Subgroups — D₅₃×#2242 #27539383

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276949	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276948	C₂	1	2	yes	maximal, complemented
5	#20276947	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276946	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276945	C₂	1	2	yes	maximal, complemented
8	#358×D₅₃	C₂	1	2	yes	maximal, complemented
9	#357×D₅₃	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₅₃	—	2	1	no	—
14	C₂×D₈×D₅₃	C₂²	1	1	yes	characteristic
15	#125×D₅₃	—	2	1	no	—
16	#124×D₅₃	—	2	1	no	—
17	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
18	#16811025	C₂²	1	1	yes	characteristic, complemented
19	#16811024	C₂²	1	2	yes	complemented
20	#16811023	—	2	1	no	—
21	#16811022	—	2	1	no	—
22	#16811008	C₂²	1	1	yes	characteristic
23	#16811007	C₂²	1	1	yes	characteristic
24	#16811006	C₂²	1	1	yes	characteristic
25	#16811005	—	2	1	no	—
26	#16811004	—	2	1	no	—
27	#107×D₅₃	C₂²	1	2	yes	—
28	#106×D₅₃	—	2	1	no	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	#16810428	C₂²	1	2	yes	complemented
31	#16810427	C₂²	1	2	yes	complemented
32	#16810426	C₂²	1	2	yes	complemented
33	#16810425	C₂²	1	2	yes	—
34	#16810424	C₂²	1	2	yes	—
35	#16810423	C₂²	1	2	yes	—
36	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
38	C₅₃×#499	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#495	C₂²	1	1	yes	characteristic
40	C₅₃×#358	C₂²	1	2	yes	complemented
41	C₅₃×#357	C₂²	1	2	yes	—
42	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₅₃	—	2	1	no	—
44	C₂×D₄×D₅₃	—	2	1	no	—
45	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
46	C₂×#11719925	—	2	1	no	—
47	Q₁₆×D₅₃	—	2	4	no	—
48	C₂×#11719913	—	2	1	no	—
49	C₂×#11719923	—	2	1	no	—
50	C₂×#11719924	—	2	1	no	—
51	SD₁₆×D₅₃	—	4	4	no	—
52	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
53	C₂×#11719922	—	2	1	no	—
54	D₈×D₅₃	—	2	4	no	—
55	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
57	#13133209	—	2	1	no	—
58	#13133208	—	2	2	no	—
59	#13133207	—	4	2	no	—
60	#13133206	—	2	1	no	—
61	#13133205	—	2	1	no	—
62	#13133204	—	2	1	no	—
63	#13133203	—	4	2	no	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	C₄⋊C₄×D₅₃	—	4	2	no	—
66	C₂²⋊C₄×D₅₃	—	4	2	no	—
67	C₄²×D₅₃	D₄	1	1	yes	characteristic
68	C₂×#11719908	C₂³	1	2	yes	—
69	C₁₆×D₅₃	—	2	4	no	—
70	#13133151	C₂³	1	2	yes	—
71	#13133150	—	2	2	no	—
72	#13133149	—	2	1	no	—
73	#13133148	—	2	2	no	—
74	C₄×#88322	D₄	1	1	yes	characteristic
75	C₈×Dic₅₃	D₄	1	1	yes	characteristic
76	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₅₃	—	2	1	no	—
79	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
80	#125×C₅₃	—	2	1	no	—
81	#124×C₅₃	—	2	1	no	—
82	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
83	#107×C₅₃	C₂³	1	2	yes	—
84	#106×C₅₃	—	2	1	no	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂³×D₅₃	—	4	1	no	—
