Subgroups — D₅₃×#2264 #27542228

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20277035	C₂	1	1	yes	maximal, characteristic, complemented
4	#20277034	C₂	1	2	yes	maximal, complemented
5	#20277033	C₂	1	1	yes	maximal, characteristic, complemented
6	#20277032	C₂	1	1	yes	maximal, characteristic, complemented
7	#20277031	C₂	1	2	yes	maximal, complemented
8	Q₆₄×D₅₃	C₂	1	2	yes	maximal, complemented
9	SD₆₄×D₅₃	C₂	1	2	yes	maximal, complemented
10	M₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2264	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₅₃	—	2	1	no	—
13	C₂×Q₁₆×D₅₃	—	2	1	no	—
14	#16811056	C₂²	1	1	yes	characteristic
15	#16811055	C₂²	1	1	yes	characteristic
16	#16811054	—	2	1	no	—
17	#16811053	—	2	1	no	—
18	#16811052	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13133163	—	2	1	no	—
21	Q₃₂×D₅₃	—	2	1	no	—
22	Q₃₂×D₅₃	—	2	1	no	—
23	Q₃₂×D₅₃	C₂²	1	2	yes	—
24	Q₃₂×D₅₃	C₂²	1	1	yes	characteristic
25	Q₃₂×D₅₃	C₂²	1	2	yes	complemented
26	SD₃₂×D₅₃	—	2	1	no	—
27	D₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
29	Dic₈₄₈	C₂²	1	2	yes	complemented
30	#16810463	C₂²	1	2	yes	complemented
31	#16810463	C₂²	1	2	yes	complemented
32	#16810462	C₂²	1	2	yes	complemented
33	#16810461	C₂²	1	2	yes	complemented
34	#16810460	C₂²	1	2	yes	complemented
35	#16810457	C₂²	1	1	yes	characteristic, complemented
36	#16810456	C₂²	1	2	yes	complemented
37	C₃₂×D₅₃	C₂²	1	2	yes	—
38	C₅₃×#508	C₂²	1	1	yes	characteristic
39	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×Q₆₄	C₂²	1	2	yes	complemented
41	C₅₃×SD₆₄	C₂²	1	2	yes	complemented
42	C₅₃×M₆₄	C₂²	1	1	yes	characteristic, complemented
43	#41×D₅₃	—	4	1	no	—
44	C₂×Q₈×D₅₃	—	4	1	no	—
45	#13133240	—	2	1	no	—
46	#13133239	—	2	1	no	—
47	#13133238	—	4	1	no	—
48	#13133237	—	4	1	no	—
49	#13133236	—	2	1	no	—
50	C₂×Dic₂₁₂	—	2	1	no	—
51	C₂×#11719925	—	4	1	no	—
52	Q₁₆×D₅₃	—	2	1	no	—
53	Q₁₆×D₅₃	—	2	2	no	—
54	Q₁₆×D₅₃	—	2	2	no	—
55	Q₁₆×D₅₃	—	4	1	no	—
56	Q₁₆×D₅₃	—	4	1	no	—
57	SD₁₆×D₅₃	—	4	1	no	—
58	D₈×D₅₃	—	2	1	no	—
59	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
60	Dic₄₂₄	—	2	1	no	—
61	Dic₄₂₄	C₂³	1	2	yes	—
62	Dic₄₂₄	C₂³	1	1	yes	characteristic
63	#13133163	—	2	2	no	—
64	#13133163	—	2	2	no	—
65	#13133163	—	2	1	no	—
66	#13133162	—	2	1	no	—
67	#13133161	—	2	1	no	—
68	#13133160	—	2	1	no	—
69	D₈₄₈	C₂³	1	1	yes	characteristic
70	#13133158	—	2	1	no	—
71	C₂×#11719908	D₄	1	1	yes	characteristic
72	C₁₆×D₅₃	D₄	1	1	yes	characteristic
