Subgroups — #12070001

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	#10614862	C₂	1	2	yes	maximal, complemented
3	#10602546	C₂	1	2	yes	maximal, complemented
4	#10601699	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601695	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601692	C₂	1	2	yes	maximal, complemented
7	SD₁₆×#168	C₂	1	2	yes	maximal, complemented
8	#10601675	C₂	1	1	yes	maximal, characteristic, complemented
9	#10601671	C₂	1	1	yes	maximal, characteristic, complemented
10	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
11	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#529	C₂²	1	2	yes	—
13	C₅×#499	C₄	1	1	yes	characteristic
14	C₅×#495	C₄	1	1	yes	characteristic
15	C₅×#488	C₄	1	2	yes	—
16	C₅×#467	C₄	1	1	yes	characteristic
17	C₅×#466	C₄	1	1	yes	characteristic
18	C₅×#465	C₄	1	2	yes	—
19	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₅×SD₁₆	C₂²	1	2	yes	—
21	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	#65313	C₂²	1	2	yes	—
24	Q₈×#168	C₂²	1	2	yes	—
25	#65301	—	2	2	no	—
26	#65301	C₂²	1	2	yes	—
27	#65300	—	2	2	no	—
28	D₄×#168	C₂²	1	2	yes	—
29	#65289	—	2	2	no	—
30	#65288	—	2	2	no	—
31	#65214	C₂²	1	1	yes	characteristic
32	#65214	C₂²	1	1	yes	characteristic
33	#65213	C₂²	1	2	yes	—
34	Q₈⋊#168	C₂²	1	2	yes	—
35	Q₈⋊#168	C₂²	1	2	yes	—
36	D₄⋊#168	C₂²	1	2	yes	—
37	D₄⋊#168	C₂²	1	2	yes	—
38	#64677	C₂²	1	1	yes	characteristic
39	#64677	C₂²	1	1	yes	characteristic
40	#64677	C₂²	1	2	yes	—
41	#64675	C₂²	1	1	yes	characteristic
42	#64675	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17974	—	5	1	no	maximal, Sylow
46	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
48	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₅×#126	—	2	2	no	—
52	C₅×#125	C₂×C₄	1	2	yes	—
53	C₅×#124	C₂×C₄	1	2	yes	—
54	C₅×#123	—	2	2	no	—
55	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
56	C₅×#121	C₂×C₄	1	2	yes	—
57	C₄×C₅×Q₈	C₂³	1	2	yes	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₄×C₅×D₄	C₂³	1	2	yes	—
60	C₅×#117	—	2	2	no	—
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₂×#681	—	2	2	no	—
63	C₂×#681	—	2	2	no	—
64	C₂²×#168	—	2	2	no	—
65	C₅×#107	C₂×C₄	1	1	yes	characteristic
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×#106	C₂×C₄	1	2	yes	—
68	C₅×#103	C₂×C₄	1	2	yes	—
69	C₅×#103	C₂×C₄	1	2	yes	—
70	C₅×#102	C₂×C₄	1	2	yes	—
71	C₅×#102	C₂×C₄	1	2	yes	—
72	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂²⋊#168	—	2	1	no	—
77	C₂²⋊#168	—	2	2	no	—
78	C₂²⋊#168	—	2	2	no	—
79	C₂²⋊#168	D₄	1	2	yes	—
80	C₄⋊#168	—	2	2	no	—
81	C₄⋊#168	—	2	1	no	—
82	C₄⋊#168	—	2	2	no	—
83	C₄⋊#168	—	2	2	no	—
84	C₄⋊#168	D₄	1	2	yes	—
85	C₄⋊#168	C₂³	1	2	yes	—
86	C₄⋊#168	C₂³	1	2	yes	—
87	#3935	—	2	2	no	—
88	C₄×#168	—	2	2	no	—
89	C₄×#168	C₂³	1	1	yes	characteristic
90	C₄×#168	C₂³	1	1	yes	characteristic
91	C₄×#168	C₂³	1	1	yes	characteristic
92	C₄×#168	C₂³	1	1	yes	characteristic
93	#3168	D₅	1	1	yes	characteristic
94	#2993	—	5	2	no	—
95	#1594	—	5	1	no	—
96	#1593	—	5	1	no	—
97	#1590	—	5	2	no	—
98	#1580	—	5	1	no	—
99	#1579	—	5	1	no	—
100	C₈×SD₁₆	—	5	2	no	—
101	#1566	—	5	1	no	—
102	#1495	—	5	2	no	—
103	#1450	—	5	1	no	—
104	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₅	—	2	2	no	—
108	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	#41	1	1	yes	characteristic
