Subgroups — #12068591

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611024	C₂	1	1	yes	maximal, characteristic
2	#10611007	C₂	1	1	yes	maximal, characteristic
3	#10610782	C₂	1	1	yes	maximal, characteristic
4	C₅×#1574	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601546	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601538	C₂	1	1	yes	maximal, characteristic, complemented
7	#10601283	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599367	C₂	1	2	yes	maximal, complemented
9	#10599363	C₂	1	2	yes	maximal, complemented
10	#10599327	C₂	1	2	yes	maximal, complemented
11	D₅×#334	C₂	1	2	yes	maximal, complemented
12	C₄×#709	C₂²	1	1	yes	characteristic
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₅×#423	C₂²	1	1	yes	characteristic
16	#65416	—	2	2	no	—
17	#65214	C₂²	1	1	yes	characteristic
18	#65213	C₂²	1	1	yes	characteristic
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	#65124	C₂²	1	2	yes	—
21	#65124	C₂²	1	2	yes	—
22	#65123	—	2	1	no	—
23	#65123	C₄	1	1	yes	characteristic
24	#65123	C₄	1	1	yes	characteristic
25	D₅×C₄⋊C₈	C₂²	1	2	yes	—
26	D₅×C₄⋊C₈	C₂²	1	2	yes	—
27	#65120	—	2	1	no	—
28	#65119	C₄	1	1	yes	characteristic
29	#65119	C₄	1	1	yes	characteristic
30	#65017	—	2	2	no	—
31	#65014	C₄	1	2	yes	—
32	#65014	C₄	1	2	yes	—
33	C₈×D₂₀	—	2	1	no	—
34	#64974	C₂²	1	2	yes	—
35	C₄×C₈×D₅	C₂²	1	2	yes	—
36	#64968	—	2	1	no	—
37	C₅×#335	C₂²	1	2	yes	—
38	C₅×#334	C₂²	1	2	yes	—
39	#64700	C₂²	1	1	yes	characteristic
40	#64700	C₂²	1	1	yes	characteristic
41	#64700	C₂²	1	2	yes	—
42	#64700	C₂²	1	2	yes	—
43	#64699	C₂²	1	1	yes	characteristic
44	#64699	C₂²	1	1	yes	characteristic
45	#64678	C₂²	1	2	yes	—
46	#64677	C₂²	1	2	yes	—
47	#17834	—	5	1	no	maximal, Sylow
48	C₂×#709	Q₈	1	1	yes	characteristic
49	C₂×C₅×M₁₆	—	2	1	no	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₅	D₄	1	2	yes	—
53	#4072	Q₈	1	1	yes	characteristic
54	C₂×#681	—	2	1	no	—
55	C₂²×#168	C₂×C₄	1	1	yes	characteristic
56	C₂²×#168	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅	—	2	2	no	—
58	C₂×C₈×D₅	—	2	2	no	—
59	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
61	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
62	C₄×D₂₀	D₄	1	1	yes	characteristic
63	#4018	Q₈	1	2	yes	—
64	C₄²×D₅	C₂³	1	2	yes	—
65	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀	D₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₅×C₄⋊C₈	C₂³	1	2	yes	—
72	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₅×C₂²⋊C₈	—	2	1	no	—
74	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₄×C₅×C₈	—	2	1	no	—
76	C₄×C₅×C₈	C₂³	1	2	yes	—
77	C₂²⋊#168	—	2	1	no	—
78	#3952	—	2	2	no	—
79	#3952	—	2	2	no	—
80	#3952	C₂×C₄	1	2	yes	—
81	#3952	C₂×C₄	1	2	yes	—
82	#3947	—	2	2	no	—
83	#3947	—	2	2	no	—
84	#3946	—	2	2	no	—
85	#3946	—	2	2	no	—
86	#3946	C₂×C₄	1	2	yes	—
87	#3946	C₂×C₄	1	2	yes	—
88	C₈×Dic₅	C₂×C₄	1	2	yes	—
89	C₈×Dic₅	C₂×C₄	1	2	yes	—