73	C₁₆×D₅₃	D₄	1	1	yes	characteristic
74	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
75	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
76	#13133123	C₂³	1	2	yes	—
77	#135×C₅₃	—	2	1	no	—
78	C₂×Q₁₆×C₅₃	—	2	1	no	—
79	Q₃₂×C₅₃	—	2	1	no	—
80	Q₃₂×C₅₃	C₂³	1	2	yes	—
81	Q₃₂×C₅₃	C₂³	1	1	yes	characteristic
82	SD₃₂×C₅₃	—	2	1	no	—
83	D₁₆×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
85	C₃₂×C₅₃	C₂³	1	2	yes	—
86	#41×C₅₃	—	4	1	no	—
87	C₂×Q₈×C₅₃	—	4	1	no	—
88	#11719949	—	4	1	no	—
89	Q₈×D₅₃	—	4	1	no	—
90	Q₈×D₅₃	—	4	2	no	—
91	Q₈×D₅₃	—	4	2	no	—
92	Q₈×D₅₃	—	8	1	no	—
93	Q₈×D₅₃	—	8	1	no	—
94	#11719947	—	4	1	no	—
95	#11719947	—	8	1	no	—
96	D₄×D₅₃	—	4	1	no	—
97	D₄×D₅₃	—	8	1	no	—
98	#11719945	—	4	1	no	—
99	#11719945	—	8	1	no	—
100	C₂×C₄×D₅₃	—	8	1	no	—
101	C₂×C₄×D₅₃	—	8	1	no	—
102	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
103	C₂×Dic₁₀₆	—	4	1	no	—
104	C₂×Dic₁₀₆	—	8	1	no	—
105	Q₁₆×C₅₃	—	2	1	no	—
106	Q₁₆×C₅₃	—	2	2	no	—
107	Q₁₆×C₅₃	—	4	1	no	—
108	SD₁₆×C₅₃	—	4	1	no	—
109	D₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	#11719925	—	4	2	no	—
112	#11719925	—	4	2	no	—
113	#11719925	—	4	1	no	—
114	#11719924	—	4	1	no	—
115	#11719923	—	4	1	no	—
116	#11719922	—	4	1	no	—
117	C₂×#88322	D₈	1	1	yes	characteristic
118	Dic₂₁₂	—	2	1	no	—
119	Dic₂₁₂	—	2	2	no	—
120	Dic₂₁₂	—	4	1	no	—
121	D₄₂₄	—	2	1	no	—
122	#11719913	—	4	1	no	—
123	C₈×D₅₃	D₈	1	1	yes	characteristic
124	C₈×D₅₃	D₈	1	1	yes	characteristic
125	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
126	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
129	#11719908	C₂×D₄	1	1	yes	characteristic
130	#11719908	C₂×D₄	1	1	yes	characteristic
131	C₂²×D₅₃	—	8	1	no	—
132	C₂²×D₅₃	D₁₆	1	1	yes	characteristic
133	Q₈×C₅₃	—	4	2	no	—
134	Q₈×C₅₃	—	4	1	no	—
135	Q₈×C₅₃	—	8	1	no	—
136	D₄×C₅₃	—	4	1	no	—
137	D₄×C₅₃	—	8	1	no	—
138	C₂×C₄×C₅₃	—	8	1	no	—
139	C₂×C₄×C₅₃	—	8	1	no	—
140	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂²⋊D₅₃	—	8	1	no	—
142	C₂²⋊D₅₃	—	8	1	no	—
143	C₂²⋊D₅₃	—	8	1	no	—
144	C₂×Dic₅₃	—	8	1	no	—
145	C₂×Dic₅₃	—	8	1	no	—
146	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
147	D₂₁₂	—	4	1	no	—
148	D₂₁₂	—	8	1	no	—
149	C₄×D₅₃	—	8	1	no	—
150	C₄×D₅₃	—	8	2	no	—
151	C₄×D₅₃	—	8	2	no	—
152	C₄×D₅₃	—	8	1	no	—
153	C₄×D₅₃	—	8	1	no	—
154	C₄×D₅₃	D₁₆	1	1	yes	characteristic
155	C₄×D₅₃	D₁₆	1	1	yes	characteristic
156	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
157	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