87	C₂×Q₈×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	—	2	1	no	—
89	C₂×C₂²⋊D₅₃	—	4	1	no	—
90	Q₈×D₅₃	—	4	2	no	—
91	Q₈×D₅₃	—	4	2	no	—
92	D₄×D₅₃	—	4	2	no	—
93	D₄×D₅₃	—	4	2	no	—
94	C₂×D₂₁₂	—	2	1	no	—
95	C₂×C₄×D₅₃	—	4	1	no	—
96	C₂×C₄×D₅₃	—	4	2	no	—
97	C₂×C₄×D₅₃	D₈	1	2	yes	—
98	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₁₀₆	—	2	1	no	—
100	C₂×Dic₁₀₆	—	4	1	no	—
101	Q₁₆×C₅₃	—	2	2	no	—
102	SD₁₆×C₅₃	—	4	2	no	—
103	D₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²⋊Dic₅₃	—	4	2	no	—
111	#11719925	—	4	2	no	—
112	#11719924	—	4	2	no	—
113	#11719923	—	4	2	no	—
114	#11719922	—	4	2	no	—
115	#11719921	—	4	2	no	—
116	#11719921	—	4	2	no	—
117	C₄⋊Dic₅₃	—	2	2	no	—
118	C₄⋊Dic₅₃	—	4	2	no	—
119	#11719919	—	4	2	no	—
120	#11719919	—	4	2	no	—
121	#11719919	—	4	2	no	—
122	C₄×Dic₅₃	D₈	1	2	yes	—
123	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×#88322	C₂×D₄	1	1	yes	characteristic
125	C₂×#88322	C₂×D₄	1	1	yes	characteristic
126	Dic₂₁₂	—	2	2	no	—
127	D₄₂₄	—	2	2	no	—
128	#11719913	—	4	2	no	—
129	C₈×D₅₃	SD₁₆	1	4	yes	—
130	C₈×D₅₃	#41	1	4	yes	—
131	C₁₆×C₅₃	—	2	2	no	—
132	#11719908	—	2	2	no	—
133	C₂³×C₅₃	—	4	1	no	—
134	C₂²×D₅₃	—	4	1	no	—
135	C₂²×D₅₃	—	8	2	no	—
136	C₂²×D₅₃	—	8	4	no	—
137	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₅₃	—	4	1	no	—
139	Q₈×C₅₃	—	4	1	no	—
140	D₄×C₅₃	—	4	1	no	—
141	D₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₅₃	—	8	2	no	—
147	C₂×Dic₅₃	—	4	1	no	—
148	C₂×Dic₅₃	—	4	2	no	—
149	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
150	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
151	D₂₁₂	—	4	1	no	—
152	D₂₁₂	—	4	1	no	—
153	C₄×D₅₃	—	2	4	no	—
154	C₄×D₅₃	—	8	2	no	—
155	C₄×D₅₃	—	8	4	no	—
156	C₄×D₅₃	#124	1	2	yes	—
157	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
158	Dic₁₀₆	—	4	1	no	—
159	Dic₁₀₆	—	4	1	no	—
160	Dic₁₀₆	—	8	2	no	—
161	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
162	C₈×C₅₃	C₂×#41	1	2	yes	—
163	#88322	#124	1	2	yes	—
164	#88322	C₂×SD₁₆	1	2	yes	—
165	C₂×#2242	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	—	8	1	no	—
167	C₂²×C₅₃	—	8	2	no	—
168	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₅₃	—	8	1	no	—
170	C₂×D₅₃	—	8	2	no	—
171	C₂×D₅₃	—	16	4	no	—
172	C₂×D₅₃	#486	1	2	yes	—
173	C₂×D₅₃	#508	1	4	yes	—
174	C₄×C₅₃	—	2	2	no	—
175	C₄×C₅₃	—	8	1	no	—
176	C₄×C₅₃	—	8	2	no	—
177	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
179	Dic₅₃	—	2	2	no	—
180	Dic₅₃	—	8	1	no	—
181	Dic₅₃	—	8	2	no	—
182	Dic₅₃	#486	1	2	yes	—
183	C₂×#499	—	53	1	no	—
184	C₂×#495	—	53	1	no	—
185	#2242	—	53	1	no	—
186	#2242	—	53	1	no	—
187	#2242	—	53	2	no	—
188	#2242	—	53	2	no	—
189	#2242	—	53	1	no	—
190	#2242	D₅₃	1	1	yes	characteristic, direct product factor
191	C₂×#358	—	53	2	no	—
192	C₂×#357	—	53	2	no	—
193	C₂×C₄×C₁₆	—	53	1	no	—
194	C₂×C₅₃	—	16	1	no	—
195	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#508	1	2	yes	—
