Subgroups — #12069444

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611029	C₂	1	1	yes	maximal, characteristic
2	#10611017	C₂	1	1	yes	maximal, characteristic
3	#10610997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1453	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601544	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600649	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600648	C₂	1	1	yes	maximal, characteristic
8	#10598873	C₂	1	2	yes	maximal, complemented
9	#10598872	C₂	1	2	yes	maximal, complemented
10	#10598869	C₂	1	2	yes	maximal, complemented
11	C₈×#677	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
13	#65804	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₅×#431	C₂²	1	1	yes	characteristic
16	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	#65215	C₂²	1	1	yes	characteristic
18	#65214	C₂²	1	1	yes	characteristic
19	#65131	C₄	1	2	yes	—
20	#65130	C₄	1	2	yes	—
21	#65123	—	2	2	no	—
22	#65117	—	2	2	no	—
23	#65020	C₄	1	2	yes	—
24	D₅×C₂²⋊C₈	—	2	2	no	—
25	#65010	C₄	1	2	yes	—
26	#65009	—	2	2	no	—
27	C₄×#677	C₂²	1	2	yes	—
28	C₄×#677	C₂²	1	2	yes	—
29	C₈×D₂₀	C₂²	1	2	yes	—
30	C₈×D₂₀	C₂²	1	2	yes	—
31	#64968	C₂²	1	2	yes	complemented
32	C₈×Dic₁₀	C₂²	1	2	yes	complemented
33	C₅×#322	C₂²	1	1	yes	characteristic, complemented
34	C₅×#322	C₂²	1	1	yes	characteristic, complemented
35	C₅×#322	C₂²	1	1	yes	characteristic, complemented
36	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
37	#64679	C₂²	1	2	yes	—
38	#64679	C₂²	1	2	yes	—
39	#64678	C₂²	1	2	yes	complemented
40	#64677	C₂²	1	2	yes	complemented
41	#64676	C₂²	1	2	yes	complemented
42	#64676	C₂²	1	2	yes	complemented
43	#17924	—	5	1	no	maximal, Sylow
44	C₂²×D₂₀	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×D₅	—	2	1	no	—
46	C₂×C₅×M₁₆	—	2	1	no	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×#685	—	2	1	no	—
49	#4072	—	2	1	no	—
50	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
51	#4070	—	2	1	no	—
52	#4070	C₂×C₄	1	2	yes	—
53	C₂×#677	C₂×C₄	1	2	yes	—
54	C₂×#677	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅	—	2	2	no	—
56	C₂×C₈×D₅	—	2	2	no	—
57	C₄×D₂₀	—	2	1	no	—
58	C₄×D₂₀	—	2	1	no	—
59	C₄×D₂₀	C₂×C₄	1	2	yes	—
60	C₄×D₂₀	C₂³	1	2	yes	—
61	#4016	—	2	1	no	—
62	#4015	C₂×C₄	1	1	yes	characteristic
63	#4015	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀	—	2	1	no	—
65	C₄×Dic₁₀	C₂³	1	2	yes	—
66	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₅×C₄⋊C₈	D₄	1	2	yes	—
68	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₅×C₂²⋊C₈	D₄	1	2	yes	—
70	C₅×#97	—	2	1	no	—
71	C₄×C₅×C₈	C₂³	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	C₂²⋊#168	—	2	1	no	—
78	C₂²⋊#168	—	2	1	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	#3949	C₂×C₄	1	2	yes	—
84	#3949	C₂×C₄	1	2	yes	—
