Subgroups — #27355895

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285275	C₂	1	1	yes	maximal, characteristic
2	#20283461	C₂	1	1	yes	maximal, characteristic
3	#20273615	C₂	1	2	yes	maximal, complemented
4	#20273593	C₂	1	2	yes	maximal, complemented
5	#20273570	C₂	1	2	yes	maximal, complemented
6	#20273525	C₂	1	2	yes	maximal, complemented
7	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273118	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273113	C₂	1	1	yes	maximal, characteristic
10	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	—	2	1	no	—
13	#16811173	C₂²	1	1	yes	characteristic
14	#16811019	C₄	1	1	yes	characteristic
15	#16810997	C₄	1	1	yes	characteristic
16	#16810973	C₄	1	1	yes	characteristic
17	#16810971	C₄	1	1	yes	characteristic
18	#16810854	C₄	1	2	yes	—
19	#16810854	C₄	1	2	yes	—
20	C₄×#11719924	C₂²	1	2	yes	—
21	C₄×#11719924	C₂²	1	2	yes	—
22	#16810702	C₂²	1	2	yes	—
23	#16810696	—	2	2	no	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	#16810676	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	#16810591	—	2	1	no	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	#16810375	C₂²	1	1	yes	characteristic
30	#16810375	C₂²	1	1	yes	characteristic
31	#16810375	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	1	yes	characteristic
33	Q₈⋊#88322	C₂²	1	2	yes	—
34	Q₈⋊#88322	C₂²	1	2	yes	—
35	#16810350	C₂²	1	2	yes	—
36	#16810350	C₂²	1	2	yes	—
37	#16810345	C₂²	1	2	yes	—
38	#16810345	C₂²	1	2	yes	—
39	C₅₃×#533	C₂²	1	1	yes	characteristic
40	C₅₃×#446	—	2	2	no	—
41	C₅₃×#446	C₂²	1	2	yes	—
42	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
43	C₅₃×#424	—	2	1	no	—
44	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#405	—	2	1	no	—
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719945	—	2	1	no	—
48	C₂×#11719924	C₂×C₄	1	2	yes	—
49	C₂×#11719924	C₂×C₄	1	2	yes	—
50	#13133225	—	4	4	no	—
51	C₂×#11719912	—	2	2	no	—
52	C₂²×#88322	—	2	1	no	—
53	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
54	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	—	2	2	no	—
56	#13133217	C₂×C₄	1	1	yes	characteristic
57	#13133214	C₂×C₄	1	1	yes	characteristic
58	#13133205	—	2	1	no	—
59	#13133191	—	2	1	no	—
60	#13133191	C₂×C₄	1	2	yes	—
61	C₄×D₂₁₂	—	2	1	no	—
62	C₄×D₂₁₂	C₂³	1	2	yes	—
63	#13133148	C₂×C₄	1	2	yes	—
64	#13133148	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	—	2	1	no	—
66	C₄⋊#88322	—	2	1	no	—
67	C₄⋊#88322	C₂³	1	1	yes	characteristic
68	C₄⋊#88322	C₂³	1	1	yes	characteristic
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133137	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	C₂²⋊#88322	—	2	1	no	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	2	no	—
76	C₄×#88322	C₂×C₄	1	1	yes	characteristic
77	C₄×#88322	C₂×C₄	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	2	yes	—
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
82	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	—	2	1	no	—
84	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#125×C₅₃	—	2	1	no	—
87	C₂²⋊Q₈×C₅₃	—	2	1	no	—
88	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₅₃	—	2	1	no	—
90	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
91	#117×C₅₃	—	2	1	no	—
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	—	2	2	no	—
94	C₄⋊C₈×C₅₃	—	2	2	no	—
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	#97×C₅₃	—	2	2	no	—
99	#97×C₅₃	D₄	1	2	yes	—
100	C₄×C₈×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
107	C₂×C₂²⋊D₅₃	—	2	2	no	—
108	#11719945	—	4	2	no	—
109	#11719945	—	4	2	no	—
110	C₂×D₂₁₂	—	2	1	no	—
111	C₂×D₂₁₂	—	2	1	no	—
112	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
113	C₂×C₄×D₅₃	—	2	2	no	—
114	C₂×Dic₁₀₆	—	2	1	no	—
115	M₁₆×C₅₃	—	2	2	no	—
116	M₁₆×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	4	2	no	—
120	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
134	#11719924	—	2	4	no	—
135	#11719924	—	4	2	no	—
136	#11719921	—	2	2	no	—
137	#11719921	—	2	2	no	—
138	C₄⋊Dic₅₃	—	2	1	no	—
