Subgroups — D₅₃×#2240 #27539373

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276940	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276939	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276938	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276937	C₂	1	4	yes	maximal, complemented
7	#357×D₅₃	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×D₅₃	—	2	1	no	—
11	C₂×D₈×D₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×D₅₃	—	2	1	no	—
13	#127×D₅₃	—	2	1	no	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic, complemented
16	#16811027	—	2	1	no	—
17	#16811027	C₂²	1	2	yes	complemented
18	#16811026	—	2	1	no	—
19	#16811000	C₂²	1	1	yes	characteristic
20	#16810999	—	2	1	no	—
21	#16810998	—	2	1	no	—
22	#16810998	C₂²	1	2	yes	—
23	#107×D₅₃	—	2	1	no	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810425	C₂²	1	4	yes	complemented
27	#16810424	C₂²	1	4	yes	complemented
28	#16810423	C₂²	1	4	yes	complemented
29	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
30	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
31	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#493	C₂²	1	1	yes	characteristic
33	C₅₃×#357	C₂²	1	4	yes	complemented
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×Q₈×D₅₃	—	4	1	no	—
36	C₂×D₄×D₅₃	—	2	2	no	—
37	C₂×D₄×D₅₃	—	4	1	no	—
38	C₂×D₄×D₅₃	—	4	1	no	—
39	C₂×D₄₂₄	—	2	1	no	—
40	C₂×D₄₂₄	C₂³	1	2	yes	—
41	C₂×#11719922	—	2	2	no	—
42	C₂×#11719922	—	2	2	no	—
43	D₈×D₅₃	—	2	8	no	—
44	D₈×D₅₃	—	4	4	no	—
45	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	#13133217	—	2	1	no	—
48	#13133216	—	2	2	no	—
49	#13133216	—	4	1	no	—
50	#13133215	—	4	1	no	—
51	#13133214	—	2	1	no	—
52	#13133213	—	2	2	no	—
53	#13133213	—	4	1	no	—
54	#13133213	—	4	1	no	—
55	C₄⋊C₄×D₅₃	—	2	2	no	—
56	C₄⋊C₄×D₅₃	—	4	1	no	—
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	C₂×#11719908	C₂³	1	2	yes	—
59	C₁₆×D₅₃	—	2	4	no	—
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133150	—	2	2	no	—
63	#13133150	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₅₃	—	2	1	no	—
68	C₂×D₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	#127×C₅₃	—	2	1	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	C₂³	1	2	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×D₅₃	—	4	2	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	C₂×D₄×C₅₃	—	2	2	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	1	no	—
80	C₂×C₂²⋊D₅₃	—	4	2	no	—
81	C₂×C₂²⋊D₅₃	—	4	2	no	—
82	C₂×C₂²⋊D₅₃	—	4	2	no	—
83	Q₈×D₅₃	—	8	4	no	—
84	D₄×D₅₃	—	4	4	no	—
85	D₄×D₅₃	—	4	4	no	—
86	D₄×D₅₃	—	8	4	no	—
87	D₄×D₅₃	—	8	4	no	—
88	C₂×D₂₁₂	—	2	2	no	—
89	C₂×D₂₁₂	—	4	1	no	—
90	C₂×D₂₁₂	—	4	1	no	—
91	C₂×C₄×D₅₃	—	4	2	no	—
92	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
