Subgroups — D₅₃×#2209 #27538338

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₅₃	C₂	1	2	yes	maximal
2	#494×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20276766	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276765	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276764	C₂	1	2	yes	maximal, complemented
7	#20276763	C₂	1	2	yes	maximal, complemented
8	#20276762	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
10	#358×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	—	2	1	no	—
13	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂×Q₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×D₅₃	—	2	1	no	—
16	C₄×Q₈×D₅₃	—	2	1	no	—
17	C₂×Dic₄₂₄	C₂²	1	2	yes	—
18	C₂×#13133163	C₂²	1	2	yes	—
19	C₂×#13133163	C₂²	1	2	yes	—
20	Q₃₂×D₅₃	—	2	8	no	—
21	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
22	#16811003	C₂²	1	1	yes	characteristic
23	#16811002	—	2	1	no	—
24	#16811001	—	2	1	no	—
25	#16811001	C₂²	1	2	yes	—
26	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
27	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
28	#16810717	C₂²	1	1	yes	characteristic
29	C₄×#11719925	—	2	1	no	—
30	#16810715	—	2	1	no	—
31	#107×D₅₃	C₂²	1	1	yes	characteristic
32	#103×D₅₃	—	2	1	no	—
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#11719908	C₂²	1	1	yes	characteristic
35	#16810441	C₂²	1	2	yes	—
36	#16810428	C₂²	1	2	yes	—
37	#16810427	C₂²	1	2	yes	—
38	#16810426	C₂²	1	2	yes	—
39	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
40	C₅₃×#494	C₂²	1	1	yes	characteristic
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
43	C₅₃×#358	C₂²	1	2	yes	—
44	C₂×Q₈×D₅₃	—	2	1	no	—
45	C₂×Q₈×D₅₃	—	2	2	no	—
46	C₂×Dic₂₁₂	—	2	1	no	—
47	C₂×Dic₂₁₂	C₂³	1	2	yes	—
48	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
49	C₂×#11719925	—	2	1	no	—
50	C₂×#11719925	—	2	2	no	—
51	C₂×#11719925	—	2	2	no	—
52	Q₁₆×D₅₃	—	2	2	no	—
53	Q₁₆×D₅₃	—	2	4	no	—
54	Q₁₆×D₅₃	—	2	4	no	—
55	Q₁₆×D₅₃	—	4	4	no	—
56	Q₁₆×D₅₃	D₄	1	4	yes	—
57	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
59	#13133217	—	2	1	no	—
60	#13133216	—	4	1	no	—
61	#13133216	—	4	1	no	—
62	#13133215	—	2	2	no	—
63	Q₈×Dic₅₃	—	2	1	no	—
64	C₄×Dic₁₀₆	—	2	1	no	—
65	C₄×Dic₁₀₆	—	4	1	no	—
66	#13133182	—	2	1	no	—
67	#13133182	—	4	1	no	—
68	C₄⋊C₄×D₅₃	—	2	1	no	—
69	C₄⋊C₄×D₅₃	—	4	1	no	—
70	C₄⋊C₄×D₅₃	—	4	1	no	—
71	C₄⋊C₄×D₅₃	—	4	1	no	—
72	C₄²×D₅₃	—	4	1	no	—
73	C₄²×D₅₃	D₄	1	1	yes	characteristic
74	Dic₄₂₄	—	2	4	no	—
75	#13133163	—	2	4	no	—
76	#13133163	—	2	4	no	—
77	C₂×#11719908	C₂³	1	2	yes	—
78	C₁₆×D₅₃	—	2	4	no	—
79	#13133151	C₂³	1	1	yes	characteristic
80	#13133150	—	2	1	no	—
81	#13133142	—	2	1	no	—
82	#13133141	—	2	1	no	—
83	#13133140	—	2	1	no	—
84	C₄×#88322	D₄	1	1	yes	characteristic
85	C₈×Dic₅₃	D₄	1	1	yes	characteristic
86	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₅₃	—	2	1	no	—
88	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
89	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	—	2	1	no	—
92	Q₃₂×C₅₃	—	2	4	no	—
93	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
94	#107×C₅₃	C₂³	1	1	yes	characteristic
95	#103×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₅₃	—	2	1	no	—