197	D₅₃	—	16	1	no	—
198	D₅₃	#2242	1	4	yes	direct product factor
199	C₂²×Q₁₆	—	53	1	no	—
200	C₂²×SD₁₆	—	106	1	no	—
201	C₂²×D₈	—	53	1	no	—
202	C₂×#125	—	106	1	no	—
203	C₂×#124	—	106	1	no	—
204	C₂²×C₁₆	—	53	2	no	—
205	#499	—	53	2	no	—
206	#499	—	53	1	no	—
207	#499	—	106	1	no	—
208	#499	—	106	1	no	—
209	#499	C₂×D₅₃	1	1	yes	characteristic, complemented
210	#495	—	53	1	no	—
211	#495	—	53	1	no	—
212	#495	—	53	1	no	—
213	#495	—	106	1	no	—
214	#495	—	106	1	no	—
215	#495	C₂×D₅₃	1	1	yes	characteristic
216	C₂×#107	—	53	2	no	—
217	C₂×#106	—	106	1	no	—
218	C₂×C₄×C₈	—	53	1	no	—
219	#358	—	53	2	no	—
220	#358	—	53	2	no	—
221	#358	—	53	2	no	—
222	#358	C₂×D₅₃	1	2	yes	complemented
223	#357	—	53	2	no	—
224	#357	—	53	2	no	—
225	#357	—	53	2	no	—
226	#357	C₂×D₅₃	1	2	yes	—
227	C₄×C₁₆	—	53	2	no	—
228	C₄×C₁₆	—	53	1	no	—
229	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
230	C₅₃	C₂×#2242	1	1	yes	characteristic, complemented, Sylow
231	C₂²×Q₈	—	106	1	no	—
232	C₂²×D₄	—	106	1	no	—
233	C₂×Q₁₆	—	53	1	no	—
234	C₂×Q₁₆	—	106	4	no	—
235	C₂×Q₁₆	—	106	1	no	—
236	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	106	1	no	—
239	C₂×SD₁₆	—	106	1	no	—
240	C₂×SD₁₆	—	106	1	no	—
241	C₂×SD₁₆	—	212	4	no	—
242	C₂×D₈	—	53	1	no	—
243	C₂×D₈	—	106	4	no	—
244	C₂×D₈	—	106	1	no	—
245	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₂²×C₈	—	53	1	no	—
247	C₂²×C₈	—	53	1	no	—
248	#125	—	2	1	no	—
249	#125	—	106	2	no	—
250	#125	—	106	1	no	—
251	#125	—	212	2	no	—
252	#124	—	2	1	no	—
253	#124	—	106	1	no	—
254	#124	—	106	1	no	—
255	#124	—	106	1	no	—
256	#124	—	212	2	no	—
257	C₂×C₄⋊C₄	—	106	2	no	—
258	C₂×C₄⋊C₄	—	212	2	no	—
259	C₂×C₂²⋊C₄	—	212	2	no	—
260	C₂×C₄²	—	53	1	no	—
261	C₂×C₁₆	—	53	2	no	—
262	C₂×C₁₆	—	106	4	no	—
263	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
264	#107	—	53	2	no	—
265	#107	—	106	2	no	—
266	#107	C₂²×D₅₃	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	106	2	no	—
269	#106	—	106	1	no	—
270	C₄×C₈	—	53	1	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	—	53	1	no	—
273	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
274	C₂⁴	—	212	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	106	1	no	—
277	C₂×Q₈	—	212	1	no	—
278	C₂×Q₈	—	212	2	no	—
279	C₂×Q₈	—	212	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	106	1	no	—
282	C₂×D₄	—	212	2	no	—
283	C₂×D₄	—	212	2	no	—
284	C₂×D₄	—	212	1	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	53	2	no	—
287	C₂²×C₄	—	212	1	no	—
288	C₂²×C₄	—	212	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	106	2	no	—
291	Q₁₆	—	212	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	212	2	no	—
294	SD₁₆	—	212	2	no	—
295	SD₁₆	—	212	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	106	2	no	—
298	D₈	—	212	2	no	—
299	C₂×C₈	—	53	1	no	—
300	C₂×C₈	—	53	1	no	—
301	C₂×C₈	—	53	4	no	—