93	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Dic₁₀₆	—	4	1	no	—
96	C₂×Dic₁₀₆	—	4	2	no	—
97	D₈×C₅₃	—	2	4	no	—
98	D₈×C₅₃	—	4	2	no	—
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	#11719922	—	4	4	no	—
105	#11719922	—	4	4	no	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	4	1	no	—
108	#11719919	—	4	2	no	—
109	#11719919	—	4	2	no	—
110	C₄×Dic₅₃	D₈	1	1	yes	characteristic
111	C₄×Dic₅₃	D₈	1	1	yes	characteristic
112	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×#88322	C₂×D₄	1	1	yes	characteristic
114	C₂×#88322	C₂×D₄	1	1	yes	characteristic
115	D₄₂₄	—	2	4	no	—
116	D₄₂₄	—	4	2	no	—
117	C₈×D₅₃	SD₁₆	1	4	yes	—
118	C₈×D₅₃	#41	1	4	yes	—
119	C₁₆×C₅₃	—	2	2	no	—
120	#11719908	—	2	2	no	—
121	C₂³×C₅₃	—	4	2	no	—
122	C₂²×D₅₃	—	4	2	no	—
123	C₂²×D₅₃	—	8	4	no	—
124	C₂²×D₅₃	—	8	4	no	—
125	C₂²×D₅₃	—	8	4	no	—
126	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₅₃	—	8	2	no	—
128	D₄×C₅₃	—	4	2	no	—
129	D₄×C₅₃	—	4	2	no	—
130	D₄×C₅₃	—	8	2	no	—
131	D₄×C₅₃	—	8	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₅₃	—	8	4	no	—
137	C₂²⋊D₅₃	—	8	4	no	—
138	C₂²⋊D₅₃	—	8	4	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
143	D₂₁₂	—	4	2	no	—
144	D₂₁₂	—	4	2	no	—
145	D₂₁₂	—	8	2	no	—
146	D₂₁₂	—	8	2	no	—
147	C₄×D₅₃	—	8	4	no	—
148	C₄×D₅₃	D₁₆	1	4	yes	—
149	C₄×D₅₃	SD₃₂	1	4	yes	—
150	C₄×D₅₃	#124	1	2	yes	—
151	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
152	Dic₁₀₆	—	8	2	no	—
153	Dic₁₀₆	—	8	4	no	—
154	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
155	C₈×C₅₃	C₂×#41	1	2	yes	—
156	#88322	#124	1	2	yes	—
157	#88322	C₂×SD₁₆	1	2	yes	—
158	C₂×#2240	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	8	2	no	—
160	C₂²×C₅₃	—	8	2	no	—
161	C₂²×C₅₃	—	8	2	no	—
162	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₅₃	—	8	2	no	—
164	C₂×D₅₃	—	8	2	no	—
165	C₂×D₅₃	—	8	2	no	—
166	C₂×D₅₃	—	16	8	no	—
167	C₂×D₅₃	#486	1	2	yes	—
168	C₂×D₅₃	C₂×D₁₆	1	2	yes	—
169	C₂×D₅₃	C₂×SD₃₂	1	2	yes	—
170	C₄×C₅₃	—	8	2	no	—
171	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
172	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
173	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
174	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
175	Dic₅₃	—	8	2	no	—
176	Dic₅₃	#486	1	2	yes	—
177	Dic₅₃	C₂×D₁₆	1	2	yes	—
178	Dic₅₃	C₂×SD₃₂	1	2	yes	—
179	C₂×#500	—	53	1	no	—
180	C₂×#493	—	53	1	no	—
181	#2240	—	53	1	no	—
182	#2240	—	53	1	no	—
183	#2240	—	53	4	no	—
184	#2240	—	53	1	no	—
185	#2240	D₅₃	1	1	yes	characteristic, direct product factor
186	C₂×#357	—	53	4	no	—
187	C₂×C₄×C₁₆	—	53	1	no	—
188	C₂×C₅₃	—	16	2	no	—
189	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
191	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
192	D₅₃	—	16	2	no	—
193	D₅₃	#2240	1	4	yes	direct product factor
