Subgroups — D₅₃×#1708 #27371743

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#492×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#487×D₅₃	C₂	1	2	yes	maximal, complemented
4	#486×D₅₃	C₂	1	2	yes	maximal
5	#20274318	C₂	1	2	yes	maximal, complemented
6	#20274317	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274316	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274315	C₂	1	2	yes	maximal, complemented
9	#20274314	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×D₅₃	—	2	2	no	—
13	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₅₃	C₂²	1	1	yes	characteristic
15	#127×D₅₃	C₂²	1	1	yes	characteristic
16	#16811021	C₂²	1	1	yes	characteristic, complemented
17	#16811020	—	2	2	no	—
18	#16811019	C₂²	1	2	yes	complemented
19	#16810997	C₂²	1	1	yes	characteristic
20	#16810996	C₂²	1	1	yes	characteristic
21	#16810995	C₂²	1	1	yes	characteristic
22	#16810994	—	2	2	no	—
23	#16810975	C₂²	1	2	yes	complemented
24	#16810974	C₂²	1	2	yes	complemented
25	#16810973	C₂²	1	2	yes	complemented
26	#16810972	—	2	4	no	—
27	#16810971	C₂²	1	2	yes	—
28	#16810970	C₂²	1	2	yes	—
29	#16810969	C₂²	1	2	yes	—
30	#16810968	—	2	4	no	—
31	#106×D₅₃	—	2	2	no	—
32	#103×D₅₃	—	2	4	no	—
33	#102×D₅₃	—	2	4	no	—
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₅₃	C₂²	1	2	yes	—
36	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
37	C₈×#88322	C₂²	1	2	yes	complemented
38	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#492	C₂²	1	1	yes	characteristic
40	C₅₃×#487	C₂²	1	2	yes	complemented
41	C₅₃×#486	C₂²	1	2	yes	—
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₅₃	—	2	2	no	—
44	C₂×Q₈×D₅₃	—	2	1	no	—
45	C₂×D₄×D₅₃	—	2	2	no	—
46	C₂×D₄×D₅₃	—	2	1	no	—
47	C₂×#11719913	—	2	2	no	—
48	C₂×#11719923	—	2	2	no	—
49	C₂×#11719924	—	2	2	no	—
50	SD₁₆×D₅₃	—	4	8	no	—
51	C₂×C₈×D₅₃	D₄	1	4	yes	—
52	C₂×C₈×D₅₃	D₄	1	2	yes	—
53	#13133217	C₂³	1	2	yes	—
54	#13133217	C₂³	1	1	yes	characteristic
55	#13133216	—	2	2	no	—
56	#13133216	—	2	2	no	—
57	#13133216	—	2	2	no	—
58	#13133215	—	2	2	no	—
59	#13133215	—	2	1	no	—
60	#13133214	C₂³	1	1	yes	characteristic
61	#13133213	—	2	2	no	—
62	#13133213	—	2	1	no	—
63	C₄⋊C₄×D₅₃	—	2	2	no	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	C₄⋊C₄×D₅₃	—	2	2	no	—
66	C₄²×D₅₃	C₂³	1	1	yes	characteristic
67	#13133149	—	2	2	no	—
68	#13133148	—	2	4	no	—
69	#13133142	—	2	4	no	—
70	#13133141	—	2	4	no	—
71	#13133140	—	2	4	no	—
72	#13133139	—	2	4	no	—
73	#13133138	—	2	4	no	—
74	#13133137	—	2	4	no	—
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	2	yes	—
77	C₈×Dic₅₃	D₄	1	2	yes	—
78	C₈×Dic₅₃	D₄	1	4	yes	—
79	C₂×SD₁₆×C₅₃	—	2	2	no	—
80	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	#127×C₅₃	C₂³	1	1	yes	characteristic
83	#106×C₅₃	—	2	2	no	—
84	#103×C₅₃	—	2	4	no	—
85	#102×C₅₃	—	2	4	no	—
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×D₅₃	—	4	1	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	2	1	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	1	no	—
93	C₂×C₂²⋊D₅₃	—	4	1	no	—
94	C₂×C₂²⋊D₅₃	—	4	2	no	—
95	Q₈×D₅₃	—	4	4	no	—
96	Q₈×D₅₃	—	4	8	no	—
97	D₄×D₅₃	—	4	4	no	—
98	D₄×D₅₃	—	4	8	no	—
99	C₂×D₂₁₂	—	2	2	no	—
100	C₂×D₂₁₂	—	2	1	no	—
101	C₂×C₄×D₅₃	—	4	1	no	—
102	C₂×C₄×D₅₃	—	4	2	no	—
103	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
105	C₂×Dic₁₀₆	—	2	2	no	—
106	C₂×Dic₁₀₆	—	2	1	no	—
107	C₂×Dic₁₀₆	—	4	1	no	—
108	C₂×Dic₁₀₆	—	4	2	no	—
109	SD₁₆×C₅₃	—	4	4	no	—