139	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	1	no	—
142	C₂×#88322	—	2	1	no	—
143	C₂×#88322	—	2	2	no	—
144	C₂×#88322	—	4	1	no	—
145	C₂×#88322	—	4	2	no	—
146	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
147	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
148	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
149	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
150	#11719912	—	4	4	no	—
151	C₈×D₅₃	—	4	4	no	—
152	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂²×D₅₃	—	2	2	no	—
154	Q₈×C₅₃	—	2	2	no	—
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	8	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₅₃	#137	1	2	yes	—
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
170	C₂²⋊D₅₃	—	4	4	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	D₂₁₂	—	2	2	no	—
173	D₂₁₂	—	2	2	no	—
174	D₂₁₂	—	4	2	no	—
175	D₂₁₂	—	8	1	no	—
176	C₄×D₅₃	—	4	2	no	—
177	C₄×D₅₃	—	4	2	no	—
178	Dic₁₀₆	—	4	2	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	4	1	no	—
182	C₈×C₅₃	—	4	1	no	—
183	#88322	—	2	1	no	—
184	#88322	—	2	1	no	—
185	#88322	—	2	1	no	—
186	#88322	—	4	2	no	—
187	#88322	—	4	1	no	—
188	#88322	—	4	1	no	—
189	#88322	#131	1	2	yes	complemented
190	#18071	—	53	1	no	maximal, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	#440	1	2	yes	—
194	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂×D₅₃	—	4	2	no	—
196	C₂×D₅₃	—	4	2	no	—
197	C₂×D₅₃	—	4	2	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	—	4	2	no	—
203	C₄×C₅₃	#440	1	2	yes	—
204	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
205	C₄×C₅₃	#568	1	1	yes	characteristic
206	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
208	Dic₅₃	—	4	2	no	—
209	#3167	—	53	1	no	—
210	#2962	D₅₃	1	1	yes	characteristic, complemented
211	#2935	—	53	1	no	—
212	#1592	—	53	2	no	—
213	#1588	—	53	2	no	—
214	#1585	—	53	2	no	—
215	SD₁₆⋊C₈	—	53	2	no	—
216	C₂×#335	—	53	1	no	—
217	#1481	—	53	1	no	—
218	#1480	—	53	1	no	—
219	#1451	—	53	1	no	—
220	C₂×C₅₃	—	2	2	no	—
221	C₂×C₅₃	#2935	1	1	yes	characteristic
222	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
223	C₂×C₅₃	#2957	1	1	yes	characteristic
224	D₅₃	—	8	2	no	—
225	C₂×#131	—	106	1	no	—
226	#533	C₂×D₅₃	1	1	yes	characteristic
227	#532	—	53	1	no	—
228	#498	—	53	1	no	—
229	#492	—	53	1	no	—
230	#487	—	53	1	no	—
231	#486	—	53	1	no	—
232	#465	—	53	2	no	—
233	#465	—	53	2	no	—
234	#446	—	2	2	no	—
235	#446	C₂×D₅₃	1	2	yes	—
236	C₈×Q₈	C₂×D₅₃	1	2	yes	—
237	C₄×SD₁₆	—	53	2	no	—
238	C₄×SD₁₆	—	53	2	no	—
239	#436	—	53	2	no	—
240	#435	—	106	2	no	—
241	C₈×D₄	—	53	2	no	—
242	#431	—	53	1	no	—
243	#424	—	2	1	no	—
244	C₂×C₄⋊C₈	—	53	1	no	—
245	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
246	#408	—	106	1	no	—
247	#405	—	2	1	no	—
248	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
249	C₂×C₄×C₈	—	53	1	no	—
250	#335	—	53	1	no	—
251	#335	—	53	1	no	—
252	#335	—	53	1	no	—
253	#335	—	53	1	no	—
254	Q₈⋊C₈	—	53	2	no	—
255	Q₈⋊C₈	—	53	2	no	—
256	D₄⋊C₈	—	53	2	no	—
257	D₄⋊C₈	—	53	2	no	—
258	#322	—	53	2	no	—
259	#322	—	53	2	no	—
260	C₅₃	#18071	1	1	yes	characteristic, complemented, Sylow
261	C₂×#41	—	106	1	no	—
262	C₂×SD₁₆	—	53	2	no	—
263	C₂×SD₁₆	—	53	2	no	—
264	#131	—	212	4	no	—
265	C₂×M₁₆	—	106	2	no	—
266	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₂²×C₈	—	53	1	no	—
271	C₂²×C₈	—	106	1	no	—
272	C₂²×C₈	—	106	2	no	—
273	C₄⋊Q₈	—	53	1	no	—
274	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
275	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
276	#127	—	53	1	no	—
277	#125	—	2	1	no	—
278	#124	—	106	1	no	—
279	C₂²⋊Q₈	—	2	1	no	—
280	C₂²⋊Q₈	C₄×D₅₃	1	2	yes	—
281	#121	—	53	2	no	—
282	#121	—	106	1	no	—
283	C₄×Q₈	—	2	1	no	—
284	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
285	C₄×D₄	—	53	2	no	—
286	C₄×D₄	—	106	1	no	—
287	#117	—	2	1	no	—
288	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
289	#106	—	53	2	no	—
290	#106	—	53	2	no	—
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	53	1	no	—
295	C₄⋊C₈	—	53	1	no	—
296	C₄⋊C₈	—	106	1	no	—
