Subgroups — #12068564

Order 1280.

588 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	C₂	1	1	yes	maximal, characteristic
2	#10614246	C₂	1	2	yes	maximal
3	#10601546	C₂	1	2	yes	maximal, complemented
4	#10601545	C₂	1	2	yes	maximal, complemented
5	#10601544	C₂	1	2	yes	maximal, complemented
6	#10600644	C₂	1	2	yes	maximal, complemented
7	#10600643	C₂	1	2	yes	maximal, complemented
8	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#41	C₂²	1	1	yes	characteristic
11	C₅×#446	C₄	1	1	yes	characteristic
12	C₅×#446	C₄	1	1	yes	characteristic
13	C₅×#436	C₄	1	1	yes	characteristic
14	C₅×#436	C₄	1	1	yes	characteristic
15	C₅×#435	C₄	1	2	yes	—
16	C₅×#435	C₄	1	2	yes	—
17	C₅×#431	C₂²	1	1	yes	characteristic
18	C₅×#431	C₂²	1	2	yes	—
19	C₄×C₅×M₁₆	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#65523	—	2	2	no	—
22	#65289	—	2	2	no	—
23	#65289	—	2	2	no	—
24	#65286	—	2	2	no	—
25	#65215	C₂²	1	2	yes	—
26	#65215	C₂²	1	2	yes	—
27	#65214	C₂²	1	2	yes	—
28	#65214	C₂²	1	2	yes	—
29	#65213	C₂²	1	2	yes	—
30	C₂×C₄⋊#168	C₂²	1	2	yes	—
31	#64678	C₂²	1	2	yes	—
32	#64678	C₂²	1	2	yes	—
33	#64678	C₂²	1	2	yes	—
34	#64678	C₂²	1	2	yes	—
35	#64677	C₂²	1	2	yes	—
36	#64677	C₂²	1	2	yes	—
37	#64677	C₂²	1	1	yes	characteristic
38	#64677	C₂²	1	1	yes	characteristic
39	#64677	C₂²	1	1	yes	characteristic
40	#64677	C₂²	1	1	yes	characteristic
41	#17831	—	5	1	no	maximal, Sylow
42	C₂×C₅×#41	D₄	1	1	yes	characteristic
43	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Q₈	Q₈	1	1	yes	characteristic
49	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
50	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×D₄	Q₈	1	1	yes	characteristic
52	C₄×C₅×D₄	Q₈	1	2	yes	—
53	C₅×#117	D₄	1	2	yes	—
54	C₅×#117	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂³	1	2	yes	—
57	C₂×#681	—	2	2	no	—
58	C₂²×#168	—	2	2	no	—
59	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₅×#97	C₂×C₄	1	2	yes	—
67	C₅×#97	C₂×C₄	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	2	yes	—
73	C₂²⋊#168	—	2	2	no	—
74	C₂²⋊#168	—	2	2	no	—
75	C₂²⋊#168	—	2	2	no	—
76	C₂²⋊#168	—	2	2	no	—
77	C₄⋊#168	—	2	2	no	—
78	C₄⋊#168	—	2	2	no	—
79	C₄⋊#168	C₂³	1	2	yes	—
80	C₄⋊#168	C₂³	1	2	yes	—
81	C₄⋊#168	C₂³	1	2	yes	—
82	C₄⋊#168	C₂³	1	2	yes	—
83	#3935	—	2	2	no	—
84	#3935	—	2	2	no	—
85	C₄×#168	—	2	2	no	—
86	C₄×#168	—	2	2	no	—
87	#2965	D₅	1	1	yes	characteristic
88	#2925	—	5	2	no	—
89	#1574	—	5	2	no	—
90	#1573	—	5	2	no	—
91	#1572	—	5	2	no	—
92	#1569	—	5	2	no	—
93	#1568	—	5	1	no	—
94	#1565	—	5	2	no	—
95	#1564	—	5	1	no	—
96	C₅×#41	—	4	2	no	—
97	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
99	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
106	C₅×M₁₆	—	2	4	no	—
107	C₅×M₁₆	—	4	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	4	1	no	—
110	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
117	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
118	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
119	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
120	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
121	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
122	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
123	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅	C₂²×C₄	1	2	yes	—
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₄	1	2	yes	—
127	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
128	C₄²×C₅	C₂×Q₈	1	2	yes	—
129	C₄²×C₅	C₂⁴	1	1	yes	characteristic
130	#681	—	4	4	no	—
131	C₂×#168	—	2	2	no	—
132	C₂×#168	—	2	2	no	—
133	C₂×#168	—	2	2	no	—
134	C₂×#168	—	2	2	no	—
135	C₂×#168	—	4	4	no	—
136	C₄×#41	C₂×D₅	1	1	yes	characteristic
137	#446	Dic₅	1	1	yes	characteristic
138	#446	Dic₅	1	1	yes	characteristic
139	#436	Dic₅	1	1	yes	characteristic
140	#436	Dic₅	1	1	yes	characteristic
141	#435	Dic₅	1	2	yes	—
142	#435	Dic₅	1	2	yes	—
143	#433	—	10	2	no	—
144	#433	—	10	2	no	—
145	#432	—	5	2	no	—
146	#432	—	5	2	no	—
147	#431	—	5	2	no	—
148	#431	—	5	2	no	—
149	#431	C₂×D₅	1	1	yes	characteristic
150	#431	C₂×D₅	1	2	yes	—
151	#423	—	5	2	no	—
152	C₂×C₄⋊C₈	—	5	2	no	—
153	#408	—	10	2	no	—
154	#405	—	10	2	no	—
155	C₄×M₁₆	C₂×D₅	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
157	#335	—	5	2	no	—
158	#335	—	5	2	no	—
159	#335	—	5	2	no	—
160	#335	—	5	2	no	—
161	#334	—	5	1	no	—
162	#334	—	5	1	no	—
163	#334	—	5	2	no	—
164	#334	—	5	1	no	—
165	#334	—	5	1	no	—
166	#334	—	5	2	no	—
167	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
169	C₅×Q₈	—	4	1	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	1	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
185	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
187	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	#131	1	2	yes	—
196	#168	—	4	2	no	—
197	#168	—	4	2	no	—
198	#168	—	4	2	no	—
199	#168	—	4	2	no	—
200	C₂×#41	D₂₀	1	1	yes	characteristic
