Subgroups — #12082463

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	#10614947	C₂	1	1	yes	maximal, characteristic
3	#10614946	C₂	1	1	yes	maximal, characteristic
4	#10610189	C₂	1	1	yes	maximal, characteristic
5	#10602567	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602566	C₂	1	1	yes	maximal, characteristic, complemented
7	#10602028	C₂	1	4	yes	maximal, complemented
8	#10602026	C₂	1	4	yes	maximal, complemented
9	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#4047	—	2	1	no	—
13	#65811	C₂²	1	1	yes	characteristic
14	C₅×#487	—	2	1	no	—
15	C₅×#487	—	2	1	no	—
16	C₅×#487	C₂²	1	2	yes	—
17	C₅×#487	C₂²	1	2	yes	—
18	C₂×C₅×#103	—	2	1	no	—
19	C₂×C₅×#103	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#65399	—	2	1	no	—
22	#65377	—	2	1	no	—
23	#65377	—	2	1	no	—
24	#65376	C₂²	1	4	yes	—
25	#65373	C₂²	1	4	yes	—
26	#65368	—	2	1	no	—
27	#65367	—	2	1	no	—
28	#65257	—	2	2	no	—
29	#65256	—	2	2	no	—
30	#65214	C₂²	1	1	yes	characteristic
31	#65214	C₂²	1	1	yes	characteristic
32	#65213	C₂²	1	1	yes	characteristic
33	#64977	—	2	1	no	—
34	#64977	C₂²	1	2	yes	—
35	C₈×D₂₀	—	2	1	no	—
36	C₈×D₂₀	C₂²	1	2	yes	—
37	#64767	C₂²	1	4	yes	—
38	#64766	C₂²	1	4	yes	—
39	#64678	C₂²	1	2	yes	—
40	#64677	C₂²	1	2	yes	—
41	#19199	—	5	1	no	maximal, Sylow
42	C₂²×C₅×Q₈	—	2	1	no	—
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂×#709	—	2	1	no	—
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×C₄⋊Q₈	—	2	1	no	—
53	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	—	2	1	no	—
57	C₂×C₅×C₄⋊C₄	—	2	1	no	—
58	C₂×C₅×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	#4076	—	2	1	no	—
61	#4076	D₄	1	2	yes	—
62	C₂×#681	—	2	1	no	—
63	#4047	—	4	4	no	—
64	C₂×#676	—	2	2	no	—
65	C₂×C₈×D₅	—	2	2	no	—
66	#4021	—	2	1	no	—
67	#4020	D₄	1	1	yes	characteristic
68	C₄×D₂₀	—	2	1	no	—
69	C₄×D₂₀	C₂³	1	2	yes	—
70	#4015	D₄	1	1	yes	characteristic
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#103	—	2	2	no	—
74	C₅×#103	—	2	2	no	—
75	C₄×C₅×C₈	D₄	1	1	yes	characteristic
76	C₄×C₅×C₈	D₄	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₂²⋊#168	—	2	1	no	—
80	C₂²⋊#168	—	2	1	no	—
81	#3952	—	2	2	no	—
82	#3952	—	2	2	no	—
83	#3941	—	2	4	no	—
84	#3941	—	2	4	no	—
85	#3940	—	2	4	no	—
86	#3939	—	2	4	no	—
87	C₄⋊#168	—	2	1	no	—
88	C₄⋊#168	C₂³	1	2	yes	—
89	C₄⋊#168	C₂³	1	2	yes	—
90	C₄⋊#168	C₂³	1	2	yes	—
91	C₄×#168	—	2	1	no	—
92	C₄×#168	—	2	1	no	—
93	#3227	—	5	1	no	—
94	#3226	—	5	1	no	—
95	C₂×#487	D₅	1	1	yes	characteristic, complemented
96	#2931	—	5	1	no	—
97	#1703	—	5	4	no	—
98	#1683	—	5	4	no	—
99	#1564	—	5	1	no	—
100	#1548	—	5	1	no	—
101	#1546	—	5	1	no	—
102	C₂×C₅×Q₈	—	2	2	no	—
103	C₂×C₅×Q₈	—	2	2	no	—
104	C₂×C₅×Q₈	—	4	1	no	—
105	C₂×C₅×Q₈	—	4	1	no	—
106	C₂×C₅×Q₈	—	4	2	no	—
107	C₂×C₅×Q₈	—	4	2	no	—
108	C₂×C₅×Q₈	—	8	2	no	—
109	C₂²×C₄×C₅	—	4	1	no	—
110	C₂²×C₄×C₅	—	4	1	no	—
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅	#41	1	1	yes	characteristic
114	C₂×C₂²⋊D₅	—	2	2	no	—
115	#709	—	4	2	no	—
116	#709	—	4	2	no	—
117	C₂×D₂₀	—	2	1	no	—
118	C₂×D₂₀	—	2	1	no	—
119	C₂×D₂₀	C₂×D₄	1	2	yes	—
120	C₂×C₄×D₅	—	2	2	no	—
121	C₂×Dic₁₀	—	2	1	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	—	2	2	no	—
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
128	C₅×C₄⋊C₄	—	2	2	no	—
129	C₅×C₄⋊C₄	—	2	2	no	—
130	C₅×C₄⋊C₄	—	2	2	no	—
131	C₅×C₄⋊C₄	—	2	2	no	—
132	C₅×C₄⋊C₄	—	4	1	no	—
