Subgroups — #12071089

Order 1280.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610242	C₂	1	2	yes	maximal, complemented
2	#10610200	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601553	C₂	1	2	yes	maximal, complemented
5	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598878	C₂	1	2	yes	maximal, complemented
7	#10598872	C₂	1	2	yes	maximal, complemented
8	#10598849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#10598845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65813	C₂²	1	2	yes	—
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65417	—	2	2	no	—
17	#65416	—	2	2	no	—
18	#65405	C₄	1	1	yes	characteristic
19	#65404	C₄	1	1	yes	characteristic
20	#65403	C₄	1	2	yes	—
21	#65214	C₂²	1	2	yes	—
22	#64998	—	2	2	no	—
23	#64997	—	2	2	no	—
24	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
25	#64986	C₄	1	2	yes	—
26	#64985	C₄	1	1	yes	characteristic
27	C₄×D₄₀	C₂²	1	1	yes	characteristic
28	C₄×#677	C₂²	1	2	yes	—
29	#64977	C₂²	1	2	yes	—
30	C₈×D₂₀	C₂²	1	2	yes	—
31	#64972	C₄	1	1	yes	characteristic
32	#64968	C₂²	1	2	yes	—
33	C₈×Dic₁₀	C₂²	1	2	yes	—
34	C₅×#322	C₂²	1	1	yes	characteristic
35	C₅×#322	C₂²	1	1	yes	characteristic
36	C₅×C₈²	C₂²	1	1	yes	characteristic
37	C₅×C₈²	C₂²	1	1	yes	characteristic
38	#64679	C₂²	1	2	yes	—
39	#64679	C₂²	1	2	yes	—
40	#64678	C₂²	1	1	yes	characteristic
41	#64678	C₂²	1	1	yes	characteristic
42	#64677	C₂²	1	2	yes	—
43	#64676	C₂²	1	2	yes	—
44	#64676	C₂²	1	2	yes	—
45	#18069	—	5	1	no	maximal, Sylow
46	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
47	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	#4076	C₂×C₄	1	2	yes	—
52	#4075	—	2	2	no	—
53	C₄×C₂²⋊D₅	—	2	2	no	—
54	#4070	C₂×C₄	1	2	yes	—
55	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
56	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
57	C₂×#677	C₂×C₄	1	2	yes	—
58	C₂×#676	—	2	2	no	—
59	C₂×C₈×D₅	—	2	2	no	—
60	#4022	—	2	2	no	—
61	#4021	C₂×C₄	1	2	yes	—
62	C₄×D₂₀	C₂³	1	2	yes	—
63	#4018	—	2	2	no	—
64	#4016	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₀	C₂³	1	2	yes	—
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₂²⋊C₈	—	2	1	no	—
68	C₅×#97	D₄	1	1	yes	characteristic
69	C₅×#97	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂²⋊#168	—	2	2	no	—
79	#3953	C₂×C₄	1	2	yes	—
80	#3953	C₂×C₄	1	2	yes	—
81	#3952	—	2	2	no	—
82	#3952	—	2	2	no	—
83	#3950	C₂×C₄	1	1	yes	characteristic
84	#3950	C₂×C₄	1	1	yes	characteristic
85	#3949	C₂×C₄	1	2	yes	—
86	#3948	C₂×C₄	1	2	yes	—
87	#3948	C₂×C₄	1	2	yes	—
88	#3947	—	2	2	no	—
89	#3946	—	2	2	no	—
90	#3946	—	2	2	no	—
91	C₈×Dic₅	—	2	2	no	—
92	C₄⋊#168	C₂³	1	2	yes	—
93	C₄⋊#168	C₂³	1	2	yes	—
94	C₄×#168	—	2	2	no	—
95	#3168	—	5	1	no	—
96	#2969	—	5	2	no	—
97	#1591	—	5	1	no	—
98	#1589	—	5	1	no	—
99	#1583	—	5	2	no	—
100	#1582	—	5	2	no	—
101	C₈×Q₁₆	—	5	1	no	—
102	C₈×D₈	—	5	1	no	—
103	#1564	—	5	1	no	—
104	#1482	—	5	2	no	—
105	C₈×M₁₆	D₅	1	1	yes	characteristic, complemented
106	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	#41	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	2	2	no	—
110	C₂×D₂₀	C₂²×C₄	1	2	yes	—
111	C₂×C₄×D₅	—	2	2	no	—
112	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
113	C₅×M₁₆	—	2	2	no	—
114	C₅×M₁₆	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	4	1	no	—
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅	#41	1	1	yes	characteristic
130	C₄²×C₅	C₂⁴	1	1	yes	characteristic
131	C₂²⋊Dic₅	—	2	2	no	—
132	#685	—	2	2	no	—
133	#685	—	2	2	no	—
134	#685	—	2	2	no	—
135	#685	—	2	2	no	—