111	C₅×Q₁₆	—	2	2	no	—
112	C₅×SD₁₆	—	2	4	no	—
113	C₅×D₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
125	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₅×C₂²⋊C₄	—	2	2	no	—
128	C₅×C₂²⋊C₄	—	2	2	no	—
129	C₅×C₂²⋊C₄	—	2	2	no	—
130	C₅×C₂²⋊C₄	—	2	2	no	—
131	C₄²×C₅	—	2	2	no	—
132	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅	#41	1	1	yes	characteristic
135	C₄²×C₅	C₂⁴	1	1	yes	characteristic
136	#681	—	8	2	no	—
137	#681	—	8	2	no	—
138	C₂×#168	—	2	2	no	—
139	C₂×#168	—	2	2	no	—
140	C₂×#168	—	2	1	no	—
141	C₂×#168	—	2	1	no	—
142	C₂×#168	—	8	2	no	—
143	C₂×#168	C₂×D₄	1	2	yes	—
144	C₂×#168	C₂×D₄	1	2	yes	—
145	#529	C₂×D₅	1	2	yes	—
146	#499	Dic₅	1	1	yes	characteristic
147	#495	Dic₅	1	1	yes	characteristic
148	#488	Dic₅	1	2	yes	—
149	#467	Dic₅	1	1	yes	characteristic
150	#466	Dic₅	1	1	yes	characteristic
151	#465	Dic₅	1	2	yes	—
152	#446	—	5	2	no	—
153	C₈×Q₈	—	5	2	no	—
154	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
155	C₄×SD₁₆	C₂×D₅	1	2	yes	—
156	C₄×D₈	C₂×D₅	1	1	yes	characteristic
157	#436	—	5	2	no	—
158	#436	—	10	2	no	—
159	#435	—	10	2	no	—
160	C₈×D₄	—	5	2	no	—
161	#433	—	10	2	no	—
162	#431	—	5	1	no	—
163	#431	—	5	1	no	—
164	#424	—	10	2	no	—
165	#423	—	5	2	no	—
166	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
167	#334	—	5	2	no	—
168	#334	—	5	1	no	—
169	#334	—	5	1	no	—
170	Q₈⋊C₈	—	5	2	no	—
171	Q₈⋊C₈	—	5	2	no	—
172	D₄⋊C₈	—	5	2	no	—
173	D₄⋊C₈	—	5	2	no	—
174	#322	—	5	1	no	—
175	#322	—	5	1	no	—
176	C₈²	—	5	1	no	—
177	C₈²	—	5	1	no	—
178	C₂³×C₅	—	2	2	no	—
179	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
180	C₅×Q₈	—	2	2	no	—
181	C₅×Q₈	—	2	2	no	—
182	C₅×D₄	—	2	2	no	—
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	8	2	no	—
185	C₂×C₄×C₅	—	2	1	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	4	1	no	—
192	C₂×C₄×C₅	—	4	2	no	—
193	C₂×C₄×C₅	—	4	2	no	—
194	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
200	C₅×C₈	—	2	1	no	—
201	C₅×C₈	—	2	1	no	—
202	C₅×C₈	#131	1	2	yes	—
203	C₅×C₈	#131	1	2	yes	—
204	#168	—	2	2	no	—
205	#168	—	2	2	no	—
206	#168	—	4	1	no	—
207	#168	—	4	1	no	—
208	#168	—	4	2	no	—
209	#168	—	4	2	no	—
210	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
211	C₂×SD₁₆	C₂×Dic₅	1	2	yes	—
212	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
213	C₂×M₁₆	—	10	2	no	—
214	C₂×M₁₆	—	10	2	no	—
215	C₂²×C₈	—	10	2	no	—
216	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
217	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
218	#126	—	2	2	no	—
219	#125	C₂×Dic₅	1	2	yes	—
220	#124	C₂×Dic₅	1	2	yes	—
221	#123	—	2	2	no	—
222	C₂²⋊Q₈	C₂×Dic₅	1	2	yes	—
223	#121	C₂×Dic₅	1	2	yes	—
224	C₄×Q₈	C₂²×D₅	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	C₂²×D₅	1	2	yes	—
227	#117	—	2	2	no	—
228	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
229	#107	C₂×Dic₅	1	1	yes	characteristic
230	#107	C₂×Dic₅	1	1	yes	characteristic
231	#106	C₂×Dic₅	1	2	yes	—
232	C₄⋊C₈	—	5	2	no	—
233	C₄⋊C₈	—	5	2	no	—
234	C₄⋊C₈	—	5	2	no	—
235	C₄⋊C₈	—	10	2	no	—
236	C₄⋊C₈	—	10	2	no	—
237	C₄⋊C₈	—	10	2	no	—
238	C₄⋊C₈	—	10	1	no	—
239	#103	C₂×Dic₅	1	2	yes	—
240	#103	C₂×Dic₅	1	2	yes	—
241	#102	C₂×Dic₅	1	2	yes	—
242	#102	C₂×Dic₅	1	2	yes	—
243	C₂²⋊C₈	—	5	2	no	—
244	C₂²⋊C₈	—	10	2	no	—
245	C₂²⋊C₈	—	10	2	no	—
246	C₂²⋊C₈	—	10	1	no	—