133	C₅×C₄⋊C₄	—	4	1	no	—
134	C₅×C₄⋊C₄	—	4	1	no	—
135	C₅×C₄⋊C₄	—	4	1	no	—
136	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₅	#41	1	1	yes	characteristic
139	C₄²×C₅	C₂⁴	1	1	yes	characteristic
140	#685	—	2	2	no	—
141	#685	—	2	2	no	—
142	C₄⋊Dic₅	—	2	1	no	—
143	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
144	#681	—	4	2	no	—
145	C₂×#168	—	2	2	no	—
146	C₂×#168	—	2	2	no	—
147	C₂×#168	—	2	2	no	—
148	#676	—	4	4	no	—
149	C₈×D₅	—	4	4	no	—
150	C₂×#131	—	10	1	no	—
151	#532	—	5	1	no	—
152	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
153	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
154	#500	—	10	1	no	—
155	#498	—	10	1	no	—
156	#487	—	2	1	no	—
157	#487	—	2	1	no	—
158	#487	C₂×D₅	1	2	yes	—
159	#487	C₂×D₅	1	2	yes	—
160	#486	—	10	1	no	—
161	#486	—	10	1	no	—
162	#481	—	10	2	no	—
163	#480	—	10	2	no	—
164	#476	—	5	4	no	—
165	#474	—	5	4	no	—
166	#436	—	5	2	no	—
167	#436	—	10	1	no	—
168	C₈×D₄	—	5	2	no	—
169	C₈×D₄	—	10	1	no	—
170	#431	—	5	1	no	—
171	#431	—	5	1	no	—
172	#423	—	5	1	no	—
173	C₂×#103	—	2	1	no	—
174	C₂×#103	—	2	1	no	—
175	#408	—	10	1	no	—
176	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
177	#335	—	5	2	no	—
178	#334	—	5	2	no	—
179	#333	—	5	4	no	—
180	#332	—	5	4	no	—
181	C₂³×C₅	#121	1	1	yes	characteristic
182	C₂²×D₅	—	2	2	no	—
183	C₅×Q₈	—	4	2	no	—
184	C₅×Q₈	—	4	2	no	—
185	C₅×Q₈	—	8	2	no	—
186	C₅×Q₈	—	8	2	no	—
187	C₅×Q₈	—	8	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	—	8	2	no	—
193	C₂×C₄×C₅	—	8	2	no	—
194	C₂×C₄×C₅	#121	1	1	yes	characteristic
195	C₂×C₄×C₅	#121	1	1	yes	characteristic
196	C₂×C₄×C₅	#121	1	1	yes	characteristic
197	C₂×C₄×C₅	#135	1	2	yes	—
198	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
199	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
202	C₂²⋊D₅	—	4	4	no	—
203	C₂×Dic₅	—	2	2	no	—
204	D₂₀	—	2	4	no	—
205	D₂₀	—	4	2	no	—
206	D₂₀	—	4	2	no	—
207	C₄×D₅	—	4	2	no	—
208	C₄×D₅	—	4	2	no	—
209	Dic₁₀	—	4	2	no	—
210	C₅×C₈	—	2	1	no	—
211	C₅×C₈	—	2	1	no	—
212	C₅×C₈	—	2	1	no	—
213	C₅×C₈	—	2	1	no	—
214	#168	—	4	2	no	—
215	#168	—	4	2	no	—
216	#168	—	4	2	no	—
217	C₂×#41	—	10	1	no	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×Q₈	—	2	1	no	—
220	#131	—	20	4	no	—
221	C₂×M₁₆	—	10	1	no	—
222	C₂×M₁₆	—	10	2	no	—
223	C₂²×C₈	—	10	2	no	—
224	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
225	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	5	1	no	—
233	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
234	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
235	#127	—	5	1	no	—
236	#124	—	10	1	no	—
237	#121	—	5	2	no	—
238	#121	—	10	1	no	—
239	C₄×D₄	—	5	2	no	—
240	C₄×D₄	—	10	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
246	#107	—	10	4	no	—
247	#106	—	10	4	no	—
248	C₄⋊C₈	—	5	2	no	—
249	C₄⋊C₈	—	5	2	no	—
250	C₄⋊C₈	—	5	2	no	—
251	C₄⋊C₈	—	10	1	no	—
252	#103	—	2	2	no	—
253	#103	—	2	2	no	—
254	#103	—	2	2	no	—
255	#103	—	2	2	no	—
256	#102	—	10	4	no	—
257	#102	—	10	4	no	—
258	C₂²⋊C₈	—	10	1	no	—
259	C₂²⋊C₈	—	10	1	no	—
260	C₂²⋊C₈	—	10	2	no	—
261	C₂²⋊C₈	—	10	2	no	—
262	C₄×C₈	—	10	1	no	—
263	C₄×C₈	—	10	1	no	—
264	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
265	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
268	C₂²×C₅	—	2	1	no	—
269	C₂²×C₅	—	2	1	no	—
270	C₂²×C₅	#463	1	2	yes	—
271	C₂²×C₅	C₂×#121	1	1	yes	characteristic
272	C₂×D₅	—	4	2	no	—