98	C₂×Q₈×C₅₃	—	2	2	no	—
99	Q₈×D₅₃	—	2	4	no	—
100	Q₈×D₅₃	—	4	2	no	—
101	Q₈×D₅₃	—	4	4	no	—
102	Q₈×D₅₃	—	4	4	no	—
103	C₂×C₄×D₅₃	—	4	1	no	—
104	C₂×C₄×D₅₃	—	4	1	no	—
105	C₂×C₄×D₅₃	—	4	2	no	—
106	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
107	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₁₀₆	—	2	1	no	—
110	C₂×Dic₁₀₆	—	2	2	no	—
111	C₂×Dic₁₀₆	—	4	1	no	—
112	C₂×Dic₁₀₆	—	4	2	no	—
113	Q₁₆×C₅₃	—	2	2	no	—
114	Q₁₆×C₅₃	—	2	1	no	—
115	Q₁₆×C₅₃	—	2	2	no	—
116	Q₁₆×C₅₃	—	4	2	no	—
117	Q₁₆×C₅₃	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	4	1	no	—
122	C₄⋊C₄×C₅₃	—	4	1	no	—
123	C₄⋊C₄×C₅₃	—	4	1	no	—
124	C₄²×C₅₃	—	4	1	no	—
125	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
126	#11719925	—	2	4	no	—
127	#11719925	—	4	4	no	—
128	#11719925	—	4	4	no	—
129	C₄⋊Dic₅₃	—	2	1	no	—
130	C₄⋊Dic₅₃	—	4	1	no	—
131	C₄⋊Dic₅₃	—	4	1	no	—
132	C₄⋊Dic₅₃	—	4	1	no	—
133	#11719919	—	4	1	no	—
134	#11719919	—	4	1	no	—
135	#11719919	—	4	1	no	—
136	#11719919	—	4	2	no	—
137	#11719919	—	4	2	no	—
138	C₄×Dic₅₃	—	4	1	no	—
139	C₄×Dic₅₃	—	4	1	no	—
140	C₄×Dic₅₃	—	4	1	no	—
141	C₄×Dic₅₃	D₈	1	1	yes	characteristic
142	C₄×Dic₅₃	D₈	1	1	yes	characteristic
143	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×#88322	C₂×D₄	1	1	yes	characteristic
145	C₂×#88322	C₂×D₄	1	1	yes	characteristic
146	Dic₂₁₂	—	2	2	no	—
147	Dic₂₁₂	—	2	1	no	—
148	Dic₂₁₂	—	2	2	no	—
149	Dic₂₁₂	—	4	2	no	—
150	Dic₂₁₂	C₂×D₄	1	2	yes	—
151	C₈×D₅₃	—	2	2	no	—
152	C₈×D₅₃	C₂×D₄	1	2	yes	—
153	C₈×D₅₃	#41	1	2	yes	—
154	C₁₆×C₅₃	—	2	2	no	—
155	#11719908	—	2	2	no	—
156	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
157	Q₈×C₅₃	—	2	2	no	—
158	Q₈×C₅₃	—	4	1	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
164	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×Dic₅₃	—	4	1	no	—
168	C₂×Dic₅₃	—	4	2	no	—
169	C₂×Dic₅₃	—	4	1	no	—
170	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
171	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
172	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
173	C₄×D₅₃	—	2	2	no	—
174	C₄×D₅₃	—	4	4	no	—
175	C₄×D₅₃	—	8	4	no	—
176	C₄×D₅₃	—	8	2	no	—
177	C₄×D₅₃	Q₃₂	1	4	yes	complemented
178	C₄×D₅₃	#121	1	2	yes	—
179	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
180	Dic₁₀₆	—	2	2	no	—
181	Dic₁₀₆	—	4	1	no	—
182	Dic₁₀₆	—	4	4	no	—
183	Dic₁₀₆	—	4	2	no	—
184	Dic₁₀₆	—	4	2	no	—
185	Dic₁₀₆	—	8	4	no	—
186	C₈×C₅₃	—	2	1	no	—
187	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
188	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
189	#88322	—	2	1	no	—
190	#88322	#121	1	2	yes	—
191	C₂×#2209	—	53	1	no	maximal, Sylow
192	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
193	C₂×D₅₃	#459	1	2	yes	—
194	C₂×D₅₃	C₂×Q₃₂	1	2	yes	—
195	C₂×D₅₃	#510	1	2	yes	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	4	2	no	—
198	C₄×C₅₃	—	8	1	no	—
199	C₄×C₅₃	—	8	2	no	—
200	C₄×C₅₃	C₂×Q₃₂	1	2	yes	complemented
201	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
202	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	Dic₅₃	—	2	1	no	—
204	Dic₅₃	—	4	2	no	—
205	Dic₅₃	—	8	1	no	—
206	Dic₅₃	—	8	2	no	—
207	Dic₅₃	#459	1	2	yes	—
208	Dic₅₃	C₂×Q₃₂	1	2	yes	complemented