110	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
111	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	#11719924	—	4	4	no	—
117	#11719923	—	4	4	no	—
118	C₄⋊Dic₅₃	—	2	2	no	—
119	C₄⋊Dic₅₃	—	2	2	no	—
120	C₄⋊Dic₅₃	—	2	2	no	—
121	#11719919	—	4	2	no	—
122	#11719919	—	4	2	no	—
123	#11719919	—	4	2	no	—
124	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
125	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×#88322	C₂×D₄	1	4	yes	—
127	C₂×#88322	C₂×D₄	1	2	yes	—
128	#11719913	—	4	4	no	—
129	C₈×D₅₃	—	2	8	no	—
130	C₈×D₅₃	SD₁₆	1	8	yes	—
131	C₂³×C₅₃	—	4	1	no	—
132	C₂²×D₅₃	—	4	1	no	—
133	C₂²×D₅₃	—	8	2	no	—
134	C₂²×D₅₃	—	8	4	no	—
135	C₂²×D₅₃	#127	1	1	yes	characteristic
136	Q₈×C₅₃	—	4	2	no	—
137	Q₈×C₅₃	—	4	4	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
144	C₂²⋊D₅₃	—	8	2	no	—
145	C₂²⋊D₅₃	—	8	4	no	—
146	C₂×Dic₅₃	—	4	1	no	—
147	C₂×Dic₅₃	—	4	2	no	—
148	C₂×Dic₅₃	#127	1	1	yes	characteristic
149	C₂×Dic₅₃	#127	1	2	yes	—
150	D₂₁₂	—	4	2	no	—
151	D₂₁₂	—	4	4	no	—
152	C₄×D₅₃	—	8	2	no	—
153	C₄×D₅₃	—	8	4	no	—
154	C₄×D₅₃	C₂×SD₁₆	1	4	yes	—
155	C₄×D₅₃	#135	1	8	yes	—
156	Dic₁₀₆	—	4	2	no	—
157	Dic₁₀₆	—	4	4	no	—
158	Dic₁₀₆	—	8	2	no	—
159	Dic₁₀₆	—	8	4	no	—
160	C₈×C₅₃	—	2	4	no	—
161	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
162	#88322	—	2	4	no	—
163	#88322	C₂×SD₁₆	1	4	yes	—
164	C₂×#1708	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	8	1	no	—
166	C₂²×C₅₃	—	8	2	no	—
167	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
168	C₂×D₅₃	—	8	1	no	—
169	C₂×D₅₃	—	8	2	no	—
170	C₂×D₅₃	—	16	4	no	—
171	C₂×D₅₃	#492	1	2	yes	—
172	C₂×D₅₃	#495	1	4	yes	—
173	C₄×C₅₃	—	8	1	no	—
174	C₄×C₅₃	—	8	2	no	—
175	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
176	C₄×C₅₃	C₂×#135	1	4	yes	—
177	Dic₅₃	—	8	1	no	—
178	Dic₅₃	—	8	2	no	—
179	Dic₅₃	#492	1	2	yes	—
180	Dic₅₃	#495	1	4	yes	—
181	C₂×#498	—	53	1	no	—
182	C₂×#492	—	53	1	no	—
183	C₂×#487	—	53	2	no	—
184	C₂×#486	—	53	2	no	—
185	#1708	—	53	1	no	—
186	#1708	—	53	1	no	—
187	#1708	—	53	2	no	—
188	#1708	—	53	2	no	—
189	#1708	—	53	1	no	—
190	#1708	D₅₃	1	1	yes	characteristic, direct product factor
191	C₂×C₈²	—	53	1	no	—
192	C₂×C₅₃	—	16	1	no	—
193	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
194	C₂×C₅₃	C₂×#495	1	2	yes	—
195	D₅₃	—	16	1	no	—
196	D₅₃	#1708	1	4	yes	direct product factor
197	C₂²×SD₁₆	—	106	2	no	—
198	C₂×C₄⋊Q₈	—	53	2	no	—
199	C₂×C₄⋊Q₈	—	53	1	no	—
200	C₂×#127	—	53	1	no	—
201	#498	—	53	2	no	—
202	#498	—	53	1	no	—
203	#498	—	106	2	no	—
204	#498	C₂×D₅₃	1	1	yes	characteristic, complemented
205	#492	—	53	1	no	—
206	#492	—	53	1	no	—
207	#492	—	53	1	no	—
208	#492	—	106	2	no	—
209	#492	C₂×D₅₃	1	1	yes	characteristic
210	#487	—	53	2	no	—
211	#487	—	53	2	no	—
212	#487	—	53	2	no	—
213	#487	—	106	4	no	—
214	#487	C₂×D₅₃	1	2	yes	complemented
215	#486	—	53	2	no	—
216	#486	—	53	2	no	—
217	#486	—	53	2	no	—
218	#486	—	106	4	no	—
219	#486	C₂×D₅₃	1	2	yes	—
220	C₂×#106	—	106	2	no	—
221	C₂×#103	—	106	4	no	—
222	C₂×#102	—	106	4	no	—
223	C₂×C₄×C₈	—	53	2	no	—
224	C₂×C₄×C₈	—	53	1	no	—
225	C₈²	—	53	1	no	—
226	C₈²	—	53	2	no	—
227	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
228	C₅₃	C₂×#1708	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	106	1	no	—
230	C₂²×Q₈	—	106	2	no	—
231	C₂²×D₄	—	106	1	no	—
232	C₂²×D₄	—	106	2	no	—
233	C₂×SD₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	106	2	no	—
235	C₂×SD₁₆	—	106	2	no	—
236	C₂×SD₁₆	—	106	2	no	—
