78	C₄×#168	—	2	1	no	—
79	#2920	—	5	1	no	—
80	#2920	—	5	1	no	—
81	#2920	—	5	1	no	—
82	#1299	—	5	1	no	—
83	#1299	—	5	1	no	—
84	#1299	—	5	1	no	—
85	#1299	D₅	1	1	yes	characteristic, complemented
86	C₂³×D₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
90	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₅	—	4	1	no	—
94	C₂×C₄×D₅	—	4	1	no	—
95	C₂×C₄×D₅	—	4	2	no	—
96	C₂×C₄×D₅	—	4	2	no	—
97	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
98	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
99	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
109	C₅×M₁₆	—	4	2	no	—
110	C₅×M₁₆	—	4	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₅	C₄²	1	1	yes	characteristic
119	C₄×Dic₅	C₄²	1	1	yes	characteristic
120	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
129	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
130	#681	—	4	2	no	—
131	#681	—	4	2	no	—
132	C₂×#168	—	2	2	no	—
133	C₂×#168	—	2	2	no	—
134	C₂×#168	—	2	2	no	—
135	#676	—	4	4	no	—
136	C₂²×C₄²	—	5	1	no	—
137	#432	—	5	2	no	—
138	#432	—	5	2	no	—
139	#432	—	5	2	no	—
140	#432	—	10	1	no	—
141	#432	—	10	1	no	—
142	#432	—	10	1	no	—
143	#431	—	5	1	no	—
144	#431	—	10	1	no	—
145	#431	—	10	1	no	—
146	#431	—	10	1	no	—
147	#431	—	10	1	no	—
148	#431	—	10	1	no	—
149	#431	C₂×D₅	1	1	yes	characteristic
150	#423	—	5	2	no	—
151	#423	—	5	1	no	—
152	#423	—	10	1	no	—
153	#423	C₂×D₅	1	1	yes	characteristic
154	#407	—	5	2	no	—
155	#407	—	10	1	no	—
156	#407	—	10	1	no	—
157	C₄×M₁₆	—	5	1	no	—
158	C₄×M₁₆	—	5	2	no	—
159	C₄×M₁₆	—	10	1	no	—
160	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
161	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×D₅	—	4	1	no	—
163	C₂²×D₅	—	4	1	no	—
164	C₂²×D₅	—	4	1	no	—
165	C₂²×D₅	#95	1	1	yes	characteristic
166	C₂²×D₅	#95	1	1	yes	characteristic
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×Dic₅	—	4	1	no	—
178	C₂×Dic₅	—	4	1	no	—
179	C₂×Dic₅	—	4	1	no	—
180	C₂×Dic₅	#95	1	1	yes	characteristic
181	C₂×Dic₅	#95	1	1	yes	characteristic
182	C₂×Dic₅	#95	1	1	yes	characteristic
183	C₂×Dic₅	#95	1	1	yes	characteristic
184	C₂×Dic₅	#95	1	1	yes	characteristic
185	C₂×Dic₅	#95	1	1	yes	characteristic
186	C₄×D₅	—	2	1	no	—
187	C₄×D₅	—	2	1	no	—
188	C₄×D₅	—	2	1	no	—
189	C₄×D₅	—	2	1	no	—
190	C₄×D₅	—	2	2	no	—
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	2	2	no	—
193	C₄×D₅	—	2	2	no	—
194	C₄×D₅	—	4	1	no	—
195	C₄×D₅	—	4	1	no	—
196	C₄×D₅	—	4	1	no	—
197	C₄×D₅	—	4	1	no	—
198	C₄×D₅	—	4	1	no	—
199	C₄×D₅	—	4	1	no	—
200	C₅×C₈	—	4	2	no	—
201	C₅×C₈	—	4	2	no	—
202	C₅×C₈	—	4	2	no	—
203	#168	—	4	2	no	—
204	#168	—	4	2	no	—
205	#168	—	4	2	no	—
206	C₂³×C₄	—	5	1	no	—
207	C₂³×C₄	—	5	1	no	—
208	C₂³×C₄	—	5	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	5	2	no	—
