Subgroups — #11989592

Order 1280.

676 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₄×#4072	C₂	1	1	yes	maximal, characteristic
3	#10613780	C₂	1	1	yes	maximal, characteristic
4	#10598237	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×#4072	—	2	1	no	—
7	C₅×#431	—	2	1	no	—
8	C₅×#431	—	2	1	no	—
9	C₅×#431	C₄	1	2	yes	complemented
10	C₅×#431	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₄×C₅×C₈	—	2	1	no	—
14	C₄×C₂²⋊Dic₅	—	2	1	no	—
15	C₄×C₂²⋊Dic₅	C₄	1	2	yes	—
16	#65496	—	2	1	no	—
17	C₄×C₄⋊Dic₅	—	2	1	no	—
18	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
19	#65221	C₄	1	2	yes	—
20	C₄²×Dic₅	—	2	1	no	—
21	#65215	—	2	1	no	—
22	#65215	C₄	1	2	yes	complemented
23	#65214	—	2	1	no	—
24	#65214	C₄	1	2	yes	complemented
25	#65213	C₂²	1	2	yes	—
26	C₄×#681	—	2	1	no	—
27	#8092	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊Dic₅	—	2	2	no	—
45	#4072	—	2	8	no	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	2	no	—
48	C₂×C₄×Dic₅	—	2	2	no	—
49	C₂×C₄×Dic₅	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₅×C₂²⋊C₈	—	2	2	no	—
54	C₅×C₂²⋊C₈	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	#3963	—	2	2	no	—
58	#3963	C₂×C₄	1	4	yes	—
59	C₂²⋊#168	—	2	2	no	—
60	C₂²⋊#168	—	2	2	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₄⋊#168	C₂×C₄	1	4	yes	—
63	#3935	—	2	2	no	—
64	C₄×#168	—	2	2	no	—
65	#2920	—	5	1	no	—
66	C₂×#431	D₅	1	1	yes	characteristic
67	C₄×#117	—	5	1	no	—
68	#1303	—	5	4	no	—
69	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₅	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	4	4	no	—
94	C₄²×C₅	C₄²	1	1	yes	characteristic
95	C₄²×C₅	C₄²	1	1	yes	characteristic
96	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
110	C₂²⋊Dic₅	—	2	8	no	—
111	C₄⋊Dic₅	—	2	8	no	—
112	C₄×Dic₅	—	2	8	no	—
113	C₄×Dic₅	—	4	4	no	—
114	C₄×Dic₅	—	4	4	no	—
115	#681	—	4	4	no	—
116	C₂×#168	—	2	4	no	—
117	C₂×#117	—	10	1	no	—
118	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
119	#432	—	5	2	no	—
120	#432	—	10	1	no	—
121	#431	—	2	1	no	—
122	#431	—	2	1	no	—
123	#431	—	5	2	no	—
124	#431	—	10	1	no	—
125	#431	Dic₅	1	2	yes	complemented
126	#431	C₂×D₅	1	2	yes	—
127	#423	—	5	2	no	—
128	C₂×C₄⋊C₈	Dic₅	1	2	yes	complemented
129	#407	—	10	1	no	—
130	C₂×C₂²⋊C₈	—	2	1	no	—
131	C₄×M₁₆	—	10	1	no	—
132	C₂×C₄×C₈	—	2	1	no	—
133	C₄×C₄⋊C₄	—	5	2	no	—
134	C₄×C₄⋊C₄	—	10	1	no	—
135	C₄×C₂²⋊C₄	—	5	2	no	—
136	C₄×C₂²⋊C₄	—	10	1	no	—
137	#376	—	5	2	no	—
138	C₄³	—	10	1	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	2	no	—
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	#95	1	1	yes	characteristic
143	C₂³×C₅	#95	1	1	yes	characteristic
144	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	#95	1	1	yes	characteristic
162	C₂×C₄×C₅	#95	1	1	yes	characteristic
163	C₂×C₄×C₅	#95	1	1	yes	characteristic
164	C₂×C₄×C₅	#95	1	1	yes	characteristic
165	C₂×C₄×C₅	#95	1	1	yes	characteristic
166	C₂×C₄×C₅	#95	1	1	yes	characteristic
167	C₂×C₄×C₅	#104	1	4	yes	—
168	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	#135	1	4	yes	—
175	C₂×Dic₅	—	2	8	no	—
176	C₂×Dic₅	—	4	4	no	—
177	C₂×Dic₅	—	4	4	no	—
178	C₅×C₈	—	4	4	no	—
179	#168	—	4	4	no	—
180	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
181	C₂³×C₄	D₂₀	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×M₁₆	—	10	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	10	8	no	—
186	C₂×C₄⋊C₄	—	10	2	no	—
187	C₂×C₂²⋊C₄	—	10	2	no	—
188	C₂×C₄²	—	10	2	no	—
189	C₂×C₄²	—	10	2	no	—
190	C₂×C₄²	—	10	2	no	—
191	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
192	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
196	C₂×C₄²	D₂₀	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	5	4	no	—
205	C₄⋊C₈	—	10	2	no	—
206	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
207	C₂²⋊C₈	—	2	2	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₂²⋊C₈	—	10	2	no	—
211	#97	—	10	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	10	2	no	—