85	#3948	C₂×C₄	1	2	yes	—
86	#3948	C₂×C₄	1	2	yes	—
87	#3947	—	2	2	no	—
88	#3946	—	2	2	no	—
89	#3946	—	2	2	no	—
90	C₈×Dic₅	—	2	2	no	—
91	C₄⋊#168	C₂³	1	2	yes	—
92	C₄⋊#168	C₂³	1	2	yes	—
93	#3935	—	2	1	no	—
94	C₄×#168	—	2	1	no	—
95	#3101	—	5	1	no	—
96	#2985	—	5	1	no	—
97	#2972	—	5	1	no	—
98	#1585	—	5	2	no	—
99	#1582	—	5	2	no	—
100	SD₁₆⋊C₈	—	5	2	no	—
101	C₈×SD₁₆	—	5	2	no	—
102	#1572	—	5	1	no	—
103	#1490	—	5	1	no	—
104	#1485	—	5	1	no	—
105	#1453	D₅	1	1	yes	characteristic, complemented
106	C₂³×D₅	—	2	1	no	—
107	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	#41	1	1	yes	characteristic
110	C₂²×Dic₅	—	2	1	no	—
111	C₂×D₂₀	—	2	1	no	—
112	C₂×D₂₀	—	2	4	no	—
113	C₂×D₂₀	C₂²×C₄	1	2	yes	—
114	C₂×C₄×D₅	—	2	2	no	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×C₄×D₅	—	8	2	no	—
117	C₂×Dic₁₀	—	2	1	no	—
118	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
119	C₅×M₁₆	—	4	2	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
126	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	#41	1	1	yes	characteristic
129	C₄²×C₅	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₅	—	2	2	no	—
131	#685	—	2	2	no	—
132	#685	—	2	2	no	—
133	C₄⋊Dic₅	—	2	1	no	—
134	C₄⋊Dic₅	—	2	1	no	—
135	C₄⋊Dic₅	—	2	1	no	—
136	C₄⋊Dic₅	—	2	1	no	—
137	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
138	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
139	#683	—	2	2	no	—
140	#683	—	2	2	no	—
141	C₄×Dic₅	—	2	2	no	—
142	C₂×#168	—	2	2	no	—
143	C₂×#168	—	2	2	no	—
144	#677	—	2	4	no	—
145	#677	—	2	4	no	—
146	C₈×D₅	—	8	2	no	—
147	C₈×D₅	—	8	2	no	—
148	#533	—	5	1	no	—
149	C₂×C₄×D₄	—	5	1	no	—
150	#483	—	5	2	no	—
151	#476	—	5	2	no	—
152	#461	—	5	2	no	—
153	#450	—	5	2	no	—
154	#446	—	5	2	no	—
155	#446	—	10	2	no	—
156	C₈×Q₈	—	5	2	no	—
157	C₄×SD₁₆	—	5	2	no	—
158	C₄×SD₁₆	—	5	2	no	—
159	C₈×D₄	—	5	2	no	—
160	C₈×D₄	—	5	2	no	—
161	C₈×D₄	—	10	2	no	—
162	#433	—	10	2	no	—
163	#432	—	5	1	no	—
164	#431	—	5	1	no	—
165	#431	C₂×D₅	1	1	yes	characteristic
166	#431	C₂×D₅	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	—	10	2	no	—
168	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
169	#335	—	5	2	no	—
170	#334	—	5	2	no	—
171	Q₈⋊C₈	—	5	2	no	—
172	Q₈⋊C₈	—	5	2	no	—
173	D₄⋊C₈	—	5	2	no	—
174	D₄⋊C₈	—	5	2	no	—
175	#322	C₂×D₅	1	1	yes	characteristic, complemented
176	#322	C₂×D₅	1	1	yes	characteristic, complemented
177	#322	C₂×D₅	1	1	yes	characteristic, complemented
178	C₈²	C₂×D₅	1	1	yes	characteristic, complemented
179	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
180	C₂²×D₅	—	2	2	no	—
181	C₂²×D₅	—	2	4	no	—
182	C₂²×D₅	—	8	1	no	—
183	C₂²×D₅	—	8	1	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
193	C₂×Dic₅	—	2	2	no	—
194	C₂×Dic₅	—	2	2	no	—
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	—	8	1	no	—
