Subgroups — #12069834

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611314	C₂	1	1	yes	maximal, characteristic
2	#10611132	C₂	1	2	yes	maximal, complemented
3	C₅×C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601674	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#687	C₂	1	2	yes	maximal, complemented
6	#10599363	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599357	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598970	C₂	1	2	yes	maximal, complemented
9	#10598954	C₂	1	2	yes	maximal, complemented
10	C₈×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
12	#65891	C₂²	1	2	yes	—
13	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₅×C₈×D₄	C₂²	1	2	yes	—
15	#65446	C₄	1	1	yes	characteristic
16	#65444	C₄	1	1	yes	characteristic
17	C₈×C₂²⋊D₅	—	2	2	no	—
18	#65390	—	2	2	no	—
19	C₄×#687	C₂²	1	2	yes	—
20	#65301	C₂²	1	2	yes	—
21	#65176	C₄	1	1	yes	characteristic
22	#65167	C₄	1	1	yes	characteristic
23	#65124	C₂²	1	2	yes	—
24	#65120	C₂²	1	2	yes	—
25	#65067	C₄	1	2	yes	—
26	#65057	C₄	1	2	yes	—
27	#65017	—	2	2	no	—
28	#65009	—	2	2	no	—
29	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₀	C₂²	1	2	yes	—
34	C₅×#335	C₂²	1	1	yes	characteristic
35	C₅×D₄⋊C₈	C₂²	1	2	yes	—
36	C₅×C₈²	C₂²	1	1	yes	characteristic
37	D₄⋊#168	C₂²	1	2	yes	—
38	#64704	C₂²	1	2	yes	—
39	#64700	C₂²	1	2	yes	—
40	#64678	C₂²	1	1	yes	characteristic
41	#64676	C₂²	1	2	yes	—
42	C₈×#168	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17959	—	5	1	no	maximal, Sylow
46	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	—	2	2	no	—
48	C₄×C₅×D₄	C₂³	1	2	yes	—
49	#4084	C₂×C₄	1	2	yes	—
50	#4082	C₂×C₄	1	2	yes	—
51	C₂×#687	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₅	—	2	2	no	—
53	#4072	—	2	2	no	—
54	C₂×#681	—	2	2	no	—
55	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅	—	2	1	no	—
57	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅	D₄	1	1	yes	characteristic
60	C₂×C₈×D₅	D₄	1	1	yes	characteristic
61	#4043	C₂×C₄	1	2	yes	—
62	#4036	C₂×C₄	1	2	yes	—
63	#4029	—	2	2	no	—
64	#4025	—	2	2	no	—
65	C₄²×D₅	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₀	C₂³	1	2	yes	—
67	C₅×#107	C₂×C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×#102	C₂×C₄	1	2	yes	—
70	C₅×C₂²⋊C₈	—	2	2	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	#3964	C₂×C₄	1	2	yes	—
75	C₂²⋊#168	—	2	2	no	—
76	#3952	—	2	2	no	—
77	#3950	C₂×C₄	1	1	yes	characteristic
78	#3948	C₂×C₄	1	2	yes	—
79	#3947	—	2	2	no	—
80	#3947	—	2	2	no	—
81	#3946	—	2	2	no	—
82	#3946	—	2	2	no	—
83	C₈×Dic₅	—	2	2	no	—
84	C₈×Dic₅	—	2	1	no	—
85	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₅	D₄	1	1	yes	characteristic
88	C₈×Dic₅	D₄	1	1	yes	characteristic
89	#3942	C₂×C₄	1	2	yes	—
90	#3940	C₂×C₄	1	2	yes	—