209	C₂²×Q₃₂	—	53	2	no	—
210	C₂×#494	—	53	1	no	—
211	C₂×C₄×Q₁₆	—	53	1	no	—
212	#2209	—	53	2	no	—
213	#2209	—	53	1	no	—
214	#2209	—	53	1	no	—
215	#2209	—	53	1	no	—
216	#2209	—	53	2	no	—
217	#2209	D₅₃	1	1	yes	characteristic, direct product factor
218	C₂×C₄⋊C₁₆	—	53	1	no	—
219	C₂×#358	—	53	2	no	—
220	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
221	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
222	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
223	D₅₃	#2209	1	4	yes	direct product factor
224	C₂²×Q₁₆	—	53	1	no	—
225	C₂²×Q₁₆	—	53	2	no	—
226	C₂²×Q₁₆	—	106	1	no	—
227	C₂×C₄⋊Q₈	—	106	1	no	—
228	C₂×C₄×Q₈	—	106	1	no	—
229	C₂×Q₃₂	—	53	2	no	—
230	C₂×Q₃₂	—	53	2	no	—
231	C₂×Q₃₂	—	53	2	no	—
232	C₂×Q₃₂	—	106	8	no	—
233	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
234	C₂²×C₁₆	—	53	2	no	—
235	#494	—	53	2	no	—
236	#494	—	53	1	no	—
237	#494	—	106	1	no	—
238	#494	—	106	1	no	—
239	#494	C₂×D₅₃	1	1	yes	characteristic
240	C₄×Q₁₆	—	53	1	no	—
241	C₄×Q₁₆	—	53	1	no	—
242	C₄×Q₁₆	—	53	1	no	—
243	C₄×Q₁₆	—	106	1	no	—
244	C₄×Q₁₆	—	106	1	no	—
245	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
246	C₂×#107	—	53	1	no	—
247	C₂×#103	—	106	1	no	—
248	C₂×C₄×C₈	—	53	1	no	—
249	C₄⋊C₁₆	—	53	2	no	—
250	C₄⋊C₁₆	—	53	1	no	—
251	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
252	#358	—	53	2	no	—
253	#358	—	53	2	no	—
254	#358	—	53	2	no	—
255	#358	C₂×D₅₃	1	2	yes	—
256	C₅₃	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
257	C₂²×Q₈	—	106	1	no	—
258	C₂²×Q₈	—	106	2	no	—
259	C₂×Q₁₆	—	2	1	no	—
260	C₂×Q₁₆	—	53	4	no	—
261	C₂×Q₁₆	—	53	1	no	—
262	C₂×Q₁₆	—	53	2	no	—
263	C₂×Q₁₆	—	106	2	no	—
264	C₂×Q₁₆	—	106	4	no	—
265	C₂×Q₁₆	—	106	4	no	—
266	C₂×Q₁₆	—	106	1	no	—
267	C₂×Q₁₆	—	106	2	no	—
268	C₂×Q₁₆	—	106	2	no	—
269	C₂×Q₁₆	—	106	1	no	—
270	C₂×Q₁₆	—	212	4	no	—
271	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
272	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
273	C₂²×C₈	—	53	1	no	—
274	C₂²×C₈	—	53	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	106	2	no	—
277	C₄⋊Q₈	—	106	1	no	—
278	C₄⋊Q₈	—	212	1	no	—
279	C₄⋊Q₈	—	212	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	—	106	1	no	—
282	C₄×Q₈	—	106	1	no	—
283	C₄×Q₈	—	106	1	no	—
284	C₄×Q₈	—	212	1	no	—
285	C₄×Q₈	—	212	1	no	—
286	C₂×C₄⋊C₄	—	106	1	no	—
287	C₂×C₄⋊C₄	—	212	1	no	—
288	C₂×C₄⋊C₄	—	212	1	no	—
289	C₂×C₄⋊C₄	—	212	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	212	1	no	—
292	Q₃₂	—	2	4	no	—
293	Q₃₂	—	106	4	no	—
294	Q₃₂	—	106	4	no	—
295	Q₃₂	—	106	4	no	—
296	C₂×C₁₆	—	53	2	no	—
297	C₂×C₁₆	—	106	4	no	—
298	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
299	#107	—	53	1	no	—
300	#107	—	106	1	no	—
301	#107	C₂²×D₅₃	1	1	yes	characteristic
302	#103	—	2	1	no	—
303	#103	—	106	1	no	—
304	#103	—	106	1	no	—
305	#103	—	106	1	no	—
306	C₄×C₈	—	53	1	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	106	4	no	—
313	C₂×Q₈	—	106	1	no	—
314	C₂×Q₈	—	106	2	no	—
315	C₂×Q₈	—	212	2	no	—
316	C₂×Q₈	—	212	1	no	—
317	C₂×Q₈	—	212	4	no	—
318	C₂×Q₈	—	212	4	no	—
319	C₂×Q₈	—	212	2	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	212	1	no	—
324	C₂²×C₄	—	212	2	no	—
