Subgroups — #12065038

Order 1280.

798 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#10614248	C₂	1	1	yes	maximal, characteristic
3	#10614248	C₂	1	1	yes	maximal, characteristic
4	#10600642	C₂	1	2	yes	maximal, complemented
5	#10600641	C₂	1	2	yes	maximal, complemented
6	M₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
7	M₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600519	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600518	C₂	1	2	yes	maximal, complemented
10	C₈×#681	C₂	1	2	yes	maximal, complemented
11	C₂×#64674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#131	C₄	1	2	yes	—
13	C₄×C₅×#41	C₂²	1	1	yes	characteristic
14	C₅×#405	C₄	1	4	yes	—
15	C₅×#405	C₄	1	2	yes	—
16	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₅×M₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	#65313	C₄	1	2	yes	—
21	#65313	C₄	1	2	yes	—
22	#65301	C₄	1	2	yes	—
23	#65301	C₄	1	2	yes	—
24	#65300	C₄	1	4	yes	—
25	#65300	C₄	1	4	yes	—
26	#65214	C₂²	1	2	yes	—
27	#65214	C₂²	1	2	yes	—
28	#65214	C₂²	1	1	yes	characteristic
29	#65214	C₂²	1	1	yes	characteristic
30	C₄×#681	C₂²	1	2	yes	—
31	C₄×#681	C₂²	1	2	yes	—
32	C₂×C₄×#168	C₂²	1	1	yes	characteristic
33	C₂×C₄×#168	C₂²	1	1	yes	characteristic
34	#64675	C₂²	1	2	yes	—
35	#64675	C₂²	1	2	yes	—
36	#64674	C₂²	1	1	yes	characteristic
37	#64674	C₂²	1	1	yes	characteristic
38	#64674	C₂²	1	1	yes	characteristic
39	#64674	C₂²	1	1	yes	characteristic
40	#64674	C₂²	1	2	yes	—
41	#64674	C₂²	1	2	yes	—
42	C₈×#168	C₂²	1	2	yes	—
43	C₈×#168	C₂²	1	2	yes	—
44	#17337	—	5	1	no	maximal, Sylow
45	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
46	C₅×#131	—	2	8	no	—
47	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
48	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
49	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
54	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
55	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
57	C₅×#117	C₂×C₄	1	2	yes	—
58	C₅×#117	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅	C₂³	1	2	yes	—
61	C₂×#681	C₂×C₄	1	4	yes	—
62	C₂×#681	C₂×C₄	1	4	yes	—
63	C₂²×#168	C₂×C₄	1	2	yes	—
64	C₂²×#168	C₂×C₄	1	2	yes	—
65	C₅×#97	C₂×C₄	1	4	yes	—
66	C₅×#97	C₂×C₄	1	2	yes	—
67	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂²⋊#168	C₂×C₄	1	2	yes	—
73	C₂²⋊#168	C₂×C₄	1	2	yes	—
74	C₂²⋊#168	C₂×C₄	1	4	yes	—
75	C₂²⋊#168	C₂×C₄	1	4	yes	—
76	C₄⋊#168	C₂×C₄	1	4	yes	—
77	C₄⋊#168	C₂×C₄	1	4	yes	—
78	C₄⋊#168	C₂×C₄	1	2	yes	—
79	C₄⋊#168	C₂×C₄	1	2	yes	—
80	#3935	C₂×C₄	1	4	yes	—
81	#3935	C₂×C₄	1	4	yes	—
82	C₄×#168	C₂×C₄	1	2	yes	—
83	C₄×#168	C₂×C₄	1	2	yes	—
84	C₄×#168	C₂³	1	1	yes	characteristic
85	C₄×#168	C₂³	1	1	yes	characteristic
86	C₄×#168	C₂³	1	2	yes	—
87	C₄×#168	C₂³	1	2	yes	—
88	C₄×#168	C₂³	1	1	yes	characteristic
89	C₄×#168	C₂³	1	1	yes	characteristic
90	#2965	—	5	1	no	—
91	#2965	—	5	1	no	—
92	C₄×#131	D₅	1	1	yes	characteristic
93	#1455	—	5	2	no	—
94	#1453	—	5	2	no	—
95	#1452	—	5	1	no	—
96	#1451	—	5	2	no	—
97	C₈×M₁₆	—	5	1	no	—
98	C₈×M₁₆	—	5	1	no	—
99	C₈×M₁₆	—	5	2	no	—
100	C₂×#322	—	5	1	no	—
101	C₅×#41	—	2	4	no	—
102	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
103	C₂×C₅×D₄	C₄²	1	2	yes	—
104	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₅×M₁₆	—	2	8	no	—
112	C₅×M₁₆	M₁₆	1	8	yes	complemented
113	C₂×C₅×C₈	—	2	4	no	—
114	C₂×C₅×C₈	—	2	8	no	—
115	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
118	C₅×C₄⋊C₄	C₄²	1	2	yes	—
119	C₅×C₄⋊C₄	C₄²	1	2	yes	—
120	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