136	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
137	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
138	#683	—	2	2	no	—
139	#683	—	2	2	no	—
140	#683	—	2	2	no	—
141	#683	—	2	2	no	—
142	C₄×Dic₅	—	2	2	no	—
143	C₂×#168	—	2	2	no	—
144	C₂×#168	—	2	2	no	—
145	Dic₂₀	—	2	2	no	—
146	D₄₀	—	2	2	no	—
147	#677	—	4	2	no	—
148	#676	—	4	4	no	—
149	C₈×D₅	—	4	4	no	—
150	#529	—	5	2	no	—
151	#499	—	5	1	no	—
152	#495	—	5	1	no	—
153	#488	—	5	2	no	—
154	#467	—	5	1	no	—
155	#466	—	5	1	no	—
156	#465	—	5	2	no	—
157	#446	—	5	2	no	—
158	C₈×Q₈	—	5	2	no	—
159	C₄×Q₁₆	—	5	1	no	—
160	C₄×SD₁₆	—	5	2	no	—
161	C₄×D₈	—	5	1	no	—
162	#436	—	5	2	no	—
163	#436	—	10	2	no	—
164	#435	—	10	2	no	—
165	C₈×D₄	—	5	2	no	—
166	#433	—	10	2	no	—
167	#431	—	5	2	no	—
168	#431	C₂×D₅	1	1	yes	characteristic
169	#405	—	10	2	no	—
170	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
172	#335	—	5	1	no	—
173	#335	—	5	1	no	—
174	#334	—	5	2	no	—
175	Q₈⋊C₈	—	5	2	no	—
176	Q₈⋊C₈	—	5	2	no	—
177	D₄⋊C₈	—	5	2	no	—
178	D₄⋊C₈	—	5	2	no	—
179	#322	C₂×D₅	1	1	yes	characteristic
180	#322	C₂×D₅	1	1	yes	characteristic
181	C₈²	C₂×D₅	1	1	yes	characteristic
182	C₈²	C₂×D₅	1	1	yes	characteristic
183	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
184	C₂²×D₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	1	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	—	2	1	no	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
196	C₂²⋊D₅	—	8	2	no	—
197	C₂×Dic₅	—	2	2	no	—
198	C₂×Dic₅	—	2	2	no	—
199	C₂×Dic₅	—	2	2	no	—
200	D₂₀	—	2	2	no	—
201	D₂₀	—	2	2	no	—
202	C₄×D₅	—	4	2	no	—
203	C₄×D₅	—	4	2	no	—
204	Dic₁₀	—	2	2	no	—
205	Dic₁₀	—	2	2	no	—
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	2	1	no	—
208	C₅×C₈	—	2	1	no	—
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	—	2	1	no	—
211	C₅×C₈	—	4	1	no	—
212	C₅×C₈	—	4	1	no	—
213	C₅×C₈	#131	1	2	yes	—
214	C₅×C₈	#135	1	2	yes	—
215	C₅×C₈	#135	1	2	yes	—
216	#168	—	4	2	no	—
217	#168	—	4	2	no	—
218	C₂×Q₁₆	—	5	1	no	—
219	C₂×SD₁₆	—	5	2	no	—
220	C₂×D₈	—	5	1	no	—
221	C₂×M₁₆	—	10	2	no	—
222	C₂×M₁₆	D₂₀	1	1	yes	characteristic
223	C₂×M₁₆	D₂₀	1	1	yes	characteristic
224	C₂²×C₈	—	10	2	no	—
225	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
226	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
227	#126	—	10	2	no	—
228	#125	—	5	2	no	—
229	#124	—	5	2	no	—
230	#123	—	10	2	no	—
231	C₂²⋊Q₈	—	5	2	no	—
232	#121	—	5	2	no	—
233	C₄×Q₈	—	5	2	no	—
234	C₄×D₄	—	5	2	no	—
235	C₄×D₄	—	10	2	no	—
236	#117	—	10	2	no	—
237	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
238	#107	—	5	1	no	—
239	#107	—	5	1	no	—
240	#106	—	5	2	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	5	2	no	—
243	C₄⋊C₈	—	5	2	no	—
244	C₄⋊C₈	—	10	2	no	—
245	C₄⋊C₈	—	10	2	no	—
246	#103	—	5	2	no	—
247	#103	—	5	2	no	—
248	#102	—	5	2	no	—
249	#102	—	5	2	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	10	2	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	C₂²⋊C₈	—	10	2	no	—
254	#97	—	10	2	no	—
255	#97	D₂₀	1	1	yes	characteristic
256	#97	D₂₀	1	1	yes	characteristic
257	C₄×C₈	—	10	2	no	—
258	C₄×C₈	—	10	2	no	—
259	C₄×C₈	C₄×D₅	1	1	yes	characteristic
260	C₄×C₈	C₄×D₅	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
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	C₂²×C₅	—	2	1	no	—
268	C₂²×C₅	—	2	1	no	—
269	C₂²×C₅	—	2	1	no	—
270	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
271	C₂×D₅	—	4	2	no	—
272	C₂×D₅	—	4	2	no	—
273	C₂×D₅	—	4	2	no	—
274	C₄×C₅	—	2	1	no	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	2	1	no	—