325	C₂²×C₄	—	212	1	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	1	no	—
328	Q₁₆	—	2	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	53	2	no	—
331	Q₁₆	—	106	4	no	—
332	Q₁₆	—	106	2	no	—
333	Q₁₆	—	106	1	no	—
334	Q₁₆	—	106	2	no	—
335	Q₁₆	—	212	2	no	—
336	Q₁₆	—	212	4	no	—
337	Q₁₆	—	212	4	no	—
338	Q₁₆	D₄×D₅₃	1	2	yes	—
339	C₂×C₈	—	53	1	no	—
340	C₂×C₈	—	53	2	no	—
341	C₂×C₈	—	53	2	no	—
342	C₂×C₈	—	53	1	no	—
343	C₂×C₈	—	106	2	no	—
344	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	106	1	no	—
351	C₄⋊C₄	—	212	1	no	—
352	C₄⋊C₄	—	212	1	no	—
353	C₄⋊C₄	—	212	1	no	—
354	C₄⋊C₄	—	212	2	no	—
355	C₄⋊C₄	—	212	2	no	—
356	C₄⋊C₄	—	212	1	no	—
357	C₄⋊C₄	—	212	1	no	—
358	C₄⋊C₄	—	212	1	no	—
359	C₄²	—	4	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	—	53	1	no	—
363	C₄²	—	212	1	no	—
364	C₄²	—	212	1	no	—
365	C₄²	—	212	1	no	—
366	C₄²	D₄×D₅₃	1	1	yes	characteristic
367	C₁₆	—	2	2	no	—
368	C₁₆	—	106	2	no	—
369	C₂³	—	53	1	no	—
370	Q₈	—	2	2	no	—
371	Q₈	—	4	1	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	106	2	no	—
375	Q₈	—	424	4	no	—
376	Q₈	—	212	4	no	—
377	Q₈	—	212	2	no	—
378	Q₈	—	212	2	no	—
379	Q₈	—	212	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	53	1	no	—
384	C₂×C₄	—	53	1	no	—
385	C₂×C₄	—	53	4	no	—
386	C₂×C₄	—	53	2	no	—
387	C₂×C₄	—	53	2	no	—
388	C₂×C₄	—	53	1	no	—
389	C₂×C₄	—	106	2	no	—
390	C₂×C₄	—	424	2	no	—
391	C₂×C₄	—	424	4	no	—
392	C₂×C₄	—	212	4	no	—
393	C₂×C₄	—	212	1	no	—
394	C₂×C₄	—	212	2	no	—
395	C₂×C₄	—	212	1	no	—
396	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
397	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
399	C₈	—	2	1	no	—
400	C₈	—	53	2	no	—
401	C₈	—	106	1	no	—
402	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
403	C₈	#41×D₅₃	1	1	yes	characteristic
404	C₂²	—	53	2	no	—
405	C₂²	—	53	2	no	—
406	C₂²	—	53	2	no	—
407	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	4	2	no	—
410	C₄	—	8	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	53	2	no	—
413	C₄	—	53	2	no	—
414	C₄	—	106	1	no	—
415	C₄	—	424	1	no	—
416	C₄	—	424	2	no	—
417	C₄	—	212	2	no	—
418	C₄	Q₃₂×D₅₃	1	2	yes	complemented
419	C₄	#121×D₅₃	1	1	yes	characteristic
420	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
421	C₂	—	53	4	no	—
422	C₂	#459×D₅₃	1	1	yes	characteristic
423	C₂	C₂×Q₃₂×D₅₃	1	1	yes	characteristic
424	C₂	#510×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×#2209	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2209	D₅₃
hypocenter	C₅₃	C₂×#2209
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
4. #20276766	C₂	semidirect
5. #20276765	C₂	semidirect
6. #20276764	C₂	semidirect
7. #20276763	C₂	semidirect
8. #20276762	C₂	semidirect
11. C₅₃×#2209	C₂	semidirect
177. C₄×D₅₃	Q₃₂	semidirect
200. C₄×C₅₃	C₂×Q₃₂	semidirect
208. Dic₅₃	C₂×Q₃₂	semidirect
217. #2209	D₅₃	direct (N × H)
223. D₅₃	#2209	direct (N × H)
256. C₅₃	C₂×#2209	semidirect
418. C₄	Q₃₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₅₃	2
#494×D₅₃	2
C₄×Q₁₆×D₅₃	2
#20276766	2
#20276765	2
#20276764	2
#20276763	2
#20276762	2
C₄⋊C₁₆×D₅₃	2
#358×D₅₃	2
C₅₃×#2209	2
C₂×#2209	53

Subgroups — D53×#2209 #27538338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2209 #27538338