121	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
122	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
123	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
124	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
125	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅	C₂²×C₄	1	2	yes	—
128	C₄²×C₅	C₂²×C₄	1	2	yes	—
129	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅	C₂²×C₄	1	2	yes	—
131	C₄²×C₅	C₂⁴	1	1	yes	characteristic
132	#681	—	2	8	no	—
133	#681	—	4	4	no	—
134	C₂×#168	—	2	4	no	—
135	C₂×#168	—	4	2	no	—
136	C₂×#168	C₂²×C₄	1	2	yes	—
137	C₂×#168	C₂²×C₄	1	2	yes	—
138	C₂×#168	C₂²×C₄	1	2	yes	—
139	C₂×#168	C₂²×C₄	1	4	yes	—
140	C₂×#168	C₂²×C₄	1	4	yes	—
141	C₂×#168	C₂²×C₄	1	2	yes	—
142	C₂×#131	Dic₅	1	2	yes	—
143	C₄×#41	C₂×D₅	1	1	yes	characteristic
144	#446	—	5	2	no	—
145	#446	—	5	2	no	—
146	#436	—	5	2	no	—
147	#436	—	5	2	no	—
148	#435	—	5	4	no	—
149	#435	—	5	4	no	—
150	#431	—	5	1	no	—
151	#431	—	5	1	no	—
152	#431	—	5	2	no	—
153	#431	—	5	2	no	—
154	#405	Dic₅	1	4	yes	—
155	#405	Dic₅	1	2	yes	—
156	C₄×M₁₆	—	5	2	no	—
157	C₄×M₁₆	—	5	2	no	—
158	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
159	C₄×M₁₆	C₂×D₅	1	2	yes	—
160	C₂×C₄×C₈	—	5	1	no	—
161	C₂×C₄×C₈	—	5	1	no	—
162	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
164	#322	—	5	1	no	—
165	#322	—	5	1	no	—
166	#322	—	5	1	no	—
167	#322	—	5	1	no	—
168	#322	—	5	2	no	—
169	#322	—	5	2	no	—
170	#322	—	5	2	no	—
171	#322	—	5	2	no	—
172	C₈²	—	5	2	no	—
173	C₈²	—	5	2	no	—
174	C₂³×C₅	C₂×C₄²	1	2	yes	—
175	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
176	C₅×Q₈	—	2	2	no	—
177	C₅×D₄	—	2	4	no	—
178	C₅×D₄	—	2	2	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	4	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	4	no	—
184	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
188	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
191	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	—	2	2	no	—
196	C₅×C₈	C₂×M₁₆	1	8	yes	—
197	#168	—	2	2	no	—
198	#168	—	2	4	no	—
199	#168	—	2	2	no	—
200	#168	—	2	2	no	—
201	#168	—	2	4	no	—
202	#168	#131	1	4	yes	complemented
203	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
204	#131	—	2	8	no	—
205	C₂×M₁₆	—	5	4	no	—
206	C₂×M₁₆	—	5	4	no	—
207	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
208	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
209	C₂²×C₈	—	5	2	no	—
210	C₂²×C₈	—	5	2	no	—
211	C₂²×C₈	C₂×Dic₅	1	2	yes	—
212	C₂²×C₈	C₂×Dic₅	1	4	yes	—
213	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
214	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
215	C₄×D₄	C₄×D₅	1	2	yes	—
216	C₄×D₄	C₄×D₅	1	2	yes	—
217	C₄×D₄	C₄×D₅	1	1	yes	characteristic
218	C₄×D₄	C₄×D₅	1	1	yes	characteristic
219	#117	C₂×Dic₅	1	2	yes	—
220	#117	C₂×Dic₅	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₅	1	2	yes	—
223	C₄⋊C₈	—	5	4	no	—
224	C₄⋊C₈	—	5	4	no	—
225	C₄⋊C₈	—	5	2	no	—
226	C₄⋊C₈	—	5	2	no	—
227	C₂²⋊C₈	—	5	2	no	—
228	C₂²⋊C₈	—	5	2	no	—
229	C₂²⋊C₈	—	5	4	no	—
230	C₂²⋊C₈	—	5	4	no	—
231	#97	—	5	4	no	—
232	#97	—	5	4	no	—
233	#97	C₂×Dic₅	1	4	yes	—
234	#97	C₂×Dic₅	1	2	yes	—
235	C₄×C₈	—	5	2	no	—
236	C₄×C₈	—	5	2	no	—
237	C₄×C₈	—	5	1	no	—
238	C₄×C₈	—	5	1	no	—
239	C₄×C₈	—	5	2	no	—
240	C₄×C₈	—	5	2	no	—
241	C₄×C₈	—	5	1	no	—
242	C₄×C₈	—	5	1	no	—
243	C₄×C₈	C₂×Dic₅	1	2	yes	—
244	C₄×C₈	C₂×Dic₅	1	4	yes	—
245	C₄×C₈	C₂²×D₅	1	2	yes	—
246	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	—	2	2	no	—
252	C₂²×C₅	—	2	2	no	—