247	#97	—	10	2	no	—
248	C₄×C₈	—	5	1	no	—
249	C₄×C₈	—	5	1	no	—
250	C₄×C₈	—	5	1	no	—
251	C₄×C₈	—	5	1	no	—
252	C₄×C₈	—	10	2	no	—
253	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
254	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	—	4	2	no	—
261	C₂²×C₅	—	4	2	no	—
262	C₂²×C₅	—	4	2	no	—
263	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	4	2	no	—
269	C₄×C₅	—	4	2	no	—
270	C₄×C₅	—	4	2	no	—
271	C₄×C₅	C₂×#131	1	1	yes	characteristic
272	C₄×C₅	C₂×#131	1	1	yes	characteristic
273	C₄×C₅	#577	1	2	yes	—
274	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	C₂²×Dic₅	1	2	yes	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
279	C₂²×C₄	#711	1	1	yes	characteristic
280	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
281	Q₁₆	—	2	2	no	—
282	SD₁₆	—	2	4	no	—
283	D₈	—	2	2	no	—
284	M₁₆	—	40	2	no	—
285	M₁₆	—	40	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	5	2	no	—
289	C₂×C₈	—	5	2	no	—
290	C₂×C₈	—	10	2	no	—
291	C₂×C₈	—	10	2	no	—
292	C₂×C₈	—	10	1	no	—
293	C₂×C₈	—	10	1	no	—
294	C₂×C₈	—	40	2	no	—
295	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
296	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
297	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
298	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
304	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	D₄×D₅	1	1	yes	characteristic
312	C₄²	#711	1	1	yes	characteristic
313	C₄²	C₂²×Dic₅	1	1	yes	characteristic
314	C₄²	C₂³×D₅	1	1	yes	characteristic
315	C₂×C₅	—	4	1	no	—
316	C₂×C₅	—	8	2	no	—
317	C₂×C₅	#2931	1	1	yes	characteristic
318	C₂×C₅	#2952	1	1	yes	characteristic
319	C₂×C₅	#2957	1	1	yes	characteristic
320	C₂³	—	2	2	no	—
321	C₂³	D₄×Dic₅	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	8	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
337	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
338	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
340	C₂×C₄	C₂×#711	1	1	yes	characteristic
341	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	10	2	no	—
345	C₈	—	10	2	no	—
346	C₈	—	20	1	no	—
347	C₈	—	20	1	no	—
348	C₈	—	20	2	no	—
349	C₈	—	20	2	no	—
350	C₈	#4094	1	2	yes	—
351	C₈	#4094	1	2	yes	—
352	C₅	#17974	1	1	yes	characteristic, complemented, Sylow
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	#66104	1	2	yes	—
368	C₄	C₂×#4094	1	1	yes	characteristic
369	C₄	C₂×#4094	1	1	yes	characteristic
370	C₂	—	4	1	no	—
371	C₂	—	8	2	no	—
372	C₂	#10615006	1	1	yes	characteristic
373	C₂	#10615630	1	1	yes	characteristic
374	C₂	#10615641	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×#3168	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#466	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#3168	D₅
hypocenter	C₅	#17974
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
2. #10614862	C₂	semidirect
3. #10602546	C₂	semidirect
4. #10601699	C₂	semidirect
5. #10601695	C₂	semidirect
6. #10601692	C₂	semidirect
7. SD₁₆×#168	C₂	semidirect
8. #10601675	C₂	semidirect
9. #10601671	C₂	semidirect
10. #10600523	C₂	semidirect
11. #10600517	C₂	semidirect
352. C₅	#17974	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3168	2
#10614862	2
#10602546	2
#10601699	2
#10601695	2
#10601692	2
SD₁₆×#168	2
#10601675	2
#10601671	2
#10600523	2
#10600517	2
#17974	5