158	Dic₁₀₆	—	4	2	no	—
159	Dic₁₀₆	—	4	1	no	—
160	Dic₁₀₆	—	8	1	no	—
161	Dic₁₀₆	—	8	2	no	—
162	Dic₁₀₆	—	8	2	no	—
163	Dic₁₀₆	—	8	1	no	—
164	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
165	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	#88322	C₂×D₈	1	1	yes	characteristic
167	#88322	C₂×D₈	1	1	yes	characteristic
168	C₂×#2264	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	8	1	no	—
170	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
171	C₂×D₅₃	—	2	2	no	—
172	C₂×D₅₃	—	8	1	no	—
173	C₂×D₅₃	—	16	2	no	—
174	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
175	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
176	C₄×C₅₃	—	8	2	no	—
177	C₄×C₅₃	—	8	1	no	—
178	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
179	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
180	Dic₅₃	—	8	2	no	—
181	Dic₅₃	—	8	1	no	—
182	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
183	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
184	C₂×#508	—	53	1	no	—
185	C₂²×Q₃₂	—	53	1	no	—
186	#2264	—	53	1	no	—
187	#2264	—	53	1	no	—
188	#2264	—	53	2	no	—
189	#2264	—	53	2	no	—
190	#2264	—	53	1	no	—
191	#2264	D₅₃	1	1	yes	characteristic, direct product factor
192	C₂×Q₆₄	—	53	2	no	—
193	C₂×SD₆₄	—	53	2	no	—
194	C₂×M₆₄	—	53	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	16	1	no	—
197	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
198	D₅₃	—	2	1	no	—
199	D₅₃	—	16	1	no	—
200	D₅₃	#2264	1	2	yes	direct product factor
201	C₂×#135	—	106	1	no	—
202	C₂²×Q₁₆	—	106	1	no	—
203	#508	—	53	1	no	—
204	#508	—	53	1	no	—
205	#508	—	53	1	no	—
206	#508	—	106	1	no	—
207	#508	—	106	1	no	—
208	#508	C₂×D₅₃	1	1	yes	characteristic
209	C₂×Q₃₂	—	53	2	no	—
210	C₂×Q₃₂	—	53	1	no	—
211	C₂×Q₃₂	—	53	2	no	—
212	C₂×Q₃₂	—	53	1	no	—
213	C₂×Q₃₂	—	106	1	no	—
214	C₂×Q₃₂	—	106	1	no	—
215	C₂×Q₃₂	—	106	1	no	—
216	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
217	C₂×SD₃₂	—	106	1	no	—
218	C₂×D₁₆	—	53	1	no	—
219	C₂²×C₁₆	—	53	1	no	—
220	Q₆₄	—	53	2	no	—
221	Q₆₄	—	53	2	no	—
222	Q₆₄	—	53	2	no	—
223	Q₆₄	C₂×D₅₃	1	2	yes	complemented
224	SD₆₄	—	53	2	no	—
225	SD₆₄	—	53	2	no	—
226	SD₆₄	—	53	2	no	—
227	SD₆₄	C₂×D₅₃	1	2	yes	complemented
228	M₆₄	—	53	1	no	—
229	M₆₄	—	53	2	no	—
230	M₆₄	C₂×D₅₃	1	1	yes	characteristic, complemented
231	C₂×C₃₂	—	53	2	no	—
232	C₅₃	C₂×#2264	1	1	yes	characteristic, complemented, Sylow
233	C₂×#41	—	212	1	no	—
234	C₂²×Q₈	—	212	1	no	—
235	#135	—	2	1	no	—