91	C₄⋊#168	C₂³	1	2	yes	—
92	C₄×#168	C₂³	1	1	yes	characteristic
93	C₄×#168	C₂³	1	1	yes	characteristic
94	C₄×#168	C₂³	1	1	yes	characteristic
95	#3168	—	5	1	no	—
96	#2966	—	5	2	no	—
97	#1591	—	5	1	no	—
98	#1590	—	5	2	no	—
99	#1589	—	5	1	no	—
100	C₈×Q₁₆	—	5	1	no	—
101	C₈×SD₁₆	—	5	2	no	—
102	C₈×D₈	D₅	1	1	yes	characteristic, complemented
103	#1568	—	5	1	no	—
104	#1494	—	5	2	no	—
105	C₂×C₈²	—	5	1	no	—
106	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₅	—	2	2	no	—
108	C₂×C₂²⋊D₅	—	2	2	no	—
109	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅	#41	1	1	yes	characteristic
112	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
113	C₅×D₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	4	4	no	—
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
121	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₅	—	2	2	no	—
125	C₂²⋊Dic₅	—	2	2	no	—
126	C₂²⋊Dic₅	—	2	2	no	—
127	#687	—	2	4	no	—
128	#685	—	2	2	no	—
129	C₄⋊Dic₅	—	2	2	no	—
130	C₄⋊Dic₅	—	2	2	no	—
131	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
132	#683	—	2	2	no	—
133	#683	—	2	2	no	—
134	C₄×Dic₅	—	2	2	no	—
135	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₅	#41	1	1	yes	characteristic
138	#681	—	4	4	no	—
139	C₂×#168	—	2	1	no	—
140	C₂×#168	—	2	2	no	—
141	C₂×#168	C₂²×C₄	1	1	yes	characteristic
142	C₂×#168	C₂²×C₄	1	1	yes	characteristic
143	C₂×#168	C₂×D₄	1	1	yes	characteristic
144	C₂×#168	C₂×D₄	1	1	yes	characteristic
145	Dic₂₀	—	2	2	no	—
146	C₈×D₅	—	2	4	no	—
147	C₈×D₅	—	2	2	no	—
148	C₈×D₅	—	2	2	no	—
149	C₈×D₅	—	2	2	no	—
150	C₈×D₅	—	2	2	no	—
151	#529	—	5	2	no	—
152	#499	—	5	1	no	—
153	#495	—	5	1	no	—
154	#488	—	5	2	no	—
155	#467	—	5	1	no	—
156	#466	—	5	1	no	—
157	#465	—	5	2	no	—
158	#446	—	5	2	no	—
159	C₈×Q₈	—	5	2	no	—
160	C₄×Q₁₆	—	5	1	no	—
161	C₄×SD₁₆	—	5	2	no	—
162	C₄×D₈	C₂×D₅	1	1	yes	characteristic
163	#436	—	5	2	no	—
164	#435	—	10	2	no	—
165	C₈×D₄	—	10	2	no	—
166	C₈×D₄	C₂×D₅	1	2	yes	—
167	#433	—	10	2	no	—
168	#432	—	5	2	no	—
169	#405	—	10	2	no	—
170	C₂×C₄×C₈	—	5	1	no	—
171	C₂×C₄×C₈	—	5	1	no	—
172	C₂×C₄×C₈	—	5	1	no	—
173	#335	—	5	1	no	—
174	#335	C₂×D₅	1	1	yes	characteristic
175	#334	—	5	2	no	—
176	Q₈⋊C₈	—	5	2	no	—
177	Q₈⋊C₈	—	5	2	no	—
178	D₄⋊C₈	—	5	2	no	—
179	D₄⋊C₈	C₂×D₅	1	2	yes	—
180	C₈²	—	5	1	no	—
181	C₈²	—	5	1	no	—
182	C₈²	—	5	1	no	—
183	C₈²	C₂×D₅	1	1	yes	characteristic
184	C₂³×C₅	—	2	2	no	—
185	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
186	C₅×D₄	—	2	2	no	—
187	C₅×D₄	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
194	C₂²⋊D₅	—	4	4	no	—
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	—	2	2	no	—
197	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
198	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