90	C₄⋊#168	—	2	1	no	—
91	C₄⋊#168	C₂³	1	1	yes	characteristic
92	C₄⋊#168	C₂³	1	2	yes	—
93	C₄⋊#168	C₂³	1	1	yes	characteristic
94	C₄×#168	—	2	1	no	—
95	C₄×#168	C₂×C₄	1	1	yes	characteristic
96	C₄×#168	C₂×C₄	1	1	yes	characteristic
97	C₄×#168	C₂³	1	1	yes	characteristic
98	C₄×#168	C₂³	1	1	yes	characteristic
99	#2990	—	5	1	no	—
100	#2990	—	5	1	no	—
101	C₈×#41	—	5	1	no	—
102	#1574	—	5	1	no	—
103	#1574	D₅	1	1	yes	characteristic, complemented
104	#1573	—	5	2	no	—
105	#1573	—	5	2	no	—
106	#1568	—	5	2	no	—
107	#1566	—	5	1	no	—
108	#1564	—	5	1	no	—
109	C₂×#334	—	5	2	no	—
110	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
114	#709	—	2	4	no	—
115	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
117	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
118	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
119	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
120	C₅×M₁₆	—	8	1	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
126	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
129	C₄²×C₅	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
131	#685	C₄⋊C₄	1	2	yes	—
132	#685	C₄⋊C₄	1	2	yes	—
133	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
134	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
135	#683	C₄⋊C₄	1	2	yes	—
136	#683	C₄⋊C₄	1	2	yes	—
137	C₄×Dic₅	C₂²×C₄	1	2	yes	—
138	C₄×Dic₅	C₂×D₄	1	2	yes	—
139	C₄×Dic₅	C₂×Q₈	1	2	yes	—
140	#681	—	8	1	no	—
141	C₂×#168	—	2	2	no	—
142	C₂×#168	—	2	1	no	—
143	C₂×#168	—	2	1	no	—
144	C₂×#168	—	2	2	no	—
145	C₂×#168	—	4	1	no	—
146	C₂×#168	C₂²×C₄	1	1	yes	characteristic
147	C₂×#168	C₂²×C₄	1	1	yes	characteristic
148	C₂×#168	C₂²×C₄	1	1	yes	characteristic
149	C₂×#168	C₂²×C₄	1	1	yes	characteristic
150	C₈×D₅	—	2	4	no	—
151	C₈×D₅	—	4	2	no	—
152	C₈×D₅	—	8	2	no	—
153	C₈×D₅	—	8	2	no	—
154	C₄×#41	—	5	1	no	—
155	#446	—	10	1	no	—
156	#446	—	10	1	no	—
157	C₈×Q₈	—	5	1	no	—
158	C₈×Q₈	—	5	1	no	—
159	#435	—	10	2	no	—
160	#435	—	10	2	no	—
161	C₈×D₄	—	5	1	no	—
162	C₈×D₄	—	5	1	no	—
163	C₈×D₄	—	5	2	no	—
164	C₈×D₄	—	5	2	no	—
165	C₈×D₄	—	10	1	no	—
166	C₈×D₄	—	10	1	no	—
167	#432	—	5	2	no	—
168	#432	—	5	2	no	—
169	#431	—	5	2	no	—
170	#431	—	5	1	no	—
171	#431	C₂×D₅	1	1	yes	characteristic
172	#431	C₂×D₅	1	1	yes	characteristic
173	#423	—	5	1	no	—
174	#423	C₂×D₅	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	5	2	no	—
176	C₂×C₄⋊C₈	—	5	2	no	—
177	C₂×C₄×C₈	—	5	1	no	—
178	C₂×C₄×C₈	—	5	2	no	—
179	#335	—	5	1	no	—
180	#335	—	5	1	no	—
181	#335	—	5	2	no	—
182	#335	C₂×D₅	1	2	yes	—
183	#334	—	5	1	no	—
184	#334	—	5	2	no	—
185	#334	—	5	1	no	—
186	#334	—	5	2	no	—
187	#334	—	5	2	no	—
188	#334	C₂×D₅	1	2	yes	—
189	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	2	1	no	—
194	C₂×C₄×C₅	—	2	1	no	—
195	C₂×C₄×C₅	—	2	2	no	—
196	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
202	C₂²⋊D₅	—	2	4	no	—
203	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
