Subgroups — #12071087

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	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	#10614860	C₂	1	2	yes	maximal, complemented
3	#10602543	C₂	1	2	yes	maximal, complemented
4	#10601698	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601689	C₂	1	2	yes	maximal, complemented
7	#10601688	C₂	1	2	yes	maximal, complemented
8	#10601674	C₂	1	1	yes	maximal, characteristic, complemented
9	D₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
10	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#681	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	#65286	—	2	2	no	—
31	#65214	C₂²	1	2	yes	—
32	#65214	C₂²	1	1	yes	characteristic
33	C₄×#681	C₂²	1	1	yes	characteristic
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	#64678	C₂²	1	1	yes	characteristic
39	#64678	C₂²	1	1	yes	characteristic
40	#64677	C₂²	1	2	yes	—
41	#64674	C₂²	1	1	yes	characteristic
42	#64674	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#18069	—	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	D₄	1	1	yes	characteristic
64	C₂×#681	D₄	1	1	yes	characteristic
65	C₂²×#168	—	2	2	no	—
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×#107	C₂×C₄	1	1	yes	characteristic
68	C₅×#106	C₂×C₄	1	2	yes	—
69	C₅×#103	C₂×C₄	1	2	yes	—
70	C₅×#103	C₂×C₄	1	2	yes	—
71	C₅×#102	C₂×C₄	1	2	yes	—
72	C₅×#102	C₂×C₄	1	2	yes	—
73	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂×C₄	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	2	no	—
78	C₂²⋊#168	—	2	1	no	—
79	C₂²⋊#168	—	2	2	no	—
80	C₂²⋊#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	C₂³	1	2	yes	—
85	C₄⋊#168	C₂³	1	2	yes	—
86	#3935	—	2	2	no	—
87	#3935	D₄	1	1	yes	characteristic
88	#3935	D₄	1	1	yes	characteristic
89	C₄×#168	—	2	2	no	—
90	C₄×#168	—	2	2	no	—
91	C₄×#168	C₂³	1	1	yes	characteristic
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	D₅	1	1	yes	characteristic
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	—	5	1	no	—
106	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₅	—	2	2	no	—
110	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	#41	1	1	yes	characteristic
113	C₅×Q₁₆	—	2	2	no	—
114	C₅×SD₁₆	—	4	2	no	—
115	C₅×D₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	4	1	no	—
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
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	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
128	C₅×C₂²⋊C₄	—	2	2	no	—
129	C₅×C₂²⋊C₄	—	2	2	no	—
130	C₅×C₂²⋊C₄	—	2	2	no	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₅×C₂²⋊C₄	—	2	2	no	—
133	C₄²×C₅	—	2	2	no	—
134	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅	#41	1	1	yes	characteristic
137	C₄²×C₅	C₂⁴	1	1	yes	characteristic
138	#681	—	2	2	no	—
139	#681	—	2	2	no	—
140	#681	—	4	4	no	—
141	C₂×#168	—	2	2	no	—
142	C₂×#168	—	2	2	no	—
143	C₂×#168	—	2	1	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	4	4	no	—
146	C₂×#168	C₂×D₄	1	1	yes	characteristic
147	C₂×#168	C₂×D₄	1	1	yes	characteristic
148	C₂×#168	C₂×D₄	1	1	yes	characteristic
149	C₂×#168	C₂×D₄	1	1	yes	characteristic
150	#529	C₂×D₅	1	2	yes	—
151	#499	Dic₅	1	1	yes	characteristic
152	#495	Dic₅	1	1	yes	characteristic
153	#488	Dic₅	1	2	yes	—
154	#467	Dic₅	1	1	yes	characteristic
155	#466	Dic₅	1	1	yes	characteristic
156	#465	Dic₅	1	2	yes	—
157	#446	—	5	2	no	—
158	C₈×Q₈	—	5	2	no	—
159	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
160	C₄×SD₁₆	C₂×D₅	1	2	yes	—
161	C₄×D₈	C₂×D₅	1	1	yes	characteristic
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	1	no	—
168	#431	—	5	2	no	—
169	#405	—	10	2	no	—
170	C₄×M₁₆	—	5	1	no	—
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	—	5	1	no	—
180	#322	—	5	1	no	—
181	C₈²	—	5	1	no	—
182	C₈²	—	5	1	no	—
183	C₂³×C₅	—	2	2	no	—
184	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
185	C₅×Q₈	—	2	2	no	—
186	C₅×Q₈	—	2	2	no	—
187	C₅×D₄	—	2	2	no	—
188	C₅×D₄	—	2	2	no	—
189	C₅×D₄	—	8	2	no	—
190	C₂×C₄×C₅	—	2	1	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	2	1	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	2	2	no	—
196	C₂×C₄×C₅	—	2	2	no	—
197	C₂×C₄×C₅	—	4	1	no	—
198	C₂×C₄×C₅	—	4	2	no	—
199	C₂×C₄×C₅	—	4	2	no	—
200	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	2	1	no	—
208	C₅×C₈	—	2	1	no	—
209	C₅×C₈	#131	1	2	yes	—
210	#168	—	2	1	no	—
211	#168	—	2	1	no	—
212	#168	—	4	1	no	—
213	#168	—	4	1	no	—
214	#168	—	4	2	no	—
215	#168	—	4	2	no	—
216	#168	#135	1	2	yes	—
217	#168	#135	1	2	yes	—
218	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