236	#135	—	106	1	no	—
237	#135	—	106	1	no	—
238	#135	—	106	1	no	—
239	#135	—	212	1	no	—
240	#135	—	212	1	no	—
241	C₂×Q₁₆	—	2	1	no	—
242	C₂×Q₁₆	—	106	2	no	—
243	C₂×Q₁₆	—	106	2	no	—
244	C₂×Q₁₆	—	106	1	no	—
245	C₂×Q₁₆	—	106	1	no	—
246	C₂×Q₁₆	—	212	1	no	—
247	C₂×Q₁₆	—	212	1	no	—
248	C₂×Q₁₆	—	212	1	no	—
249	C₂×SD₁₆	—	212	1	no	—
250	C₂×D₈	—	106	1	no	—
251	C₂²×C₈	—	53	1	no	—
252	Q₃₂	—	2	1	no	—
253	Q₃₂	—	53	2	no	—
254	Q₃₂	—	53	1	no	—
255	Q₃₂	—	106	2	no	—
256	Q₃₂	—	106	2	no	—
257	Q₃₂	—	106	1	no	—
258	Q₃₂	—	106	1	no	—
259	Q₃₂	C₂²×D₅₃	1	2	yes	—
260	Q₃₂	C₂²×D₅₃	1	1	yes	characteristic
261	SD₃₂	—	2	1	no	—
262	SD₃₂	—	106	1	no	—
263	SD₃₂	—	106	1	no	—
264	SD₃₂	—	106	1	no	—
265	D₁₆	—	53	1	no	—
266	D₁₆	—	106	1	no	—
267	D₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×C₁₆	—	53	1	no	—
269	C₂×C₁₆	—	53	1	no	—
270	C₂×C₁₆	—	53	1	no	—
271	C₂×C₁₆	—	53	1	no	—
272	C₂×C₁₆	—	53	1	no	—
273	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
274	C₃₂	—	53	2	no	—
275	C₃₂	C₂²×D₅₃	1	2	yes	—
276	#41	—	4	1	no	—
277	#41	—	424	1	no	—
278	#41	—	424	1	no	—
279	#41	—	212	1	no	—
280	#41	—	212	1	no	—
281	#41	—	212	1	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×Q₈	—	424	1	no	—
284	C₂×Q₈	—	424	1	no	—
285	C₂×Q₈	—	424	1	no	—
286	C₂×Q₈	—	212	1	no	—
287	C₂×Q₈	—	212	2	no	—
288	C₂×Q₈	—	212	2	no	—
289	C₂×Q₈	—	212	1	no	—
290	C₂×D₄	—	424	1	no	—
291	C₂×D₄	—	212	1	no	—
292	C₂²×C₄	—	53	1	no	—
293	C₂²×C₄	—	424	1	no	—
294	C₂²×C₄	—	424	1	no	—
295	Q₁₆	—	2	1	no	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	4	1	no	—
298	Q₁₆	—	106	1	no	—
299	Q₁₆	—	106	2	no	—
300	Q₁₆	—	212	2	no	—
301	Q₁₆	—	212	2	no	—
302	Q₁₆	—	212	1	no	—
303	Q₁₆	—	212	1	no	—
304	SD₁₆	—	4	1	no	—
305	SD₁₆	—	212	1	no	—
306	SD₁₆	—	212	1	no	—
307	SD₁₆	—	212	1	no	—
308	D₈	—	2	1	no	—
309	D₈	—	106	1	no	—
310	D₈	—	212	1	no	—
311	C₂×C₈	—	53	1	no	—
312	C₂×C₈	—	53	1	no	—
313	C₂×C₈	—	53	1	no	—
314	C₂×C₈	—	53	1	no	—
315	C₂×C₈	—	53	1	no	—
316	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
317	C₁₆	—	53	1	no	—
318	C₁₆	—	53	1	no	—
319	C₁₆	D₄×D₅₃	1	1	yes	characteristic
320	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
321	C₂³	—	53	1	no	—
322	C₂³	—	424	1	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	8	1	no	—
326	Q₈	—	424	2	no	—
327	Q₈	—	424	2	no	—
328	Q₈	—	424	1	no	—
329	Q₈	—	424	1	no	—
330	Q₈	—	212	1	no	—
331	Q₈	—	212	2	no	—
332	D₄	—	4	1	no	—
333	D₄	—	8	1	no	—
334	D₄	—	424	1	no	—
335	D₄	—	424	1	no	—
336	D₄	—	424	1	no	—