199	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
200	C₄×D₅	—	2	2	no	—
201	C₄×D₅	—	2	1	no	—
202	C₄×D₅	—	2	1	no	—
203	C₄×D₅	—	2	1	no	—
204	C₄×D₅	—	2	1	no	—
205	Dic₁₀	—	2	2	no	—
206	Dic₁₀	—	2	2	no	—
207	C₅×C₈	—	2	2	no	—
208	C₅×C₈	—	2	1	no	—
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	—	4	2	no	—
211	C₅×C₈	#131	1	2	yes	—
212	C₅×C₈	#135	1	2	yes	—
213	#168	—	2	2	no	—
214	#168	—	2	1	no	—
215	#168	—	2	1	no	—
216	#168	—	4	2	no	—
217	#168	#131	1	2	yes	—
218	#168	#135	1	2	yes	—
219	C₂×Q₁₆	—	5	1	no	—
220	C₂×SD₁₆	—	5	2	no	—
221	C₂×D₈	C₄×D₅	1	1	yes	characteristic
222	C₂×M₁₆	—	10	2	no	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	—	5	1	no	—
225	C₂²×C₈	—	5	1	no	—
226	C₂²×C₈	—	5	1	no	—
227	C₂²×C₈	—	5	1	no	—
228	C₂²×C₈	—	10	1	no	—
229	#126	—	10	2	no	—
230	#125	—	5	2	no	—
231	#124	—	5	2	no	—
232	#123	—	10	2	no	—
233	C₂²⋊Q₈	—	5	2	no	—
234	#121	—	5	2	no	—
235	C₄×Q₈	—	5	2	no	—
236	C₄×D₄	—	10	2	no	—
237	C₄×D₄	C₂²×D₅	1	2	yes	—
238	#117	—	10	2	no	—
239	C₂×C₄²	—	5	1	no	—
240	#107	—	5	1	no	—
241	#107	C₄×D₅	1	1	yes	characteristic
242	#106	—	5	2	no	—
243	C₄⋊C₈	—	5	2	no	—
244	C₄⋊C₈	—	10	2	no	—
245	C₄⋊C₈	—	10	2	no	—
246	C₄⋊C₈	C₂²×D₅	1	2	yes	—
247	#103	—	5	2	no	—
248	#103	—	5	2	no	—
249	#102	—	5	2	no	—
250	#102	C₄×D₅	1	2	yes	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	C₂²⋊C₈	—	10	2	no	—
254	#97	—	10	2	no	—
255	#97	—	10	2	no	—
256	C₄×C₈	—	5	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	—	5	1	no	—
260	C₄×C₈	—	5	1	no	—
261	C₄×C₈	—	5	1	no	—
262	C₄×C₈	—	5	1	no	—
263	C₄×C₈	—	10	1	no	—
264	C₄×C₈	—	10	2	no	—
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	—	4	2	no	—
269	C₂²×C₅	—	4	2	no	—
270	C₂²×C₅	—	4	2	no	—
271	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂×D₅	—	2	1	no	—
273	C₂×D₅	—	2	1	no	—
274	C₂×D₅	#443	1	2	yes	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	4	2	no	—
277	C₄×C₅	C₂×#131	1	1	yes	characteristic
278	C₄×C₅	C₂×#131	1	1	yes	characteristic
279	C₄×C₅	C₂×#135	1	1	yes	characteristic
280	C₄×C₅	C₂×#135	1	1	yes	characteristic
281	Dic₅	—	2	1	no	—
282	Dic₅	—	2	1	no	—
283	Dic₅	—	2	1	no	—
284	Dic₅	—	4	2	no	—
285	Dic₅	—	4	2	no	—
286	C₂×Q₈	—	5	2	no	—
287	C₂×D₄	—	10	2	no	—
288	C₂×D₄	C₂×C₄×D₅	1	2	yes	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	5	1	no	—
291	C₂²×C₄	—	5	1	no	—
292	C₂²×C₄	—	5	1	no	—
293	Q₁₆	—	10	2	no	—
294	SD₁₆	—	10	4	no	—
295	D₈	—	2	2	no	—
296	M₁₆	—	20	4	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	4	no	—
300	C₂×C₈	—	5	1	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	5	1	no	—
303	C₂×C₈	—	5	1	no	—
304	C₂×C₈	—	10	2	no	—
305	C₂×C₈	—	10	2	no	—
306	C₂×C₈	—	10	4	no	—
307	C₂×C₈	—	10	1	no	—
308	C₂×C₈	—	10	2	no	—
309	C₂×C₈	—	10	2	no	—
310	C₂×C₈	—	10	2	no	—
311	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