302	C₂×C₈	—	53	4	no	—
303	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	106	2	no	—
308	C₄⋊C₄	—	212	2	no	—
309	C₄⋊C₄	—	212	2	no	—
310	C₄⋊C₄	—	212	2	no	—
311	C₄⋊C₄	—	212	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	212	2	no	—
314	C₂²⋊C₄	—	212	2	no	—
315	C₂²⋊C₄	—	212	2	no	—
316	C₄²	—	53	2	no	—
317	C₄²	—	53	1	no	—
318	C₄²	D₄×D₅₃	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	106	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	53	1	no	—
323	C₂³	—	424	2	no	—
324	C₂³	—	424	4	no	—
325	C₂³	—	212	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	424	2	no	—
329	Q₈	—	212	1	no	—
330	Q₈	—	212	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	424	2	no	—
334	D₄	—	212	1	no	—
335	D₄	—	212	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	2	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	2	no	—
342	C₂×C₄	—	106	4	no	—
343	C₂×C₄	—	424	4	no	—
344	C₂×C₄	—	424	2	no	—
345	C₂×C₄	—	212	1	no	—
346	C₂×C₄	—	212	2	no	—
347	C₂×C₄	D₈×D₅₃	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
349	C₈	—	53	2	no	—
350	C₈	—	53	2	no	—
351	C₈	SD₁₆×D₅₃	1	2	yes	—
352	C₈	#41×D₅₃	1	2	yes	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	53	4	no	—
356	C₂²	—	53	2	no	—
357	C₂²	—	848	4	no	—
358	C₂²	—	424	2	no	—
359	C₂²	—	424	1	no	—
360	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	53	2	no	—
365	C₄	—	106	2	no	—
366	C₄	—	424	2	no	—
367	C₄	—	424	1	no	—
368	C₄	#124×D₅₃	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	53	4	no	—
372	C₂	—	848	1	no	—
373	C₂	#486×D₅₃	1	1	yes	characteristic
374	C₂	#508×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2242	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₈×D₅₃	C₂²
hypercenter	#2242	D₅₃
hypocenter	C₅₃	C₂×#2242
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #499×D₅₃	C₂	semidirect
3. #20276949	C₂	semidirect
4. #20276948	C₂	semidirect
5. #20276947	C₂	semidirect
6. #20276946	C₂	semidirect
7. #20276945	C₂	semidirect
8. #358×D₅₃	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2242	C₂	semidirect
18. #16811025	C₂²	semidirect
19. #16811024	C₂²	semidirect
30. #16810428	C₂²	semidirect
31. #16810427	C₂²	semidirect
32. #16810426	C₂²	semidirect
36. C₄×#11719908	C₂²	semidirect
37. C₁₆×Dic₅₃	C₂²	semidirect
38. C₅₃×#499	C₂²	semidirect
40. C₅₃×#358	C₂²	semidirect
42. C₄×C₁₆×C₅₃	C₂²	semidirect
190. #2242	D₅₃	direct (N × H)
198. D₅₃	#2242	direct (N × H)
209. #499	C₂×D₅₃	semidirect
222. #358	C₂×D₅₃	semidirect
229. C₄×C₁₆	C₂×D₅₃	semidirect
230. C₅₃	C₂×#2242	semidirect

Maximal subgroups

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

subgroup	index
#499×D₅₃	2
#495×D₅₃	2
#20276949	2
#20276948	2
#20276947	2
#20276946	2
#20276945	2
#358×D₅₃	2
#357×D₅₃	2
C₄×C₁₆×D₅₃	2
C₅₃×#2242	2
C₂×#2242	53

Subgroups — D53×#2242 #27539383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2242 #27539383