Subgroups — #11989005

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10613780	C₂	1	1	yes	maximal, characteristic
3	C₂×#65215	C₂	1	1	yes	maximal, characteristic
4	#10598159	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×M₁₆	—	2	1	no	—
8	C₄×C₅×M₁₆	—	2	1	no	—
9	C₄×C₅×M₁₆	C₄	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₅×#97	—	2	1	no	—
12	C₂×C₄×C₅×C₈	—	2	1	no	—
13	#65496	C₄	1	2	yes	—
14	C₂×C₂²⋊#168	C₄	1	2	yes	—
15	#65215	—	2	1	no	—
16	#65215	—	2	1	no	—
17	#65215	—	2	1	no	—
18	#65215	—	2	1	no	—
19	#65215	—	2	1	no	—
20	#65215	C₄	1	2	yes	—
21	#65214	—	2	1	no	—
22	#65214	C₂²	1	2	yes	—
23	#65213	—	2	1	no	—
24	C₂×C₄⋊#168	C₂²	1	2	yes	—
25	C₄×#681	—	2	1	no	—
26	C₂×#3935	C₄	1	2	yes	—
27	#7937	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	8	no	—
33	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	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	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×#681	—	2	2	no	—
48	C₂²×#168	C₂×C₄	1	4	yes	—
49	C₅×#97	—	2	2	no	—
50	C₅×#97	—	2	2	no	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	—	2	2	no	—
53	C₂²⋊#168	—	2	2	no	—
54	C₂²⋊#168	—	2	2	no	—
55	C₂²⋊#168	—	2	2	no	—
56	C₂²⋊#168	C₂×C₄	1	4	yes	—
57	C₄⋊#168	—	2	2	no	—
58	C₄⋊#168	—	2	2	no	—
59	C₄⋊#168	—	2	2	no	—
60	C₄⋊#168	—	2	2	no	—
61	#3935	—	2	2	no	—
62	#3935	—	2	2	no	—
63	#3935	—	2	2	no	—
64	C₄×#168	—	2	2	no	—
65	#2920	—	5	1	no	—
66	C₂×#432	—	5	1	no	—
67	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
68	#1283	—	5	4	no	—
69	C₂⁴×C₅	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	4	2	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	C₄²	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	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₅×M₁₆	—	2	8	no	—
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	M₁₆	1	8	yes	complemented
94	C₂×C₅×C₈	—	2	8	no	—
95	C₂×C₅×C₈	—	2	4	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
112	#681	—	4	4	no	—
113	C₂×#168	—	2	4	no	—
114	C₂×#168	—	2	4	no	—
115	C₂×#168	—	4	4	no	—
116	C₂²×M₁₆	Dic₅	1	2	yes	—
117	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
118	#432	—	5	2	no	—
119	#432	—	10	1	no	—
120	#432	—	10	1	no	—
121	#432	—	10	1	no	—
122	#432	—	10	1	no	—
123	#432	—	10	1	no	—
124	#431	—	5	2	no	—
125	#431	—	10	1	no	—
126	#423	—	10	1	no	—
127	C₂×C₄⋊C₈	—	5	2	no	—
128	#407	—	5	2	no	—
129	C₂×C₂²⋊C₈	—	5	2	no	—
130	C₄×M₁₆	—	2	1	no	—
131	C₄×M₁₆	—	2	1	no	—
132	C₄×M₁₆	—	10	1	no	—
133	C₄×M₁₆	Dic₅	1	2	yes	—
134	C₄×M₁₆	C₂×D₅	1	2	yes	—
135	C₂×#97	—	2	1	no	—
136	C₂×#97	—	5	2	no	—
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	#95	1	1	yes	characteristic
142	C₂³×C₅	#95	1	1	yes	characteristic
143	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅	—	2	2	no	—
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	—	4	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₄⋊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×M₁₆	1	4	yes	—
174	C₅×C₈	—	2	8	no	—
175	#168	—	4	4	no	—
176	#168	—	4	4	no	—
177	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
178	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
179	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	—	10	2	no	—
183	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	5	4	no	—
186	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
187	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
189	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
190	C₂×C₄²	D₂₀	1	1	yes	characteristic
191	C₂×C₄²	D₂₀	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
198	C₄⋊C₈	—	10	2	no	—
199	C₄⋊C₈	—	10	2	no	—
200	C₄⋊C₈	—	10	2	no	—
201	C₄⋊C₈	—	10	2	no	—
202	C₂²⋊C₈	—	5	4	no	—
203	C₂²⋊C₈	—	10	2	no	—
204	C₂²⋊C₈	—	10	2	no	—
205	C₂²⋊C₈	—	10	2	no	—
206	#97	—	2	2	no	—
207	#97	—	2	2	no	—
208	#97	—	10	2	no	—
