Subgroups — D₅₃×#2226 #27538697

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 416 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	C₄×D₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276863	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276862	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276861	C₂	1	2	yes	maximal, complemented
6	#20276860	C₂	1	2	yes	maximal, complemented
7	#20276859	C₂	1	1	yes	maximal, characteristic, complemented
8	#367×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₅₃	—	2	1	no	—
14	C₄×D₄×D₅₃	—	2	1	no	—
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	#16811028	C₂²	1	1	yes	characteristic, complemented
17	#16811027	—	2	1	no	—
18	#16811027	C₂²	1	2	yes	complemented
19	#16811026	—	2	1	no	—
20	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
21	D₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#16810705	C₂²	1	1	yes	characteristic
23	C₄×#11719922	—	2	1	no	—
24	#16810703	—	2	1	no	—
25	#107×D₅₃	—	2	1	no	—
26	#107×D₅₃	C₂²	1	1	yes	characteristic
27	#107×D₅₃	C₂²	1	2	yes	—
28	#102×D₅₃	—	2	1	no	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	#16810451	C₂²	1	2	yes	complemented
31	#16810450	C₂²	1	2	yes	complemented
32	#16810450	C₂²	1	2	yes	complemented
33	C₄⋊#11719908	C₂²	1	1	yes	characteristic, complemented
34	#16810441	C₂²	1	2	yes	complemented
35	#16810425	C₂²	1	2	yes	—
36	#16810424	C₂²	1	2	yes	—
37	#16810423	C₂²	1	2	yes	—
38	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#367	C₂²	1	2	yes	complemented
41	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#357	C₂²	1	2	yes	—
43	C₂×Q₈×D₅₃	—	4	1	no	—
44	C₂×D₄×D₅₃	—	2	1	no	—
45	C₂²×C₄×D₅₃	—	4	1	no	—
46	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
47	C₂×#11719922	—	2	1	no	—
48	D₈×D₅₃	—	2	2	no	—
49	D₈×D₅₃	Q₈	1	4	yes	complemented
50	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	#13133217	—	2	1	no	—
53	#13133216	—	2	2	no	—
54	#13133216	—	4	1	no	—
55	#13133215	—	4	1	no	—
56	C₄×D₂₁₂	—	2	1	no	—
57	D₄×Dic₅₃	—	2	1	no	—
58	#13133179	—	2	1	no	—
59	C₄×C₂²⋊D₅₃	—	4	1	no	—
60	#13133177	—	4	1	no	—
61	C₄⋊C₄×D₅₃	—	2	1	no	—
62	C₄⋊C₄×D₅₃	—	2	2	no	—
63	C₄⋊C₄×D₅₃	—	4	1	no	—
64	C₂²⋊C₄×D₅₃	—	4	1	no	—
65	C₄²×D₅₃	D₄	1	1	yes	characteristic
66	C₂×#11719908	C₂³	1	2	yes	—
67	C₁₆×D₅₃	—	2	4	no	—
68	#13133151	—	2	1	no	—
69	#13133151	C₂³	1	2	yes	—
70	#13133151	C₂³	1	1	yes	characteristic
71	#13133150	—	2	2	no	—
72	#13133150	—	2	2	no	—
73	#13133150	—	2	1	no	—
74	#13133139	—	2	1	no	—
75	#13133138	—	2	1	no	—
76	#13133137	—	2	1	no	—
77	C₄×#88322	D₄	1	1	yes	characteristic
78	C₈×Dic₅₃	D₄	1	1	yes	characteristic
79	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
84	#107×C₅₃	—	2	1	no	—
85	#107×C₅₃	C₂³	1	2	yes	—
86	#107×C₅₃	C₂³	1	1	yes	characteristic
87	#102×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂³×D₅₃	—	4	1	no	—
90	C₂×Q₈×C₅₃	—	4	1	no	—
91	C₂×D₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂×C₂²⋊D₅₃	—	4	1	no	—
94	C₂²×Dic₅₃	—	4	1	no	—
95	Q₈×D₅₃	—	8	4	no	—
96	D₄×D₅₃	—	2	4	no	—
97	D₄×D₅₃	—	4	2	no	—
98	C₂×D₂₁₂	—	2	1	no	—
99	C₂×C₄×D₅₃	—	4	1	no	—
100	C₂×C₄×D₅₃	—	4	2	no	—
101	C₂×C₄×D₅₃	—	4	1	no	—
102	C₂×C₄×D₅₃	—	4	1	no	—