215	#95	—	5	4	no	—
216	#95	—	10	2	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	4	4	no	—
225	C₂²×C₅	#339	1	2	yes	—
226	C₂²×C₅	C₂×#95	1	1	yes	characteristic
227	C₂²×C₅	#416	1	2	yes	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	#339	1	2	yes	—
236	C₄×C₅	#344	1	2	yes	—
237	C₄×C₅	#416	1	2	yes	—
238	C₄×C₅	C₂×#104	1	2	yes	—
239	C₄×C₅	#427	1	2	yes	—
240	Dic₅	—	4	8	no	—
241	C₂⁴	#685	1	1	yes	characteristic
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	10	4	no	—
246	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
247	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
248	C₂²×C₄	#683	1	1	yes	characteristic
249	C₂²×C₄	#683	1	1	yes	characteristic
250	C₂²×C₄	#683	1	1	yes	characteristic
251	C₂²×C₄	#683	1	1	yes	characteristic
252	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
253	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
254	C₂²×C₄	#685	1	1	yes	characteristic
255	C₂²×C₄	#685	1	1	yes	characteristic
256	C₂²×C₄	#685	1	1	yes	characteristic
257	C₂²×C₄	#685	1	1	yes	characteristic
258	C₂²×C₄	#685	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
264	M₁₆	—	20	4	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	10	4	no	—
268	C₄⋊C₄	—	10	8	no	—
269	C₂²⋊C₄	—	10	8	no	—
270	C₄²	—	10	8	no	—
271	C₄²	—	20	4	no	—
272	C₄²	—	20	4	no	—
273	C₄²	C₄×Dic₅	1	1	yes	characteristic
274	C₄²	C₄×Dic₅	1	1	yes	characteristic
275	C₄²	#683	1	1	yes	characteristic
276	C₄²	#683	1	1	yes	characteristic
277	C₄²	#683	1	1	yes	characteristic
278	C₄²	#683	1	1	yes	characteristic
279	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
281	C₄²	#685	1	1	yes	characteristic
282	C₄²	#685	1	1	yes	characteristic
283	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
284	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
285	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
287	C₄²	C₂²×Dic₅	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	4	1	no	—
292	C₂×C₅	C₂×#339	1	1	yes	characteristic
293	C₂×C₅	#1735	1	1	yes	characteristic
294	C₂×C₅	C₂×#344	1	1	yes	characteristic
295	C₂³	—	2	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	#3963	1	1	yes	characteristic
299	C₂³	#3963	1	1	yes	characteristic
300	C₂³	C₂×#685	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	10	8	no	—
318	C₂×C₄	—	20	4	no	—
319	C₂×C₄	—	20	4	no	—
320	C₂×C₄	#3963	1	1	yes	characteristic
321	C₂×C₄	#3963	1	1	yes	characteristic
322	C₂×C₄	#3963	1	1	yes	characteristic
323	C₂×C₄	#3963	1	1	yes	characteristic
324	C₂×C₄	#3963	1	1	yes	characteristic
325	C₂×C₄	#3963	1	1	yes	characteristic
326	C₂×C₄	#3969	1	4	yes	—
327	C₂×C₄	#4050	1	4	yes	—
328	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
329	C₂×C₄	C₂×#683	1	1	yes	characteristic
330	C₂×C₄	C₂×#683	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
332	C₂×C₄	C₂×#685	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
334	C₈	—	4	4	no	—
335	C₈	—	20	4	no	—
336	C₅	#8092	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	4	no	—
345	C₂²	#64752	1	2	yes	—
346	C₂²	#65402	1	2	yes	—
347	C₂²	C₂×#3963	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	20	8	no	—
356	C₄	#64752	1	2	yes	—
357	C₄	#64779	1	2	yes	—
358	C₄	#65395	1	2	yes	—
359	C₄	#65402	1	2	yes	—
360	C₄	C₂×#3969	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	C₂×#64752	1	1	yes	characteristic
365	C₂	#10602762	1	1	yes	characteristic
366	C₂	C₂×#64779	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#431	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#431	D₅
hypocenter	C₅	#8092
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #10598237	C₂	semidirect
9. C₅×#431	C₄	semidirect
11. C₂×C₅×C₄⋊C₈	C₄	semidirect
22. #65215	C₄	semidirect
24. #65214	C₄	semidirect
125. #431	Dic₅	semidirect
128. C₂×C₄⋊C₈	Dic₅	semidirect
336. C₅	#8092	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#431	2
C₄×#4072	2
#10613780	2
#10598237	2
#8092	5