219	C₂×SD₁₆	C₂×Dic₅	1	2	yes	—
220	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
221	C₂×M₁₆	—	5	1	no	—
222	C₂×M₁₆	—	5	1	no	—
223	C₂×M₁₆	—	10	2	no	—
224	C₂²×C₈	—	10	2	no	—
225	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
226	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
227	#126	—	2	2	no	—
228	#125	C₂×Dic₅	1	2	yes	—
229	#124	C₂×Dic₅	1	2	yes	—
230	#123	—	2	2	no	—
231	C₂²⋊Q₈	C₂×Dic₅	1	2	yes	—
232	#121	C₂×Dic₅	1	2	yes	—
233	C₄×Q₈	C₂²×D₅	1	2	yes	—
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	C₂²×D₅	1	2	yes	—
236	#117	—	2	2	no	—
237	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
238	#107	C₂×Dic₅	1	1	yes	characteristic
239	#107	C₂×Dic₅	1	1	yes	characteristic
240	#106	C₂×Dic₅	1	2	yes	—
241	C₄⋊C₈	—	5	2	no	—
242	C₄⋊C₈	—	5	2	no	—
243	C₄⋊C₈	—	10	2	no	—
244	C₄⋊C₈	—	10	2	no	—
245	C₄⋊C₈	—	10	1	no	—
246	#103	C₂×Dic₅	1	2	yes	—
247	#103	C₂×Dic₅	1	2	yes	—
248	#102	C₂×Dic₅	1	2	yes	—
249	#102	C₂×Dic₅	1	2	yes	—
250	C₂²⋊C₈	—	10	2	no	—
251	C₂²⋊C₈	—	10	2	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	C₂²⋊C₈	—	10	1	no	—
254	#97	—	5	1	no	—
255	#97	—	5	1	no	—
256	#97	—	10	2	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	—	10	2	no	—
262	C₄×C₈	—	10	2	no	—
263	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
264	C₄×C₈	C₂×Dic₅	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	—	4	2	no	—
271	C₂²×C₅	—	4	2	no	—
272	C₂²×C₅	—	4	2	no	—
273	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
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	—	4	2	no	—
279	C₄×C₅	—	4	2	no	—
280	C₄×C₅	—	4	2	no	—
281	C₄×C₅	C₂×#131	1	1	yes	characteristic
282	C₄×C₅	#568	1	1	yes	characteristic
283	C₄×C₅	C₂×#135	1	1	yes	characteristic
284	C₄×C₅	C₂×#135	1	1	yes	characteristic
285	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	C₂²×Dic₅	1	2	yes	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
290	C₂²×C₄	#711	1	1	yes	characteristic
291	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	SD₁₆	—	4	2	no	—
294	D₈	—	2	2	no	—
295	M₁₆	—	10	2	no	—
296	M₁₆	—	10	2	no	—
297	M₁₆	—	20	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	1	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	1	no	—
307	C₂×C₈	—	10	1	no	—
308	C₂×C₈	—	20	4	no	—
309	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
310	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
311	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
312	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
318	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	D₄×D₅	1	1	yes	characteristic
326	C₄²	#711	1	1	yes	characteristic
327	C₄²	C₂²×Dic₅	1	1	yes	characteristic
328	C₄²	C₂³×D₅	1	1	yes	characteristic
329	C₂×C₅	—	4	1	no	—
330	C₂×C₅	—	8	2	no	—
331	C₂×C₅	#2935	1	1	yes	characteristic
332	C₂×C₅	C₄×#135	1	1	yes	characteristic
333	C₂×C₅	#2957	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	D₄×Dic₅	1	1	yes	characteristic
336	Q₈	—	2	2	no	—
337	Q₈	—	2	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	8	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	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
353	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
355	C₂×C₄	C₂×#711	1	1	yes	characteristic
356	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	5	2	no	—
361	C₈	—	5	2	no	—
362	C₈	—	10	1	no	—
363	C₈	—	10	1	no	—
364	C₈	—	20	1	no	—
365	C₈	—	20	1	no	—
366	C₈	—	20	2	no	—
367	C₈	—	20	2	no	—
368	C₈	#4094	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	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×D₄×Dic₅	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	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	D₅×#135	1	1	yes	characteristic
385	C₄	D₅×#135	1	1	yes	characteristic
386	C₄	C₂×#4094	1	1	yes	characteristic
387	C₄	#66153	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	8	2	no	—
390	C₂	#10615008	1	1	yes	characteristic
391	C₂	Dic₅×#135	1	1	yes	characteristic
392	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	#18069
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. #10614860	C₂	semidirect
3. #10602543	C₂	semidirect
4. #10601698	C₂	semidirect
5. Q₁₆×#168	C₂	semidirect
6. #10601689	C₂	semidirect
7. #10601688	C₂	semidirect
8. #10601674	C₂	semidirect
9. D₈×#168	C₂	semidirect
10. #10600522	C₂	semidirect
11. C₈×#681	C₂	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
C₅×#3168	2
#10614860	2
#10602543	2
#10601698	2
Q₁₆×#168	2
#10601689	2
#10601688	2
#10601674	2
D₈×#168	2
#10600522	2
C₈×#681	2
#18069	5