103	C₂×C₄×D₅₃	—	8	4	no	—
104	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
105	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₁₀₆	—	4	1	no	—
108	C₂×Dic₁₀₆	—	4	2	no	—
109	D₈×C₅₃	—	2	1	no	—
110	D₈×C₅₃	C₂×Q₈	1	2	yes	complemented
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	4	1	no	—
116	C₂²⋊C₄×C₅₃	—	4	1	no	—
117	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊Dic₅₃	—	4	1	no	—
119	#11719922	—	2	4	no	—
120	#11719921	—	4	1	no	—
121	#11719921	—	4	1	no	—
122	C₄⋊Dic₅₃	—	2	1	no	—
123	C₄⋊Dic₅₃	—	2	2	no	—
124	C₄⋊Dic₅₃	—	4	1	no	—
125	#11719919	—	4	1	no	—
126	#11719919	—	4	2	no	—
127	#11719919	—	4	2	no	—
128	C₄×Dic₅₃	D₈	1	1	yes	characteristic
129	C₄×Dic₅₃	D₈	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×#88322	C₂×D₄	1	1	yes	characteristic
132	C₂×#88322	C₂×D₄	1	1	yes	characteristic
133	D₄₂₄	—	2	1	no	—
134	D₄₂₄	C₂×Q₈	1	2	yes	complemented
135	C₈×D₅₃	—	2	2	no	—
136	C₈×D₅₃	C₂×Q₈	1	2	yes	—
137	C₈×D₅₃	#41	1	2	yes	—
138	C₁₆×C₅₃	—	2	2	no	—
139	#11719908	—	2	2	no	—
140	C₂³×C₅₃	—	4	1	no	—
141	C₂²×D₅₃	—	4	1	no	—
142	C₂²×D₅₃	—	4	4	no	—
143	C₂²×D₅₃	—	8	2	no	—
144	C₂²×D₅₃	—	8	2	no	—
145	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
146	Q₈×C₅₃	—	8	2	no	—
147	D₄×C₅₃	—	2	2	no	—
148	D₄×C₅₃	—	4	1	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	8	2	no	—
152	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂²⋊D₅₃	—	4	4	no	—
156	C₂×Dic₅₃	—	4	2	no	—
157	C₂×Dic₅₃	—	4	1	no	—
158	C₂×Dic₅₃	—	8	2	no	—
159	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
160	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
161	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
162	D₂₁₂	—	2	2	no	—
163	D₂₁₂	—	4	1	no	—
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	8	2	no	—
166	C₄×D₅₃	—	8	4	no	—
167	C₄×D₅₃	—	8	2	no	—
168	C₄×D₅₃	—	8	2	no	—
169	C₄×D₅₃	SD₃₂	1	4	yes	—
170	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
171	C₄×D₅₃	#137	1	2	yes	—
172	Dic₁₀₆	—	8	4	no	—
173	Dic₁₀₆	—	8	2	no	—
174	C₈×C₅₃	—	2	1	no	—
175	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
176	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
177	#88322	—	2	1	no	—
178	#88322	C₂²⋊Q₈	1	2	yes	—
179	C₂×#2226	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	8	1	no	—
182	C₂²×C₅₃	—	8	1	no	—
183	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
184	C₂×D₅₃	—	4	2	no	—
185	C₂×D₅₃	—	8	1	no	—
186	C₂×D₅₃	—	8	1	no	—
187	C₂×D₅₃	—	8	8	no	—
188	C₂×D₅₃	#474	1	2	yes	—
189	C₂×D₅₃	C₂×SD₃₂	1	2	yes	—
190	C₂×D₅₃	#509	1	2	yes	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	8	1	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
195	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
197	Dic₅₃	—	2	1	no	—
198	Dic₅₃	—	8	1	no	—
199	Dic₅₃	—	8	2	no	—
200	Dic₅₃	#474	1	2	yes	—
201	Dic₅₃	C₂×SD₃₂	1	2	yes	—
202	C₂×#500	—	53	1	no	—
203	C₂×C₄×D₈	—	53	1	no	—
204	#2226	—	53	1	no	—
205	#2226	—	53	1	no	—
206	#2226	—	53	2	no	—
207	#2226	—	53	1	no	—
208	#2226	—	53	2	no	—
209	#2226	D₅₃	1	1	yes	characteristic, direct product factor
210	C₂×#367	—	53	2	no	—
211	C₂×C₄⋊C₁₆	—	53	1	no	—
212	C₂×#357	—	53	2	no	—
213	C₂×C₅₃	—	8	2	no	—
214	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