194	C₂²×D₈	—	53	2	no	—
195	C₂²×D₈	—	106	1	no	—
196	C₂×C₄⋊Q₈	—	106	1	no	—
197	C₂×#127	—	106	1	no	—
198	C₂²×C₁₆	—	53	2	no	—
199	#500	—	53	2	no	—
200	#500	—	53	1	no	—
201	#500	—	106	1	no	—
202	#500	—	106	1	no	—
203	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
204	#493	—	53	2	no	—
205	#493	—	53	1	no	—
206	#493	—	106	1	no	—
207	#493	—	106	1	no	—
208	#493	C₂×D₅₃	1	1	yes	characteristic
209	C₂×#107	—	53	2	no	—
210	C₂×#107	—	106	1	no	—
211	C₂×C₄×C₈	—	53	1	no	—
212	#357	—	53	4	no	—
213	#357	—	53	4	no	—
214	#357	—	53	4	no	—
215	#357	C₂×D₅₃	1	4	yes	complemented
216	C₄×C₁₆	—	53	2	no	—
217	C₄×C₁₆	—	53	1	no	—
218	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
219	C₅₃	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	212	1	no	—
221	C₂²×D₄	—	106	2	no	—
222	C₂²×D₄	—	212	1	no	—
223	C₂²×D₄	—	212	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	53	2	no	—
226	C₂×D₈	—	106	8	no	—
227	C₂×D₈	—	106	2	no	—
228	C₂×D₈	—	106	2	no	—
229	C₂×D₈	—	106	1	no	—
230	C₂×D₈	—	212	4	no	—
231	C₂×D₈	C₂²×D₅₃	1	2	yes	—
232	C₂²×C₈	—	53	1	no	—
233	C₂²×C₈	—	53	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	106	2	no	—
236	C₄⋊Q₈	—	106	1	no	—
237	C₄⋊Q₈	—	212	1	no	—
238	C₄⋊Q₈	—	212	1	no	—
239	#127	—	2	1	no	—
240	#127	—	106	2	no	—
241	#127	—	106	1	no	—
242	#127	—	212	1	no	—
243	#127	—	212	1	no	—
244	C₂×C₄⋊C₄	—	106	2	no	—
245	C₂×C₄⋊C₄	—	212	1	no	—
246	C₂×C₄²	—	53	1	no	—
247	C₂×C₁₆	—	53	2	no	—
248	C₂×C₁₆	—	106	4	no	—
249	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
250	#107	—	2	1	no	—
251	#107	—	53	2	no	—
252	#107	—	106	2	no	—
253	#107	—	106	2	no	—
254	#107	—	106	1	no	—
255	#107	C₂²×D₅₃	1	2	yes	—
256	C₄×C₈	—	53	1	no	—
257	C₄×C₈	—	53	1	no	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₂⁴	—	212	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	424	4	no	—
263	C₂×Q₈	—	212	2	no	—
264	C₂×Q₈	—	212	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	106	2	no	—
269	C₂×D₄	—	424	4	no	—
270	C₂×D₄	—	424	4	no	—
271	C₂×D₄	—	212	4	no	—
272	C₂×D₄	—	212	4	no	—
273	C₂×D₄	—	212	2	no	—
274	C₂×D₄	—	212	2	no	—
275	C₂×D₄	—	212	2	no	—
276	C₂×D₄	—	212	1	no	—
277	C₂×D₄	—	212	1	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	53	1	no	—
280	C₂²×C₄	—	53	1	no	—
281	C₂²×C₄	—	212	2	no	—
282	D₈	—	2	4	no	—
283	D₈	—	4	2	no	—
284	D₈	—	106	4	no	—
285	D₈	—	212	4	no	—
286	D₈	—	212	4	no	—
287	D₈	—	212	2	no	—
288	C₂×C₈	—	53	1	no	—
289	C₂×C₈	—	53	1	no	—
290	C₂×C₈	—	53	4	no	—
291	C₂×C₈	—	53	4	no	—
292	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	106	2	no	—
297	C₄⋊C₄	—	212	2	no	—
298	C₄⋊C₄	—	212	2	no	—
299	C₄⋊C₄	—	212	1	no	—
300	C₄²	—	53	1	no	—
301	C₄²	—	53	1	no	—
302	C₄²	—	53	1	no	—
303	C₄²	D₄×D₅₃	1	1	yes	characteristic
304	C₁₆	—	2	2	no	—