197	D₂₀	—	2	4	no	—
198	D₂₀	—	2	2	no	—
199	D₂₀	#131	1	4	yes	—
200	C₄×D₅	—	4	2	no	—
201	C₄×D₅	—	4	2	no	—
202	C₄×D₅	—	8	2	no	—
203	Dic₁₀	—	2	2	no	—
204	Dic₁₀	—	2	2	no	—
205	Dic₁₀	—	8	1	no	—
206	C₅×C₈	—	2	2	no	—
207	C₅×C₈	—	2	2	no	—
208	C₅×C₈	—	2	2	no	—
209	C₅×C₈	—	2	2	no	—
210	C₅×C₈	—	2	2	no	—
211	C₅×C₈	—	4	1	no	—
212	#168	—	4	2	no	—
213	#168	—	4	2	no	—
214	C₂²×D₄	—	5	1	no	—
215	C₂³×C₄	—	10	1	no	—
216	C₂×SD₁₆	—	5	2	no	—
217	C₂×SD₁₆	—	5	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂²×C₈	—	10	2	no	—
220	C₂²×C₈	—	10	2	no	—
221	C₄⋊Q₈	—	5	1	no	—
222	C₄⋊Q₈	—	5	1	no	—
223	#125	—	10	1	no	—
224	C₂²⋊Q₈	—	5	2	no	—
225	C₂²⋊Q₈	—	10	1	no	—
226	C₄×Q₈	—	5	2	no	—
227	C₄×Q₈	—	10	1	no	—
228	C₄×D₄	—	5	2	no	—
229	C₄×D₄	—	5	2	no	—
230	C₄×D₄	—	10	1	no	—
231	C₄×D₄	—	10	1	no	—
232	#117	—	10	1	no	—
233	C₂×C₄⋊C₄	—	5	1	no	—
234	C₂×C₂²⋊C₄	—	10	1	no	—
235	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
236	#106	—	5	2	no	—
237	#106	—	5	2	no	—
238	C₄⋊C₈	—	5	2	no	—
239	C₄⋊C₈	—	5	2	no	—
240	C₄⋊C₈	—	10	2	no	—
241	C₄⋊C₈	—	10	2	no	—
242	C₄⋊C₈	C₄×D₅	1	2	yes	—
243	C₄⋊C₈	D₂₀	1	2	yes	—
244	#103	—	5	2	no	—
245	#103	—	5	2	no	—
246	#102	—	5	2	no	—
247	#102	—	5	2	no	—
248	C₂²⋊C₈	—	10	1	no	—
249	C₂²⋊C₈	—	10	1	no	—
250	C₂²⋊C₈	—	10	2	no	—
251	C₂²⋊C₈	—	10	2	no	—
252	C₂²⋊C₈	C₄×D₅	1	2	yes	—
253	C₂²⋊C₈	D₂₀	1	2	yes	—
254	#97	—	2	1	no	—
255	#97	—	10	2	no	—
256	#97	—	10	1	no	—
257	C₄×C₈	—	10	2	no	—
258	C₄×C₈	—	10	1	no	—
259	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
260	C₄×C₈	C₂²×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	C₂²×C₅	—	2	1	no	—
266	C₂²×C₅	—	2	1	no	—
267	C₂²×C₅	—	2	1	no	—
268	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
269	C₂×D₅	—	2	4	no	—
270	C₂×D₅	—	4	2	no	—
271	C₂×D₅	—	4	2	no	—
272	C₂×D₅	—	8	4	no	—
273	C₄×C₅	—	2	1	no	—
274	C₄×C₅	—	2	1	no	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	2	1	no	—
277	C₄×C₅	C₂×#131	1	2	yes	—
278	C₄×C₅	#577	1	2	yes	—
279	Dic₅	—	4	2	no	—
280	Dic₅	—	4	2	no	—
281	Dic₅	—	4	2	no	—
282	C₂⁴	—	10	1	no	—
283	C₂×Q₈	—	5	2	no	—
284	C₂×Q₈	—	10	1	no	—
285	C₂×D₄	—	5	2	no	—
286	C₂×D₄	—	10	4	no	—
287	C₂×D₄	—	10	1	no	—
288	C₂²×C₄	—	10	1	no	—
289	C₂²×C₄	—	10	2	no	—
290	C₂²×C₄	—	10	2	no	—
291	C₂²×C₄	—	40	2	no	—
292	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
294	C₂²×C₄	#709	1	1	yes	characteristic
295	SD₁₆	—	10	4	no	—
296	SD₁₆	—	10	4	no	—
297	M₁₆	—	4	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	10	2	no	—
301	C₂×C₈	—	10	2	no	—
302	C₂×C₈	—	40	2	no	—
303	C₂×C₈	—	40	2	no	—
304	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
305	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
306	C₂×C₈	C₂×D₂₀	1	2	yes	—
307	C₂×C₈	C₂×D₂₀	1	2	yes	—
308	C₄⋊C₄	—	5	2	no	—
309	C₄⋊C₄	—	5	2	no	—
310	C₄⋊C₄	—	10	2	no	—
311	C₄⋊C₄	—	10	1	no	—
312	C₄⋊C₄	—	10	1	no	—