215	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
216	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
217	D₅₃	—	8	2	no	—
218	D₅₃	#2226	1	4	yes	direct product factor
219	C₂²×D₈	—	53	1	no	—
220	C₂×C₄⋊Q₈	—	106	1	no	—
221	C₂×C₄×D₄	—	106	1	no	—
222	C₂²×C₁₆	—	53	2	no	—
223	#500	—	53	2	no	—
224	#500	—	53	1	no	—
225	#500	—	106	1	no	—
226	#500	—	106	1	no	—
227	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
228	C₄×D₈	—	53	1	no	—
229	C₄×D₈	—	53	1	no	—
230	C₄×D₈	—	53	1	no	—
231	C₄×D₈	—	106	1	no	—
232	C₄×D₈	—	106	1	no	—
233	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
234	C₂×#107	—	53	2	no	—
235	C₂×#107	—	53	1	no	—
236	C₂×#107	—	106	1	no	—
237	C₂×#102	—	106	1	no	—
238	C₂×C₄×C₈	—	53	1	no	—
239	#367	—	53	2	no	—
240	#367	—	53	2	no	—
241	#367	—	53	2	no	—
242	#367	C₂×D₅₃	1	2	yes	complemented
243	C₄⋊C₁₆	—	53	2	no	—
244	C₄⋊C₁₆	—	53	1	no	—
245	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
246	#357	—	53	2	no	—
247	#357	—	53	2	no	—
248	#357	—	53	2	no	—
249	#357	C₂×D₅₃	1	2	yes	—
250	C₅₃	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
251	C₂²×Q₈	—	212	1	no	—
252	C₂²×D₄	—	106	1	no	—
253	C₂³×C₄	—	212	1	no	—
254	C₂×D₈	—	53	4	no	—
255	C₂×D₈	—	53	1	no	—
256	C₂×D₈	—	106	2	no	—
257	C₂×D₈	—	106	1	no	—
258	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	53	1	no	—
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	—	106	2	no	—
263	C₄⋊Q₈	—	106	1	no	—
264	C₄⋊Q₈	—	212	1	no	—
265	C₄⋊Q₈	—	212	1	no	—
266	C₄×D₄	—	2	1	no	—
267	C₄×D₄	—	106	1	no	—
268	C₄×D₄	—	106	1	no	—
269	C₄×D₄	—	106	1	no	—
270	C₄×D₄	—	212	1	no	—
271	C₄×D₄	—	212	1	no	—
272	C₂×C₄⋊C₄	—	106	1	no	—
273	C₂×C₄⋊C₄	—	106	2	no	—
274	C₂×C₄⋊C₄	—	212	1	no	—
275	C₂×C₂²⋊C₄	—	212	1	no	—
276	C₂×C₄²	—	53	1	no	—
277	C₂×C₁₆	—	53	2	no	—
278	C₂×C₁₆	—	106	4	no	—
279	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
280	#107	—	2	1	no	—
281	#107	—	53	2	no	—
282	#107	—	53	1	no	—
283	#107	—	106	1	no	—
284	#107	—	106	2	no	—
285	#107	—	106	2	no	—
286	#107	—	106	1	no	—
287	#107	C₂²×D₅₃	1	2	yes	—
288	#107	C₂²×D₅₃	1	1	yes	characteristic
289	#102	—	2	1	no	—
290	#102	—	106	1	no	—
291	#102	—	106	1	no	—
292	#102	—	106	1	no	—
293	C₄×C₈	—	53	1	no	—
294	C₄×C₈	—	53	1	no	—
295	C₄×C₈	—	53	1	no	—
296	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₂⁴	—	212	1	no	—
298	C₂×Q₈	—	4	1	no	—
299	C₂×Q₈	—	424	4	no	—
300	C₂×Q₈	—	212	2	no	—
301	C₂×Q₈	—	212	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	106	4	no	—
304	C₂×D₄	—	106	1	no	—
305	C₂×D₄	—	212	2	no	—
306	C₂×D₄	—	212	1	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	424	4	no	—
312	C₂²×C₄	—	212	1	no	—
313	C₂²×C₄	—	212	1	no	—
314	C₂²×C₄	—	212	2	no	—
315	C₂²×C₄	—	212	1	no	—
316	C₂²×C₄	—	212	1	no	—
317	D₈	—	2	1	no	—
318	D₈	—	53	2	no	—
319	D₈	—	106	4	no	—
320	D₈	—	106	1	no	—
321	D₈	Q₈×D₅₃	1	2	yes	complemented
322	C₂×C₈	—	53	1	no	—
323	C₂×C₈	—	53	2	no	—
324	C₂×C₈	—	53	2	no	—
325	C₂×C₈	—	53	1	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	106	1	no	—
333	C₄⋊C₄	—	106	2	no	—
334	C₄⋊C₄	—	212	1	no	—
335	C₄⋊C₄	—	212	2	no	—
336	C₄⋊C₄	—	212	2	no	—
337	C₄⋊C₄	—	212	1	no	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₂²⋊C₄	—	212	1	no	—