211	C₂×M₁₆	—	10	2	no	—
212	C₂×M₁₆	—	10	1	no	—
213	C₂×M₁₆	C₄×D₅	1	2	yes	—
214	C₂×C₄²	—	5	1	no	—
215	C₂×C₄²	—	5	1	no	—
216	C₂×C₄²	—	5	1	no	—
217	C₂×C₄²	—	5	1	no	—
218	C₂×C₄²	—	5	1	no	—
219	C₂×C₄²	—	5	1	no	—
220	C₂×C₄²	—	5	1	no	—
221	C₂×C₄²	—	5	1	no	—
222	C₂×C₄²	—	5	1	no	—
223	C₂×C₄²	—	5	1	no	—
224	C₂×C₄²	—	5	1	no	—
225	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	5	2	no	—
229	C₄⋊C₈	—	10	2	no	—
230	C₄⋊C₈	—	10	2	no	—
231	C₄⋊C₈	—	10	2	no	—
232	C₄⋊C₈	—	10	1	no	—
233	C₄⋊C₈	—	10	1	no	—
234	C₄⋊C₈	C₄×D₅	1	2	yes	—
235	C₂²⋊C₈	—	2	1	no	—
236	C₂²⋊C₈	—	10	2	no	—
237	C₂²⋊C₈	—	10	2	no	—
238	C₂²⋊C₈	—	10	2	no	—
239	C₂²⋊C₈	—	10	2	no	—
240	C₂²⋊C₈	—	10	2	no	—
241	C₂²⋊C₈	—	10	1	no	—
242	#97	—	5	2	no	—
243	#97	—	10	2	no	—
244	#97	—	10	2	no	—
245	#97	C₄×D₅	1	2	yes	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	10	2	no	—
249	C₄×C₈	—	10	1	no	—
250	C₄×C₈	—	10	1	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	2	1	no	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	C₂×#95	1	1	yes	characteristic
255	C₂×D₅	—	2	1	no	—
256	C₂×D₅	—	2	1	no	—
257	C₂×D₅	—	2	1	no	—
258	C₂×D₅	—	2	1	no	—
259	C₂×D₅	—	2	1	no	—
260	C₂×D₅	—	2	1	no	—
261	C₂×D₅	—	4	4	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	#413	1	1	yes	characteristic
266	C₄×C₅	#413	1	1	yes	characteristic
267	C₄×C₅	#421	1	2	yes	—
268	C₄×C₅	#430	1	2	yes	—
269	Dic₅	—	2	1	no	—
270	Dic₅	—	2	1	no	—
271	Dic₅	—	2	1	no	—
272	Dic₅	—	2	1	no	—
273	Dic₅	—	2	1	no	—
274	Dic₅	—	2	1	no	—
275	C₂⁴	—	5	1	no	—
276	C₂²×C₄	—	5	1	no	—
277	C₂²×C₄	—	5	1	no	—
278	C₂²×C₄	—	5	1	no	—
279	C₂²×C₄	—	5	1	no	—
280	C₂²×C₄	—	5	1	no	—
281	C₂²×C₄	—	5	1	no	—
282	C₂²×C₄	—	5	1	no	—
283	C₂²×C₄	—	5	1	no	—
284	C₂²×C₄	—	5	1	no	—
285	C₂²×C₄	—	5	1	no	—
286	C₂²×C₄	—	5	1	no	—
287	C₂²×C₄	—	5	1	no	—
288	C₂²×C₄	—	5	1	no	—
289	C₂²×C₄	—	5	1	no	—
290	C₂²×C₄	—	5	1	no	—
291	C₂²×C₄	—	20	1	no	—
292	C₂²×C₄	—	20	1	no	—
293	C₂²×C₄	—	20	2	no	—
294	C₂²×C₄	—	20	2	no	—
295	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
297	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
298	M₁₆	—	4	2	no	—
299	M₁₆	—	4	2	no	—
300	M₁₆	—	20	4	no	—
301	M₁₆	—	20	2	no	—
302	M₁₆	—	20	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	10	2	no	—
307	C₂×C₈	—	10	2	no	—
308	C₂×C₈	—	10	2	no	—
309	C₄²	—	5	1	no	—
310	C₄²	—	5	1	no	—
311	C₄²	—	5	1	no	—
312	C₄²	—	5	1	no	—
313	C₄²	—	5	1	no	—
314	C₄²	—	5	1	no	—
315	C₄²	—	5	1	no	—
316	C₄²	—	5	1	no	—
317	C₄²	—	5	1	no	—
318	C₄²	—	5	1	no	—
319	C₄²	—	5	1	no	—
320	C₄²	—	5	1	no	—
321	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
322	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
323	C₄²	D₄×D₅	1	1	yes	characteristic
324	C₄²	D₄×D₅	1	1	yes	characteristic
325	C₂×C₅	—	4	1	no	—