253	C₂²×C₅	C₄×M₁₆	1	2	yes	—
254	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
255	C₄×C₅	—	2	2	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	2	no	—
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	2	no	—
260	C₄×C₅	C₄×M₁₆	1	2	yes	—
261	C₄×C₅	C₂²×M₁₆	1	2	yes	—
262	C₄×C₅	C₂×#131	1	2	yes	—
263	C₄×C₅	#568	1	2	yes	—
264	#41	—	2	4	no	—
265	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
266	C₂×D₄	C₄×Dic₅	1	2	yes	—
267	C₂×D₄	C₄×Dic₅	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
271	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
272	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
273	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
274	M₁₆	—	2	8	no	—
275	M₁₆	—	10	8	no	—
276	M₁₆	—	20	4	no	—
277	M₁₆	#681	1	8	yes	complemented
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	8	no	—
280	C₂×C₈	—	5	2	no	—
281	C₂×C₈	—	5	2	no	—
282	C₂×C₈	—	5	2	no	—
283	C₂×C₈	—	5	2	no	—
284	C₂×C₈	—	5	4	no	—
285	C₂×C₈	—	5	4	no	—
286	C₂×C₈	—	10	4	no	—
287	C₂×C₈	—	20	2	no	—
288	C₂×C₈	C₂²×Dic₅	1	2	yes	—
289	C₂×C₈	C₂²×Dic₅	1	2	yes	—
290	C₂×C₈	C₂²×Dic₅	1	4	yes	—
291	C₄⋊C₄	C₄×Dic₅	1	2	yes	—
292	C₄⋊C₄	C₄×Dic₅	1	2	yes	—
293	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
294	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
295	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
296	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
297	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
298	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₅	1	2	yes	—
301	C₄²	C₂×C₄×D₅	1	2	yes	—
302	C₄²	C₂²×Dic₅	1	1	yes	characteristic
303	C₄²	C₂²×Dic₅	1	2	yes	—
304	C₄²	C₂³×D₅	1	1	yes	characteristic
305	C₂×C₅	—	2	2	no	—
306	C₂×C₅	—	4	2	no	—
307	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
308	C₂×C₅	C₄×#131	1	1	yes	characteristic
309	C₂×C₅	#2875	1	1	yes	characteristic
310	C₂³	C₂×C₄×Dic₅	1	2	yes	—
311	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	D₄	—	2	4	no	—
314	D₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	C₂×#681	1	4	yes	—
321	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
322	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
324	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
329	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	5	4	no	—
333	C₈	—	10	2	no	—
334	C₈	—	10	4	no	—
335	C₈	—	10	2	no	—
336	C₈	—	10	2	no	—
337	C₈	—	10	4	no	—
338	C₈	C₂×#681	1	8	yes	—
339	C₅	#17337	1	1	yes	characteristic, complemented, Sylow
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₄×#681	1	2	yes	—
346	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	C₄×#681	1	2	yes	—
353	C₄	D₅×#131	1	2	yes	—
354	C₄	#66084	1	2	yes	—
355	C₄	C₂²×#681	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×C₄×#681	1	1	yes	characteristic
359	C₂	Dic₅×#131	1	1	yes	characteristic
360	C₂	#10615513	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₄×C₅×#131	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#17337
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
4. #10600642	C₂	semidirect
5. #10600641	C₂	semidirect
6. M₁₆×#168	C₂	semidirect
7. M₁₆×#168	C₂	semidirect
8. #10600519	C₂	semidirect
9. #10600518	C₂	semidirect
10. C₈×#681	C₂	semidirect
11. C₂×#64674	C₂	semidirect
112. C₅×M₁₆	M₁₆	semidirect
202. #168	#131	semidirect
277. M₁₆	#681	semidirect
339. C₅	#17337	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#10614248	2
#10600642	2
#10600641	2
M₁₆×#168	2
#10600519	2
#10600518	2
C₈×#681	2
C₂×#64674	2
#17337	5