340	C₂²⋊C₄	—	212	1	no	—
341	C₂²⋊C₄	—	212	1	no	—
342	C₄²	—	53	1	no	—
343	C₄²	—	53	1	no	—
344	C₄²	—	53	1	no	—
345	C₄²	D₄×D₅₃	1	1	yes	characteristic
346	C₁₆	—	2	2	no	—
347	C₁₆	—	106	2	no	—
348	C₂³	—	4	1	no	—
349	C₂³	—	53	1	no	—
350	C₂³	—	424	2	no	—
351	C₂³	—	424	2	no	—
352	C₂³	—	212	4	no	—
353	C₂³	—	212	1	no	—
354	Q₈	—	8	2	no	—
355	Q₈	—	424	2	no	—
356	Q₈	—	424	4	no	—
357	D₄	—	2	2	no	—
358	D₄	—	4	1	no	—
359	D₄	—	106	2	no	—
360	D₄	—	212	4	no	—
361	D₄	—	212	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	8	2	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	1	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	1	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	424	2	no	—
373	C₂×C₄	—	424	2	no	—
374	C₂×C₄	—	424	4	no	—
375	C₂×C₄	—	424	2	no	—
376	C₂×C₄	—	424	2	no	—
377	C₂×C₄	—	212	1	no	—
378	C₂×C₄	—	212	2	no	—
379	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
380	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	53	2	no	—
384	C₈	—	106	1	no	—
385	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
386	C₈	#41×D₅₃	1	1	yes	characteristic
387	C₂²	—	4	2	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	53	2	no	—
391	C₂²	—	53	2	no	—
392	C₂²	—	53	2	no	—
393	C₂²	—	424	8	no	—
394	C₂²	—	424	1	no	—
395	C₂²	—	424	1	no	—
396	C₂²	—	212	2	no	—
397	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	8	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	53	2	no	—
402	C₄	—	53	2	no	—
403	C₄	—	106	1	no	—
404	C₄	—	424	1	no	—
405	C₄	—	424	2	no	—
406	C₄	SD₃₂×D₅₃	1	2	yes	—
407	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
408	C₄	#137×D₅₃	1	1	yes	characteristic
409	C₂	—	8	2	no	—
410	C₂	—	53	4	no	—
411	C₂	—	424	2	no	—
412	C₂	#474×D₅₃	1	1	yes	characteristic
413	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic
414	C₂	#509×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×#2226	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₈×D₅₃	C₂²
hypercenter	#2226	D₅₃
hypocenter	C₅₃	C₂×#2226
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. #500×D₅₃	C₂	semidirect
3. #20276863	C₂	semidirect
4. #20276862	C₂	semidirect
5. #20276861	C₂	semidirect
6. #20276860	C₂	semidirect
7. #20276859	C₂	semidirect
8. #367×D₅₃	C₂	semidirect
9. C₄⋊C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2226	C₂	semidirect
16. #16811028	C₂²	semidirect
18. #16811027	C₂²	semidirect
30. #16810451	C₂²	semidirect
31. #16810450	C₂²	semidirect
32. #16810450	C₂²	semidirect
33. C₄⋊#11719908	C₂²	semidirect
34. #16810441	C₂²	semidirect
38. C₅₃×#500	C₂²	semidirect
40. C₅₃×#367	C₂²	semidirect
41. C₅₃×C₄⋊C₁₆	C₂²	semidirect
49. D₈×D₅₃	Q₈	semidirect
110. D₈×C₅₃	C₂×Q₈	semidirect
134. D₄₂₄	C₂×Q₈	semidirect
209. #2226	D₅₃	direct (N × H)
218. D₅₃	#2226	direct (N × H)
227. #500	C₂×D₅₃	semidirect
242. #367	C₂×D₅₃	semidirect
245. C₄⋊C₁₆	C₂×D₅₃	semidirect
250. C₅₃	C₂×#2226	semidirect
321. D₈	Q₈×D₅₃	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
C₄×D₈×D₅₃	2
#20276863	2
#20276862	2
#20276861	2
#20276860	2
#20276859	2
#367×D₅₃	2
C₄⋊C₁₆×D₅₃	2
#357×D₅₃	2
C₅₃×#2226	2
C₂×#2226	53

Subgroups — D53×#2226 #27538697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2226 #27538697