201	C₂×M₁₆	—	10	2	no	—
202	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
203	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
204	C₂²×C₈	—	10	2	no	—
205	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
207	C₄×Q₈	Dic₁₀	1	1	yes	characteristic
208	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
209	C₄×D₄	Dic₁₀	1	1	yes	characteristic
210	C₄×D₄	Dic₁₀	1	2	yes	—
211	C₄×D₄	C₂×Dic₅	1	2	yes	—
212	C₄×D₄	C₂×Dic₅	1	1	yes	characteristic
213	#117	D₂₀	1	2	yes	—
214	#117	D₂₀	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₅	1	2	yes	—
217	C₄⋊C₈	—	5	2	no	—
218	C₄⋊C₈	—	5	2	no	—
219	C₄⋊C₈	—	5	2	no	—
220	C₄⋊C₈	—	5	2	no	—
221	C₄⋊C₈	—	10	2	no	—
222	C₄⋊C₈	—	10	2	no	—
223	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
224	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
225	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
226	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
227	C₂²⋊C₈	—	10	2	no	—
228	C₂²⋊C₈	—	10	2	no	—
229	C₂²⋊C₈	—	10	2	no	—
230	C₂²⋊C₈	—	10	2	no	—
231	C₂²⋊C₈	C₂×Dic₅	1	2	yes	—
232	C₂²⋊C₈	C₂×Dic₅	1	2	yes	—
233	C₂²⋊C₈	C₂×Dic₅	1	2	yes	—
234	#97	—	10	2	no	—
235	#97	—	10	2	no	—
236	#97	C₂×Dic₅	1	2	yes	—
237	#97	C₂×Dic₅	1	2	yes	—
238	C₄×C₈	—	10	2	no	—
239	C₄×C₈	—	10	2	no	—
240	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
241	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅	1	2	yes	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	2	2	no	—
247	C₂²×C₅	—	2	2	no	—
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	—	2	2	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	2	2	no	—
259	C₄×C₅	C₂×#131	1	1	yes	characteristic
260	C₄×C₅	#568	1	1	yes	characteristic
261	C₄×C₅	#575	1	2	yes	—
262	#41	—	4	2	no	—
263	C₂×Q₈	C₄⋊Dic₅	1	1	yes	characteristic
264	C₂×D₄	C₄⋊Dic₅	1	2	yes	—
265	C₂×D₄	C₄⋊Dic₅	1	1	yes	characteristic
266	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
268	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₂₀	1	2	yes	—
270	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	20	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	10	2	no	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₂×C₈	—	10	2	no	—
281	C₂×C₈	—	20	4	no	—
282	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
283	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
284	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
285	C₂×C₈	C₂²×Dic₅	1	2	yes	—
286	C₂×C₈	C₂²×Dic₅	1	2	yes	—
287	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
288	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
289	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
290	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
291	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
292	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
293	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
294	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
295	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
296	C₄²	C₂×Dic₁₀	1	2	yes	—
297	C₄²	C₂×D₂₀	1	1	yes	characteristic
298	C₄²	C₂×D₂₀	1	2	yes	—
299	C₄²	C₂²×Dic₅	1	1	yes	characteristic
300	C₄²	C₂²×Dic₅	1	2	yes	—
301	C₄²	C₂³×D₅	1	1	yes	characteristic
302	C₂×C₅	—	4	1	no	—
303	C₂×C₅	—	4	2	no	—
304	C₂×C₅	#2906	1	1	yes	characteristic
305	C₂×C₅	#2911	1	1	yes	characteristic
306	C₂×C₅	#2916	1	1	yes	characteristic
307	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
309	Q₈	—	4	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
323	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
327	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
329	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	—	20	2	no	—
336	C₈	—	20	2	no	—
337	C₈	—	20	2	no	—
338	C₈	—	20	2	no	—
339	C₈	#4094	1	2	yes	—
340	C₅	#17831	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	2	no	—
347	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	#66082	1	2	yes	—
356	C₄	C₂×#4094	1	1	yes	characteristic
357	C₄	#66153	1	1	yes	characteristic
358	C₂	—	4	2	no	—
359	C₂	—	4	1	no	—
360	C₂	#10614243	1	1	yes	characteristic
361	C₂	#10615483	1	1	yes	characteristic
362	C₂	#10615520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#2965	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×C₅×#41	D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2965	D₅
hypocenter	C₅	#17831
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2965

Complements

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

N	H ≅ G/N	type
3. #10601546	C₂	semidirect
4. #10601545	C₂	semidirect
5. #10601544	C₂	semidirect
6. #10600644	C₂	semidirect
7. #10600643	C₂	semidirect
8. #10600525	C₂	semidirect
9. #10600522	C₂	semidirect
340. C₅	#17831	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2965	2
#10614246	2
#10601546	2
#10601545	2
#10601544	2
#10600644	2
#10600643	2
#10600525	2
#10600522	2
#17831	5