297	C₄⋊C₈	—	106	1	no	—
298	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
299	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
300	#103	—	53	2	no	—
301	#103	—	53	2	no	—
302	#102	—	53	2	no	—
303	#102	—	53	2	no	—
304	C₂²⋊C₈	—	106	1	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	C₂²⋊C₈	—	106	2	no	—
307	#97	—	2	2	no	—
308	#97	C₂²⋊D₅₃	1	2	yes	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	53	1	no	—
314	C₄×C₈	—	53	2	no	—
315	C₄×C₈	C₂²×D₅₃	1	2	yes	—
316	#41	—	212	2	no	—
317	#41	—	212	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	106	1	no	—
320	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
321	C₂×D₄	—	53	2	no	—
322	C₂×D₄	—	106	2	no	—
323	C₂×D₄	—	106	1	no	—
324	C₂×D₄	—	106	1	no	—
325	C₂²×C₄	—	106	2	no	—
326	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₂²×C₄	#11719945	1	1	yes	characteristic
328	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
329	SD₁₆	—	106	4	no	—
330	SD₁₆	—	212	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	212	4	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	53	1	no	—
338	C₂×C₈	—	53	1	no	—
339	C₂×C₈	—	53	1	no	—
340	C₂×C₈	—	53	1	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	—	106	1	no	—
343	C₂×C₈	—	106	1	no	—
344	C₂×C₈	—	106	2	no	—
345	C₂×C₈	—	212	4	no	—
346	C₂×C₈	—	212	1	no	—
347	C₂×C₈	—	212	2	no	—
348	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
349	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	53	2	no	—
356	C₄⋊C₄	—	106	1	no	—
357	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	106	2	no	—
360	C₂²⋊C₄	—	106	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
363	C₄²	#11719945	1	1	yes	characteristic
364	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
365	C₄²	C₂³×D₅₃	1	1	yes	characteristic
366	C₂³	—	106	2	no	—
367	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	8	1	no	—
371	Q₈	—	212	2	no	—
372	D₄	—	106	2	no	—
373	D₄	—	106	2	no	—
374	D₄	—	424	1	no	—
375	D₄	—	212	4	no	—
376	D₄	—	212	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	106	2	no	—
383	C₂×C₄	—	212	2	no	—
384	C₂×C₄	—	212	2	no	—
385	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
386	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
387	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
388	C₂×C₄	#13133242	1	2	yes	—
389	C₂×C₄	#13133248	1	2	yes	—
390	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
393	C₈	—	2	2	no	—
394	C₈	—	2	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	53	2	no	—
398	C₈	—	106	1	no	—
399	C₈	—	106	1	no	—
400	C₈	—	106	1	no	—
401	C₈	—	212	2	no	—
402	C₈	—	212	1	no	—
403	C₈	—	212	1	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	212	2	no	—
407	C₂²	—	212	2	no	—
408	C₂²	—	212	2	no	—
409	C₂²	#16810721	1	2	yes	—
410	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	212	2	no	—
417	C₄	#16810721	1	2	yes	—
418	C₄	#131×D₅₃	1	1	yes	characteristic
419	C₄	#16811388	1	1	yes	characteristic
420	C₄	C₂×#13133242	1	1	yes	characteristic
421	C₄	C₂×#13133248	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	424	2	no	—
424	C₂	#20283457	1	1	yes	characteristic
425	C₂	C₂×#16810721	1	1	yes	characteristic
426	C₂	#20283630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2962	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2962	D₅₃
hypocenter	C₅₃	#18071
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #2962

Complements

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

N	H ≅ G/N	type
3. #20273615	C₂	semidirect
4. #20273593	C₂	semidirect
5. #20273570	C₂	semidirect
6. #20273525	C₂	semidirect
7. C₂×#16810375	C₂	semidirect
8. #20273118	C₂	semidirect
10. #20273002	C₂	semidirect
11. C₅₃×#2962	C₂	semidirect
189. #88322	#131	semidirect
210. #2962	D₅₃	semidirect
260. C₅₃	#18071	semidirect

Maximal subgroups

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

subgroup	index
#20285275	2
#20283461	2
#20273615	2
#20273593	2
#20273570	2
#20273525	2
C₂×#16810375	2
#20273118	2
#20273113	2
#20273002	2
C₅₃×#2962	2
#18071	53