313	C₂×C₈	D₄×D₅	1	1	yes	characteristic
314	C₂×C₈	D₄×D₅	1	1	yes	characteristic
315	C₄⋊C₄	—	5	2	no	—
316	C₄⋊C₄	—	10	2	no	—
317	C₄⋊C₄	—	10	2	no	—
318	C₄⋊C₄	—	10	2	no	—
319	C₄⋊C₄	—	10	2	no	—
320	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	10	2	no	—
323	C₂²⋊C₄	—	10	2	no	—
324	C₂²⋊C₄	—	10	2	no	—
325	C₂²⋊C₄	—	10	2	no	—
326	C₄²	—	5	1	no	—
327	C₄²	—	5	1	no	—
328	C₄²	—	5	1	no	—
329	C₄²	—	10	2	no	—
330	C₄²	C₂³×D₅	1	1	yes	characteristic
331	C₂×C₅	—	8	2	no	—
332	C₂×C₅	#2931	1	1	yes	characteristic
333	C₂×C₅	C₄×#135	1	1	yes	characteristic
334	C₂×C₅	C₂×#443	1	1	yes	characteristic
335	D₅	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	5	1	no	—
338	Q₈	—	10	2	no	—
339	Q₈	—	10	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	20	4	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	5	1	no	—
347	C₂×C₄	—	5	1	no	—
348	C₂×C₄	—	5	1	no	—
349	C₂×C₄	—	10	2	no	—
350	C₂×C₄	—	10	1	no	—
351	C₂×C₄	—	10	1	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	—	10	1	no	—
354	C₂×C₄	—	10	2	no	—
355	C₂×C₄	—	10	2	no	—
356	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
358	C₂×C₄	D₅×#41	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	5	2	no	—
364	C₈	—	5	2	no	—
365	C₈	—	10	2	no	—
366	C₈	—	10	1	no	—
367	C₈	—	10	1	no	—
368	C₈	—	20	2	no	—
369	C₈	#4047	1	2	yes	—
370	C₈	#4058	1	2	yes	—
371	C₅	#17959	1	1	yes	characteristic, complemented, Sylow
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	5	2	no	—
376	C₂²	—	10	1	no	—
377	C₂²	—	10	1	no	—
378	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	10	1	no	—
382	C₄	—	10	1	no	—
383	C₄	—	10	1	no	—
384	C₄	—	20	2	no	—
385	C₄	—	20	2	no	—
386	C₄	C₂×#4047	1	1	yes	characteristic
387	C₄	D₅×#131	1	1	yes	characteristic
388	C₄	C₂×#4058	1	1	yes	characteristic
389	C₄	D₅×#135	1	1	yes	characteristic
390	C₂	—	8	2	no	—
391	C₂	—	10	2	no	—
392	C₂	#10611153	1	1	yes	characteristic
393	C₂	C₄×#4058	1	1	yes	characteristic
394	C₂	D₅×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×D₈	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65446	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×D₈	D₅
hypocenter	C₅	#17959
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #10611132	C₂	semidirect
3. C₅×C₈×D₈	C₂	semidirect
4. #10601674	C₂	semidirect
5. C₈×#687	C₂	semidirect
6. #10599363	C₂	semidirect
7. #10599357	C₂	semidirect
8. #10598970	C₂	semidirect
9. #10598954	C₂	semidirect
10. C₈×Dic₂₀	C₂	semidirect
11. C₈²×D₅	C₂	semidirect
102. C₈×D₈	D₅	semidirect
371. C₅	#17959	semidirect

Maximal subgroups

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

subgroup	index
#10611314	2
#10611132	2
C₅×C₈×D₈	2
#10601674	2
C₈×#687	2
#10599363	2
#10599357	2
#10598970	2
#10598954	2
C₈×Dic₂₀	2
C₈²×D₅	2
#17959	5