204	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
205	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
206	D₂₀	#108	1	4	yes	—
207	C₄×D₅	—	2	4	no	—
208	C₄×D₅	—	2	2	no	—
209	C₄×D₅	—	2	2	no	—
210	C₄×D₅	—	2	2	no	—
211	C₄×D₅	—	2	2	no	—
212	Dic₁₀	—	2	2	no	—
213	C₅×C₈	—	2	2	no	—
214	C₅×C₈	—	2	2	no	—
215	C₅×C₈	—	4	1	no	—
216	C₅×C₈	—	4	2	no	—
217	C₅×C₈	—	4	1	no	—
218	#168	—	2	1	no	—
219	#168	—	2	1	no	—
220	#168	—	2	1	no	—
221	#168	—	4	1	no	—
222	#168	—	4	2	no	—
223	#168	—	4	1	no	—
224	#168	#131	1	2	yes	—
225	C₂×#41	—	5	1	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₂×M₁₆	—	10	1	no	—
228	C₂²×C₈	—	5	1	no	—
229	C₂²×C₈	—	5	1	no	—
230	C₂²×C₈	—	5	2	no	—
231	C₂²×C₈	—	5	2	no	—
232	C₂²×C₈	—	10	2	no	—
233	C₂²×C₈	—	10	2	no	—
234	C₄×Q₈	—	5	1	no	—
235	C₄×Q₈	—	5	1	no	—
236	C₄×D₄	—	5	2	no	—
237	C₄×D₄	—	5	2	no	—
238	C₄×D₄	—	5	1	no	—
239	C₄×D₄	—	5	1	no	—
240	#117	—	5	1	no	—
241	#117	—	5	2	no	—
242	C₂×C₄²	—	5	2	no	—
243	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	5	2	no	—
246	C₄⋊C₈	—	5	2	no	—
247	C₄⋊C₈	—	5	1	no	—
248	C₄⋊C₈	—	5	2	no	—
249	C₄⋊C₈	—	5	1	no	—
250	C₄⋊C₈	—	10	2	no	—
251	C₄⋊C₈	—	10	2	no	—
252	C₄⋊C₈	—	10	1	no	—
253	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
254	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
255	C₄⋊C₈	C₂²×D₅	1	2	yes	—
256	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
257	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	5	2	no	—
260	C₂²⋊C₈	—	5	2	no	—
261	C₂²⋊C₈	—	10	2	no	—
262	C₂²⋊C₈	—	10	2	no	—
263	C₂²⋊C₈	—	10	1	no	—
264	C₂²⋊C₈	C₄×D₅	1	2	yes	—
265	#97	—	10	2	no	—
266	#97	—	10	2	no	—
267	C₄×C₈	—	2	1	no	—
268	C₄×C₈	—	5	2	no	—
269	C₄×C₈	—	5	2	no	—
270	C₄×C₈	—	5	1	no	—
271	C₄×C₈	—	5	1	no	—
272	C₄×C₈	—	5	1	no	—
273	C₄×C₈	—	5	1	no	—
274	C₄×C₈	—	10	1	no	—
275	C₄×C₈	C₂²×D₅	1	2	yes	—
276	C₂²×C₅	—	2	1	no	—
277	C₂²×C₅	—	2	1	no	—
278	C₂²×C₅	—	2	1	no	—
279	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
280	C₂×D₅	—	2	2	no	—
281	C₂×D₅	—	2	2	no	—
282	C₂×D₅	C₂×#108	1	4	yes	—
283	C₄×C₅	—	2	1	no	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	2	1	no	—
286	C₄×C₅	C₂×#108	1	2	yes	—
287	C₄×C₅	C₂×#131	1	1	yes	characteristic
288	C₄×C₅	#568	1	1	yes	characteristic
289	C₄×C₅	#577	1	1	yes	characteristic
290	C₄×C₅	#577	1	1	yes	characteristic
291	Dic₅	—	2	2	no	—
292	Dic₅	—	2	2	no	—
293	Dic₅	—	2	2	no	—
294	#41	—	10	4	no	—
295	C₂×Q₈	—	5	1	no	—
296	C₂×D₄	—	5	2	no	—
297	C₂×D₄	—	5	1	no	—
298	C₂²×C₄	—	5	2	no	—
299	C₂²×C₄	—	5	2	no	—
300	C₂²×C₄	—	5	2	no	—
301	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
302	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
303	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
304	M₁₆	—	8	1	no	—
305	M₁₆	—	40	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	5	1	no	—
310	C₂×C₈	—	5	1	no	—
311	C₂×C₈	—	5	1	no	—
312	C₂×C₈	—	5	1	no	—
313	C₂×C₈	—	10	4	no	—