237	C₂×SD₁₆	—	212	8	no	—
238	C₂²×C₈	—	53	2	no	—
239	C₂²×C₈	—	53	4	no	—
240	C₄⋊Q₈	—	53	2	no	—
241	C₄⋊Q₈	—	53	1	no	—
242	C₄⋊Q₈	—	106	2	no	—
243	C₄⋊Q₈	—	106	2	no	—
244	C₄⋊Q₈	—	106	2	no	—
245	C₄⋊Q₈	—	106	2	no	—
246	C₄⋊Q₈	—	106	1	no	—
247	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
248	C₄⋊Q₈	C₂²×D₅₃	1	1	yes	characteristic
249	#127	—	53	1	no	—
250	#127	—	106	2	no	—
251	#127	—	106	1	no	—
252	#127	C₂²×D₅₃	1	1	yes	characteristic
253	C₂×C₄⋊C₄	—	106	2	no	—
254	C₂×C₄⋊C₄	—	106	2	no	—
255	C₂×C₄⋊C₄	—	106	2	no	—
256	C₂×C₄²	—	53	1	no	—
257	#106	—	2	2	no	—
258	#106	—	106	4	no	—
259	#106	—	106	2	no	—
260	#103	—	2	4	no	—
261	#103	—	106	4	no	—
262	#103	—	106	4	no	—
263	#103	—	106	4	no	—
264	#102	—	2	4	no	—
265	#102	—	106	4	no	—
266	#102	—	106	4	no	—
267	#102	—	106	4	no	—
268	C₄×C₈	—	53	4	no	—
269	C₄×C₈	—	53	2	no	—
270	C₄×C₈	—	53	2	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	C₂²×D₅₃	1	2	yes	—
273	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
274	C₂⁴	—	212	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	106	2	no	—
278	C₂×Q₈	—	106	1	no	—
279	C₂×Q₈	—	212	2	no	—
280	C₂×Q₈	—	212	1	no	—
281	C₂×Q₈	—	212	4	no	—
282	C₂×Q₈	—	212	8	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	106	2	no	—
286	C₂×D₄	—	106	1	no	—
287	C₂×D₄	—	212	4	no	—
288	C₂×D₄	—	212	8	no	—
289	C₂×D₄	—	212	1	no	—
290	C₂×D₄	—	212	2	no	—
291	C₂²×C₄	—	53	1	no	—
292	C₂²×C₄	—	53	2	no	—
293	C₂²×C₄	—	212	2	no	—
294	C₂²×C₄	—	212	1	no	—
295	SD₁₆	—	4	4	no	—
296	SD₁₆	—	212	4	no	—
297	SD₁₆	—	212	4	no	—
298	SD₁₆	—	212	4	no	—
299	C₂×C₈	—	53	2	no	—
300	C₂×C₈	—	53	8	no	—
301	C₂×C₈	—	53	4	no	—
302	C₂×C₈	—	106	8	no	—
303	C₂×C₈	D₄×D₅₃	1	4	yes	—
304	C₂×C₈	D₄×D₅₃	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	106	2	no	—
309	C₄⋊C₄	—	106	2	no	—
310	C₄⋊C₄	—	106	2	no	—
311	C₄⋊C₄	—	212	2	no	—
312	C₄⋊C₄	—	212	2	no	—
313	C₄⋊C₄	—	212	2	no	—
314	C₄²	—	53	1	no	—
315	C₄²	—	53	2	no	—
316	C₄²	C₂³×D₅₃	1	1	yes	characteristic
317	C₂³	—	4	1	no	—
318	C₂³	—	53	1	no	—
319	C₂³	—	424	2	no	—
320	C₂³	—	424	4	no	—
321	C₂³	—	212	1	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	424	2	no	—
325	Q₈	—	424	4	no	—
326	Q₈	—	212	2	no	—
327	Q₈	—	212	4	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	4	no	—
330	D₄	—	424	2	no	—
331	D₄	—	424	4	no	—
332	D₄	—	212	2	no	—
333	D₄	—	212	4	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	53	1	no	—
337	C₂×C₄	—	53	2	no	—
338	C₂×C₄	—	53	4	no	—
339	C₂×C₄	—	53	8	no	—
340	C₂×C₄	—	424	2	no	—
341	C₂×C₄	—	424	4	no	—
342	C₂×C₄	—	212	1	no	—
343	C₂×C₄	—	212	2	no	—
344	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
346	C₈	—	2	4	no	—
347	C₈	—	53	4	no	—
348	C₈	—	106	4	no	—
349	C₈	SD₁₆×D₅₃	1	4	yes	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	53	2	no	—
353	C₂²	—	53	4	no	—
354	C₂²	—	848	4	no	—
355	C₂²	—	424	1	no	—
356	C₂²	—	424	2	no	—
357	C₂²	#127×D₅₃	1	1	yes	characteristic
358	C₄	—	8	1	no	—
359	C₄	—	8	2	no	—
360	C₄	—	53	2	no	—
361	C₄	—	53	4	no	—
362	C₄	—	424	1	no	—
363	C₄	—	424	2	no	—
364	C₄	C₂×SD₁₆×D₅₃	1	2	yes	—
365	C₄	#135×D₅₃	1	4	yes	—
366	C₂	—	16	1	no	—
367	C₂	—	53	4	no	—
368	C₂	—	848	1	no	—
369	C₂	#492×D₅₃	1	1	yes	characteristic
370	C₂	#495×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1708	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	#127×D₅₃	C₂²
hypercenter	#1708	D₅₃
hypocenter	C₅₃	C₂×#1708
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1708