313	C₄⋊C₄	—	10	1	no	—
314	C₄⋊C₄	—	10	2	no	—
315	C₄⋊C₄	—	10	1	no	—
316	C₂²⋊C₄	—	10	2	no	—
317	C₂²⋊C₄	—	10	2	no	—
318	C₂²⋊C₄	—	10	2	no	—
319	C₄²	—	10	2	no	—
320	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
321	C₄²	C₂×D₂₀	1	1	yes	characteristic
322	C₄²	#709	1	1	yes	characteristic
323	C₄²	C₂³×D₅	1	1	yes	characteristic
324	C₂×C₅	—	4	1	no	—
325	C₂×C₅	#2929	1	1	yes	characteristic
326	C₂×C₅	#2949	1	1	yes	characteristic
327	C₂×C₅	#2956	1	1	yes	characteristic
328	D₅	—	4	4	no	—
329	C₂³	—	10	4	no	—
330	C₂³	—	10	2	no	—
331	C₂³	—	40	1	no	—
332	C₂³	—	40	1	no	—
333	C₂³	C₄×D₂₀	1	1	yes	characteristic
334	Q₈	—	10	2	no	—
335	Q₈	—	10	2	no	—
336	Q₈	—	40	1	no	—
337	D₄	—	5	4	no	—
338	D₄	—	10	4	no	—
339	D₄	—	10	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	10	2	no	—
344	C₂×C₄	—	10	2	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	20	2	no	—
347	C₂×C₄	—	20	2	no	—
348	C₂×C₄	—	40	2	no	—
349	C₂×C₄	—	40	1	no	—
350	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
351	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
355	C₂×C₄	C₂×#709	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	1	no	—
362	C₈	—	20	2	no	—
363	C₈	—	20	2	no	—
364	C₅	#17924	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	10	4	no	—
369	C₂²	—	20	2	no	—
370	C₂²	—	20	2	no	—
371	C₂²	—	40	4	no	—
372	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	20	2	no	—
378	C₄	—	20	2	no	—
379	C₄	—	20	2	no	—
380	C₄	D₅×#131	1	2	yes	—
381	C₄	#66085	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	20	4	no	—
384	C₂	#10610272	1	1	yes	characteristic
385	C₂	#10610284	1	1	yes	characteristic
386	C₂	#10610303	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1453	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₂₀	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1453	D₅
hypocenter	C₅	#17924
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
3. #10610997	C₂	semidirect
4. C₅×#1453	C₂	semidirect
5. #10601544	C₂	semidirect
6. #10600649	C₂	semidirect
8. #10598873	C₂	semidirect
9. #10598872	C₂	semidirect
10. #10598869	C₂	semidirect
11. C₈×#677	C₂	semidirect
31. #64968	C₂²	semidirect
32. C₈×Dic₁₀	C₂²	semidirect
33. C₅×#322	C₂²	semidirect
34. C₅×#322	C₂²	semidirect
35. C₅×#322	C₂²	semidirect
36. C₅×C₈²	C₂²	semidirect
39. #64678	C₂²	semidirect
40. #64677	C₂²	semidirect
41. #64676	C₂²	semidirect
42. #64676	C₂²	semidirect
105. #1453	D₅	semidirect
175. #322	C₂×D₅	semidirect
176. #322	C₂×D₅	semidirect
177. #322	C₂×D₅	semidirect
178. C₈²	C₂×D₅	semidirect
364. C₅	#17924	semidirect

Maximal subgroups

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

subgroup	index
#10611029	2
#10611017	2
#10610997	2
C₅×#1453	2
#10601544	2
#10600649	2
#10600648	2
#10598873	2
#10598872	2
#10598869	2
C₈×#677	2
#17924	5