273	C₂×D₅	—	4	2	no	—
274	C₂×D₅	—	4	2	no	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	2	1	no	—
277	C₄×C₅	—	8	2	no	—
278	C₄×C₅	—	8	2	no	—
279	C₄×C₅	#463	1	2	yes	—
280	C₄×C₅	#471	1	2	yes	—
281	C₄×C₅	C₂×#135	1	1	yes	characteristic
282	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₄×C₅	#578	1	2	yes	—
284	Dic₅	—	4	2	no	—
285	#41	—	20	2	no	—
286	#41	—	20	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	4	2	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	8	2	no	—
294	C₂×Q₈	—	10	1	no	—
295	C₂×D₄	—	5	2	no	—
296	C₂×D₄	—	10	2	no	—
297	C₂×D₄	—	10	1	no	—
298	C₂×D₄	—	10	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	10	2	no	—
302	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
303	C₂²×C₄	#711	1	1	yes	characteristic
304	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
305	M₁₆	—	20	4	no	—
306	M₁₆	—	20	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	10	2	no	—
310	C₂×C₈	—	10	2	no	—
311	C₂×C₈	—	10	2	no	—
312	C₂×C₈	—	20	4	no	—
313	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	5	2	no	—
326	C₄⋊C₄	—	10	1	no	—
327	C₂²⋊C₄	—	10	2	no	—
328	C₂²⋊C₄	—	10	2	no	—
329	C₄²	D₄×D₅	1	1	yes	characteristic
330	C₄²	#711	1	1	yes	characteristic
331	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
332	C₄²	C₂³×D₅	1	1	yes	characteristic
333	C₂×C₅	—	2	2	no	—
334	C₂×C₅	C₂×#463	1	1	yes	characteristic
335	C₂×C₅	#3059	1	1	yes	characteristic
336	C₂×C₅	C₂×#471	1	1	yes	characteristic
337	D₅	—	8	2	no	—
338	C₂³	—	10	2	no	—
339	C₂³	#4084	1	1	yes	characteristic
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	20	2	no	—
346	D₄	—	10	4	no	—
347	D₄	—	20	4	no	—
348	D₄	—	20	2	no	—
349	D₄	—	20	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	8	2	no	—
356	C₂×C₄	—	10	2	no	—
357	C₂×C₄	—	20	2	no	—
358	C₂×C₄	—	20	2	no	—
359	C₂×C₄	#4084	1	1	yes	characteristic
360	C₂×C₄	#4084	1	1	yes	characteristic
361	C₂×C₄	#4084	1	1	yes	characteristic
362	C₂×C₄	#4095	1	2	yes	—
363	C₂×C₄	#4096	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
365	C₂×C₄	C₂×#711	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	20	2	no	—
372	C₈	—	20	2	no	—
373	C₈	—	20	2	no	—
374	C₅	#19199	1	1	yes	characteristic, complemented, Sylow
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	20	2	no	—
378	C₂²	—	20	2	no	—
379	C₂²	—	20	2	no	—
380	C₂²	#65346	1	2	yes	—
381	C₂²	C₂×#4084	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	20	2	no	—
387	C₄	#65346	1	2	yes	—
388	C₄	#65466	1	2	yes	—
389	C₄	#66100	1	2	yes	—
390	C₄	C₂×#4095	1	1	yes	characteristic
391	C₄	C₂×#4096	1	1	yes	characteristic
392	C₂	—	2	2	no	—
393	C₂	—	40	2	no	—
394	C₂	C₂×#65346	1	1	yes	characteristic
395	C₂	C₂×#65466	1	1	yes	characteristic
396	C₂	#10615618	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4084
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#487	C₂
socle	C₂²×C₅	C₂×#121
prime radical	#65811	C₂²
hypercenter	C₂×#487	D₅
hypocenter	C₅	#19199
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#487	C₂	semidirect
5. #10602567	C₂	semidirect
6. #10602566	C₂	semidirect
7. #10602028	C₂	semidirect
8. #10602026	C₂	semidirect
9. #10600522	C₂	semidirect
95. C₂×#487	D₅	semidirect
374. C₅	#19199	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#487	2
#10614947	2
#10614946	2
#10610189	2
#10602567	2
#10602566	2
#10602028	2
#10602026	2
#10600522	2
#19199	5