Complements

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

N	H ≅ G/N	type
1. #498×D₅₃	C₂	semidirect
3. #487×D₅₃	C₂	semidirect
5. #20274318	C₂	semidirect
6. #20274317	C₂	semidirect
7. #20274316	C₂	semidirect
8. #20274315	C₂	semidirect
9. #20274314	C₂	semidirect
10. C₈²×D₅₃	C₂	semidirect
11. C₅₃×#1708	C₂	semidirect
16. #16811021	C₂²	semidirect
18. #16811019	C₂²	semidirect
23. #16810975	C₂²	semidirect
24. #16810974	C₂²	semidirect
25. #16810973	C₂²	semidirect
36. C₈×#88322	C₂²	semidirect
37. C₈×#88322	C₂²	semidirect
38. C₅₃×#498	C₂²	semidirect
40. C₅₃×#487	C₂²	semidirect
42. C₈²×C₅₃	C₂²	semidirect
190. #1708	D₅₃	direct (N × H)
196. D₅₃	#1708	direct (N × H)
204. #498	C₂×D₅₃	semidirect
214. #487	C₂×D₅₃	semidirect
227. C₈²	C₂×D₅₃	semidirect
228. C₅₃	C₂×#1708	semidirect

Maximal subgroups

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

subgroup	index
#498×D₅₃	2
#492×D₅₃	2
#487×D₅₃	2
#486×D₅₃	2
#20274318	2
#20274317	2
#20274316	2
#20274315	2
#20274314	2
C₈²×D₅₃	2
C₅₃×#1708	2
C₂×#1708	53

Subgroups — D53×#1708 #27371743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1708 #27371743