337	D₄	—	424	1	no	—
338	D₄	—	212	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	53	1	no	—
344	C₂×C₄	—	53	1	no	—
345	C₂×C₄	—	53	1	no	—
346	C₂×C₄	—	424	1	no	—
347	C₂×C₄	—	424	1	no	—
348	C₂×C₄	—	424	2	no	—
349	C₂×C₄	—	424	2	no	—
350	C₂×C₄	—	424	1	no	—
351	C₂×C₄	—	424	1	no	—
352	C₂×C₄	—	424	1	no	—
353	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
354	C₈	—	53	1	no	—
355	C₈	—	53	1	no	—
356	C₈	D₈×D₅₃	1	1	yes	characteristic
357	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
358	C₂²	—	8	1	no	—
359	C₂²	—	53	1	no	—
360	C₂²	—	53	1	no	—
361	C₂²	—	848	2	no	—
362	C₂²	—	106	2	no	—
363	C₂²	—	424	1	no	—
364	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	53	1	no	—
368	C₄	—	53	1	no	—
369	C₄	—	424	1	no	—
370	C₄	—	424	2	no	—
371	C₄	D₁₆×D₅₃	1	1	yes	characteristic
372	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	16	1	no	—
375	C₂	—	53	2	no	—
376	C₂	—	848	1	no	—
377	C₂	—	106	1	no	—
378	C₂	C₂×D₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₆×D₅₃
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₅₃×#2264	C₂
socle	C₂×C₅₃	C₂²×D₁₆
prime radical	#508×D₅₃	C₂
hypercenter	#2264	D₅₃
hypocenter	C₅₃	C₂×#2264
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. C₂×Q₃₂×D₅₃	C₂	semidirect
3. #20277035	C₂	semidirect
4. #20277034	C₂	semidirect
5. #20277033	C₂	semidirect
6. #20277032	C₂	semidirect
7. #20277031	C₂	semidirect
8. Q₆₄×D₅₃	C₂	semidirect
9. SD₆₄×D₅₃	C₂	semidirect
10. M₆₄×D₅₃	C₂	semidirect
11. C₅₃×#2264	C₂	semidirect
19. C₂×Dic₄₂₄	C₂²	semidirect
25. Q₃₂×D₅₃	C₂²	semidirect
29. Dic₈₄₈	C₂²	semidirect
30. #16810463	C₂²	semidirect
31. #16810463	C₂²	semidirect
32. #16810462	C₂²	semidirect
33. #16810461	C₂²	semidirect
34. #16810460	C₂²	semidirect
35. #16810457	C₂²	semidirect
36. #16810456	C₂²	semidirect
39. C₂×Q₃₂×C₅₃	C₂²	semidirect
40. C₅₃×Q₆₄	C₂²	semidirect
41. C₅₃×SD₆₄	C₂²	semidirect
42. C₅₃×M₆₄	C₂²	semidirect
191. #2264	D₅₃	direct (N × H)
200. D₅₃	#2264	direct (N × H)
216. C₂×Q₃₂	C₂×D₅₃	semidirect
223. Q₆₄	C₂×D₅₃	semidirect
227. SD₆₄	C₂×D₅₃	semidirect
230. M₆₄	C₂×D₅₃	semidirect
232. C₅₃	C₂×#2264	semidirect

Maximal subgroups

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

subgroup	index
#508×D₅₃	2
C₂×Q₃₂×D₅₃	2
#20277035	2
#20277034	2
#20277033	2
#20277032	2
#20277031	2
Q₆₄×D₅₃	2
SD₆₄×D₅₃	2
M₆₄×D₅₃	2
C₅₃×#2264	2
C₂×#2264	53

Subgroups — D53×#2264 #27542228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2264 #27542228