326	C₂×C₅	#1733	1	1	yes	characteristic
327	C₂×C₅	#1738	1	1	yes	characteristic
328	C₂×C₅	#1742	1	1	yes	characteristic
329	D₅	—	4	1	no	—
330	D₅	—	4	1	no	—
331	C₂³	—	5	1	no	—
332	C₂³	—	5	1	no	—
333	C₂³	—	20	1	no	—
334	C₂³	—	20	1	no	—
335	C₂³	—	20	1	no	—
336	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	5	1	no	—
342	C₂×C₄	—	5	1	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	5	1	no	—
346	C₂×C₄	—	5	1	no	—
347	C₂×C₄	—	10	1	no	—
348	C₂×C₄	—	10	1	no	—
349	C₂×C₄	—	10	1	no	—
350	C₂×C₄	—	10	1	no	—
351	C₂×C₄	—	10	2	no	—
352	C₂×C₄	—	10	2	no	—
353	C₂×C₄	—	10	2	no	—
354	C₂×C₄	—	10	2	no	—
355	C₂×C₄	—	20	1	no	—
356	C₂×C₄	—	20	1	no	—
357	C₂×C₄	—	20	1	no	—
358	C₂×C₄	—	20	1	no	—
359	C₂×C₄	—	20	1	no	—
360	C₂×C₄	—	20	1	no	—
361	C₂×C₄	—	20	1	no	—
362	C₂×C₄	—	20	1	no	—
363	C₂×C₄	—	20	1	no	—
364	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
365	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
366	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
367	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
368	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
369	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
370	C₈	—	4	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	20	2	no	—
374	C₈	—	20	2	no	—
375	C₈	—	20	2	no	—
376	C₅	#8062	1	1	yes	characteristic, complemented, Sylow
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	10	1	no	—
381	C₂²	—	10	1	no	—
382	C₂²	—	10	1	no	—
383	C₂²	—	10	1	no	—
384	C₂²	—	10	1	no	—
385	C₂²	—	10	1	no	—
386	C₂²	—	20	4	no	—
387	C₂²	D₅×#95	1	1	yes	characteristic
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	10	1	no	—
392	C₄	—	10	1	no	—
393	C₄	—	10	1	no	—
394	C₄	—	10	1	no	—
395	C₄	—	10	1	no	—
396	C₄	—	10	1	no	—
397	C₄	#65034	1	1	yes	characteristic
398	C₄	#65040	1	1	yes	characteristic
399	C₄	#65110	1	2	yes	—
400	C₄	#65182	1	2	yes	—
401	C₂	—	4	1	no	—
402	C₂	—	20	1	no	—
403	C₂	—	20	1	no	—
404	C₂	#10599546	1	1	yes	characteristic
405	C₂	#10599731	1	1	yes	characteristic
406	C₂	#10599782	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1299	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂³×D₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1299	D₅
hypocenter	C₅	#8062
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1299

Complements

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

N	H ≅ G/N	type
4. C₅×#1299	C₂	semidirect
5. #10598234	C₂	semidirect
6. #10598201	C₂	semidirect
7. #10598160	C₂	semidirect
85. #1299	D₅	semidirect
376. C₅	#8062	semidirect

Maximal subgroups

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

subgroup	index
#10611013	2
#10610769	2
#10610248	2
C₅×#1299	2
#10598234	2
#10598201	2
#10598160	2
#8062	5