277	C₄×C₅	—	2	1	no	—
278	C₄×C₅	C₂×#131	1	1	yes	characteristic
279	C₄×C₅	#568	1	1	yes	characteristic
280	C₄×C₅	C₂×#135	1	1	yes	characteristic
281	C₄×C₅	C₂×#135	1	1	yes	characteristic
282	Dic₅	—	4	2	no	—
283	Dic₅	—	4	2	no	—
284	Dic₅	—	4	2	no	—
285	C₂×Q₈	—	5	2	no	—
286	C₂×D₄	—	5	2	no	—
287	C₂×D₄	—	10	2	no	—
288	C₂²×C₄	—	10	2	no	—
289	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
291	C₂²×C₄	#709	1	1	yes	characteristic
292	Q₁₆	—	10	2	no	—
293	SD₁₆	—	20	2	no	—
294	D₈	—	10	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	20	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	10	2	no	—
303	C₂×C₈	—	10	2	no	—
304	C₂×C₈	—	20	4	no	—
305	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
308	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
309	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
310	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
312	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
313	C₄⋊C₄	—	5	2	no	—
314	C₄⋊C₄	—	5	2	no	—
315	C₄⋊C₄	—	10	2	no	—
316	C₄⋊C₄	—	10	2	no	—
317	C₄⋊C₄	—	10	2	no	—
318	C₄⋊C₄	—	10	2	no	—
319	C₂²⋊C₄	—	10	2	no	—
320	C₂²⋊C₄	—	10	2	no	—
321	C₂²⋊C₄	—	10	2	no	—
322	C₂²⋊C₄	—	10	2	no	—
323	C₂²⋊C₄	—	10	2	no	—
324	C₄²	—	10	2	no	—
325	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
326	C₄²	C₂×D₂₀	1	1	yes	characteristic
327	C₄²	#709	1	1	yes	characteristic
328	C₄²	C₂³×D₅	1	1	yes	characteristic
329	C₂×C₅	—	4	1	no	—
330	C₂×C₅	#2935	1	1	yes	characteristic
331	C₂×C₅	C₄×#135	1	1	yes	characteristic
332	C₂×C₅	#2957	1	1	yes	characteristic
333	D₅	—	8	2	no	—
334	C₂³	—	10	2	no	—
335	C₂³	C₄×D₂₀	1	1	yes	characteristic
336	Q₈	—	10	2	no	—
337	Q₈	—	10	2	no	—
338	D₄	—	10	2	no	—
339	D₄	—	10	2	no	—
340	D₄	—	40	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	10	2	no	—
347	C₂×C₄	—	10	2	no	—
348	C₂×C₄	—	10	2	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	20	2	no	—
351	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
353	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
356	C₂×C₄	C₂×#709	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	20	2	no	—
365	C₈	—	20	2	no	—
366	C₈	#4050	1	2	yes	—
367	C₈	#4050	1	2	yes	—
368	C₈	#4053	1	2	yes	—
369	C₅	#18069	1	1	yes	characteristic, complemented, Sylow
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	20	2	no	—
374	C₂²	—	20	2	no	—
375	C₂²	—	20	2	no	—
376	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	20	2	no	—
382	C₄	—	20	2	no	—
383	C₄	—	20	2	no	—
384	C₄	C₂×#4050	1	1	yes	characteristic
385	C₄	C₂×#4050	1	1	yes	characteristic
386	C₄	C₂×#4053	1	1	yes	characteristic
387	C₄	#66079	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	40	2	no	—
390	C₂	C₄×#4050	1	1	yes	characteristic
391	C₂	#10610253	1	1	yes	characteristic
392	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65404	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#18069
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #10610242	C₂	semidirect
3. C₅×C₈×M₁₆	C₂	semidirect
4. #10601553	C₂	semidirect
5. #10600522	C₂	semidirect
6. #10598878	C₂	semidirect
7. #10598872	C₂	semidirect
8. #10598849	C₂	semidirect
9. C₈×Dic₂₀	C₂	semidirect
10. #10598845	C₂	semidirect
11. C₈×D₄₀	C₂	semidirect
105. C₈×M₁₆	D₅	semidirect
369. C₅	#18069	semidirect

Maximal subgroups

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

subgroup	index
#10610242	2
#10610200	2
C₅×C₈×M₁₆	2
#10601553	2
#10600522	2
#10598878	2
#10598872	2
#10598849	2
C₈×Dic₂₀	2
#10598845	2
C₈×D₄₀	2
#18069	5