305	C₁₆	—	106	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	53	1	no	—
308	C₂³	—	424	4	no	—
309	C₂³	—	424	4	no	—
310	C₂³	—	424	4	no	—
311	C₂³	—	212	2	no	—
312	Q₈	—	8	2	no	—
313	Q₈	—	424	4	no	—
314	Q₈	—	424	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	424	4	no	—
320	D₄	—	424	4	no	—
321	D₄	—	424	4	no	—
322	D₄	—	424	2	no	—
323	D₄	—	424	2	no	—
324	D₄	—	212	2	no	—
325	D₄	—	212	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	53	1	no	—
328	C₂×C₄	—	53	1	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	4	no	—
331	C₂×C₄	—	53	4	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	53	2	no	—
334	C₂×C₄	—	424	4	no	—
335	C₂×C₄	—	212	2	no	—
336	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
337	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
339	C₈	—	53	2	no	—
340	C₈	—	53	2	no	—
341	C₈	SD₁₆×D₅₃	1	2	yes	—
342	C₈	#41×D₅₃	1	2	yes	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	53	2	no	—
347	C₂²	—	53	2	no	—
348	C₂²	—	53	2	no	—
349	C₂²	—	848	8	no	—
350	C₂²	—	424	2	no	—
351	C₂²	—	424	2	no	—
352	C₂²	—	424	2	no	—
353	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
354	C₄	—	8	2	no	—
355	C₄	—	53	2	no	—
356	C₄	—	53	2	no	—
357	C₄	—	53	2	no	—
358	C₄	—	424	2	no	—
359	C₄	D₁₆×D₅₃	1	2	yes	—
360	C₄	SD₃₂×D₅₃	1	2	yes	—
361	C₄	#124×D₅₃	1	1	yes	characteristic
362	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
363	C₂	—	16	2	no	—
364	C₂	—	53	4	no	—
365	C₂	—	848	2	no	—
366	C₂	#486×D₅₃	1	1	yes	characteristic
367	C₂	C₂×D₁₆×D₅₃	1	1	yes	characteristic
368	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2240	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#493×D₅₃	C₂
hypercenter	#2240	D₅₃
hypocenter	C₅₃	C₂×#2240
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #500×D₅₃	C₂	semidirect
3. #20276940	C₂	semidirect
4. #20276939	C₂	semidirect
5. #20276938	C₂	semidirect
6. #20276937	C₂	semidirect
7. #357×D₅₃	C₂	semidirect
8. C₄×C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2240	C₂	semidirect
15. #16811028	C₂²	semidirect
17. #16811027	C₂²	semidirect
26. #16810425	C₂²	semidirect
27. #16810424	C₂²	semidirect
28. #16810423	C₂²	semidirect
29. C₄×#11719908	C₂²	semidirect
30. C₁₆×Dic₅₃	C₂²	semidirect
31. C₅₃×#500	C₂²	semidirect
33. C₅₃×#357	C₂²	semidirect
34. C₄×C₁₆×C₅₃	C₂²	semidirect
185. #2240	D₅₃	direct (N × H)
193. D₅₃	#2240	direct (N × H)
203. #500	C₂×D₅₃	semidirect
215. #357	C₂×D₅₃	semidirect
218. C₄×C₁₆	C₂×D₅₃	semidirect
219. C₅₃	C₂×#2240	semidirect

Maximal subgroups

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

subgroup	index
#500×D₅₃	2
#493×D₅₃	2
#20276940	2
#20276939	2
#20276938	2
#20276937	2
#357×D₅₃	2
C₄×C₁₆×D₅₃	2
C₅₃×#2240	2
C₂×#2240	53

Subgroups — D53×#2240 #27539373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2240 #27539373