209	#97	—	10	2	no	—
210	#97	—	10	2	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	10	2	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	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	—	4	4	no	—
222	C₂²×C₅	#344	1	2	yes	—
223	C₂²×C₅	C₂×#95	1	1	yes	characteristic
224	C₂²×C₅	C₄×M₁₆	1	2	yes	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
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	#407	1	2	yes	—
235	C₄×C₅	#413	1	2	yes	—
236	C₄×C₅	#423	1	2	yes	—
237	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
242	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
243	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
244	C₂²×C₄	#683	1	1	yes	characteristic
245	C₂²×C₄	#683	1	1	yes	characteristic
246	C₂²×C₄	#683	1	1	yes	characteristic
247	C₂²×C₄	#683	1	1	yes	characteristic
248	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
250	C₂²×C₄	#685	1	1	yes	characteristic
251	C₂²×C₄	#685	1	1	yes	characteristic
252	C₂²×C₄	#685	1	1	yes	characteristic
253	C₂²×C₄	#685	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
259	M₁₆	—	2	8	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	20	4	no	—
262	M₁₆	#681	1	8	yes	complemented
263	C₂×C₈	—	2	8	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	10	4	no	—
266	C₂×C₈	—	10	4	no	—
267	C₂×C₈	—	20	4	no	—
268	C₄²	#683	1	1	yes	characteristic
269	C₄²	#683	1	1	yes	characteristic
270	C₄²	#683	1	1	yes	characteristic
271	C₄²	#683	1	1	yes	characteristic
272	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
273	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
274	C₄²	#685	1	1	yes	characteristic
275	C₄²	#685	1	1	yes	characteristic
276	C₄²	#685	1	1	yes	characteristic
277	C₄²	#685	1	1	yes	characteristic
278	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
279	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
280	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
281	C₄²	C₂×D₂₀	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
284	C₂×C₅	—	2	2	no	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	4	1	no	—
287	C₂×C₅	#1728	1	1	yes	characteristic
288	C₂×C₅	#1742	1	1	yes	characteristic
289	C₂×C₅	C₂×#344	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	#3963	1	1	yes	characteristic
294	C₂³	#3963	1	1	yes	characteristic
295	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
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	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	#3963	1	1	yes	characteristic
314	C₂×C₄	#3963	1	1	yes	characteristic
315	C₂×C₄	#3963	1	1	yes	characteristic
316	C₂×C₄	#3963	1	1	yes	characteristic
317	C₂×C₄	#3963	1	1	yes	characteristic
318	C₂×C₄	#3963	1	1	yes	characteristic
319	C₂×C₄	C₂×#681	1	4	yes	—
320	C₂×C₄	C₂×#683	1	1	yes	characteristic
321	C₂×C₄	C₂×#683	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
323	C₂×C₄	C₂×#685	1	1	yes	characteristic
324	C₂×C₄	C₂×#685	1	1	yes	characteristic
325	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
326	C₈	—	2	8	no	—
327	C₈	—	20	4	no	—
328	C₈	—	20	4	no	—
329	C₅	#7937	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	#64779	1	2	yes	—
339	C₂²	C₄×#681	1	2	yes	—
340	C₂²	C₂×#3963	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	#65213	1	2	yes	—
351	C₄	#65421	1	2	yes	—
352	C₄	#65496	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	2	2	no	—
355	C₂	—	4	1	no	—
356	C₂	#10602320	1	1	yes	characteristic
357	C₂	#10602780	1	1	yes	characteristic
358	C₂	C₂×#64779	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#7937
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10598159	C₂	semidirect
93. C₅×M₁₆	M₁₆	semidirect
262. M₁₆	#681	semidirect
329. C₅	#7937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×M₁₆	2
#10613780	2
C₂×#65215	2
#10598159	2
#7937	5