314	C₂×C₈	—	10	2	no	—
315	C₂×C₈	—	10	1	no	—
316	C₂×C₈	—	10	1	no	—
317	C₂×C₈	—	10	2	no	—
318	C₂×C₈	—	20	2	no	—
319	C₂×C₈	—	20	1	no	—
320	C₂×C₈	—	40	2	no	—
321	C₂×C₈	—	40	2	no	—
322	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
324	C₄⋊C₄	—	5	2	no	—
325	C₄⋊C₄	—	5	2	no	—
326	C₄⋊C₄	—	5	1	no	—
327	C₄⋊C₄	—	5	1	no	—
328	C₂²⋊C₄	—	5	2	no	—
329	C₂²⋊C₄	—	5	2	no	—
330	C₂²⋊C₄	—	5	2	no	—
331	C₄²	—	5	2	no	—
332	C₄²	—	5	2	no	—
333	C₄²	—	5	2	no	—
334	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
335	C₄²	D₄×D₅	1	1	yes	characteristic
336	C₄²	Q₈×D₅	1	1	yes	characteristic
337	C₄²	C₂³×D₅	1	1	yes	characteristic
338	C₂×C₅	—	4	1	no	—
339	C₂×C₅	#2906	1	1	yes	characteristic
340	C₂×C₅	#2912	1	1	yes	characteristic
341	C₂×C₅	C₂²×#108	1	1	yes	characteristic
342	D₅	—	2	4	no	—
343	C₂³	—	5	2	no	—
344	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
345	Q₈	—	10	2	no	—
346	D₄	—	5	4	no	—
347	D₄	—	10	4	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	5	2	no	—
354	C₂×C₄	—	5	2	no	—
355	C₂×C₄	—	5	2	no	—
356	C₂×C₄	—	10	4	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	10	2	no	—
360	C₂×C₄	—	10	2	no	—
361	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
362	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
363	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
366	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	1	no	—
372	C₈	—	5	2	no	—
373	C₈	—	10	1	no	—
374	C₈	—	10	1	no	—
375	C₈	—	10	1	no	—
376	C₈	—	20	1	no	—
377	C₈	—	20	2	no	—
378	C₈	—	20	1	no	—
379	C₅	#17834	1	1	yes	characteristic, complemented, Sylow
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	5	4	no	—
384	C₂²	—	10	2	no	—
385	C₂²	—	10	2	no	—
386	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	10	2	no	—
391	C₄	—	10	2	no	—
392	C₄	—	10	2	no	—
393	C₄	D₅×#108	1	2	yes	—
394	C₄	D₅×#131	1	1	yes	characteristic
395	C₄	#66084	1	1	yes	characteristic
396	C₄	#66109	1	1	yes	characteristic
397	C₄	#66112	1	1	yes	characteristic
398	C₂	—	4	1	no	—
399	C₂	—	10	4	no	—
400	C₂	#10610812	1	1	yes	characteristic
401	C₂	#10610940	1	1	yes	characteristic
402	C₂	C₂×D₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1574	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	Q₈
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1574	D₅
hypocenter	C₅	#17834
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
4. C₅×#1574	C₂	semidirect
5. #10601546	C₂	semidirect
6. #10601538	C₂	semidirect
7. #10601283	C₂	semidirect
8. #10599367	C₂	semidirect
9. #10599363	C₂	semidirect
10. #10599327	C₂	semidirect
11. D₅×#334	C₂	semidirect
103. #1574	D₅	semidirect
379. C₅	#17834	semidirect

Maximal subgroups

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

subgroup	index
#10611024	2
#10611007	2
#10610782	2
C₅×#1574	2
#10601546	2
#10601538	2
#10601283	2
#10599367	2
#10599363	2
#10599327	2
D₅×#334	2
#17834	5