Subgroups — #27367539

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20286041	C₂	1	2	yes	maximal, complemented
2	#20285294	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274243	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274238	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274199	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274194	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274131	C₂	1	2	yes	maximal, complemented
8	#20274124	C₂	1	2	yes	maximal, complemented
9	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273411	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2966	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811173	C₂²	1	2	yes	—
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811000	C₂²	1	1	yes	characteristic
15	#16810970	C₂²	1	2	yes	—
16	#16810920	—	2	2	no	—
17	#16810918	C₂²	1	2	yes	—
18	#16810913	C₂²	1	2	yes	—
19	#16810859	—	2	1	no	—
20	#16810859	—	2	1	no	—
21	#16810857	—	2	2	no	—
22	#16810854	—	2	2	no	—
23	#16810835	—	2	2	no	—
24	#16810823	C₂²	1	2	yes	—
25	#16810818	C₂²	1	2	yes	—
26	C₄×D₄₂₄	—	2	1	no	—
27	C₄×D₄₂₄	—	2	1	no	—
28	C₂×C₄⋊#88322	C₂²	1	2	yes	—
29	#16810374	C₂²	1	1	yes	characteristic
30	#16810374	C₂²	1	1	yes	characteristic
31	#16810374	C₂²	1	2	yes	—
32	#16810371	C₂²	1	2	yes	—
33	#16810371	C₂²	1	2	yes	—
34	#16810366	C₂²	1	2	yes	—
35	#16810366	C₂²	1	2	yes	—
36	C₅₃×#529	C₂²	1	1	yes	characteristic
37	C₅₃×#446	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#436	C₂²	1	1	yes	characteristic
40	C₅₃×#435	—	2	1	no	—
41	C₈×D₄×C₅₃	—	2	1	no	—
42	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#433	—	2	1	no	—
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#405	—	2	1	no	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719945	—	2	2	no	—
48	C₂×#11719924	—	2	2	no	—
49	C₂×D₄₂₄	—	2	1	no	—
50	C₂×D₄₂₄	—	2	1	no	—
51	C₂×#11719922	—	2	2	no	—
52	C₂²×#88322	—	2	2	no	—
53	#13133217	C₂³	1	2	yes	—
54	#13133214	C₂³	1	2	yes	—
55	#13133205	—	2	2	no	—
56	#13133191	—	2	2	no	—
57	#13133191	—	2	2	no	—
58	C₄×D₂₁₂	—	2	2	no	—
59	C₄×D₂₁₂	—	2	2	no	—
60	#13133151	—	2	1	no	—
61	#13133151	—	2	1	no	—
62	#13133150	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄⋊#88322	—	2	2	no	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133141	—	2	2	no	—
68	#13133139	—	2	2	no	—
69	#13133139	—	2	2	no	—
70	#13133138	—	2	2	no	—
71	#13133137	—	2	2	no	—
72	#13133137	—	2	2	no	—
73	C₂×M₁₆×C₅₃	—	2	1	no	—
74	C₂²×C₈×C₅₃	—	2	1	no	—
75	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
77	#126×C₅₃	—	2	1	no	—
78	#125×C₅₃	D₄	1	1	yes	characteristic
79	#124×C₅₃	D₄	1	1	yes	characteristic
80	#123×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
82	#121×C₅₃	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₅₃	—	2	1	no	—
85	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
86	#117×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	—	2	1	no	—
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²⋊C₈×C₅₃	—	2	1	no	—
93	C₂²⋊C₈×C₅₃	—	2	1	no	—
94	C₂²⋊C₈×C₅₃	—	2	1	no	—
95	#97×C₅₃	—	2	1	no	—
96	#97×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
109	C₂×C₂²⋊D₅₃	—	4	2	no	—
110	#11719945	—	8	4	no	—
111	C₂×D₂₁₂	—	2	2	no	—
112	C₂×D₂₁₂	—	2	2	no	—
113	C₂×D₂₁₂	—	2	2	no	—
114	C₂×C₄×D₅₃	—	4	2	no	—
115	C₂×Dic₁₀₆	—	2	2	no	—
116	M₁₆×C₅₃	—	4	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	4	1	no	—
121	C₂×C₈×C₅₃	—	4	2	no	—
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₅₃	—	2	1	no	—
133	C₂²⋊C₄×C₅₃	—	2	1	no	—
134	C₂²⋊C₄×C₅₃	—	2	1	no	—
135	C₂²⋊C₄×C₅₃	—	2	1	no	—
136	C₂²⋊C₄×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	#41	1	1	yes	characteristic
141	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
142	#11719924	—	8	2	no	—
143	#11719922	—	8	2	no	—
144	#11719921	—	4	2	no	—
145	#11719921	—	4	2	no	—
146	C₄⋊Dic₅₃	—	2	2	no	—
147	C₄⋊Dic₅₃	—	2	2	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	2	no	—
150	C₂×#88322	—	8	2	no	—
151	D₄₂₄	—	4	2	no	—
152	D₄₂₄	—	4	2	no	—
153	C₂³×C₅₃	—	2	1	no	—
154	C₂³×C₅₃	#121	1	1	yes	characteristic
155	C₂²×D₅₃	—	4	2	no	—
156	Q₈×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	2	2	no	—
158	D₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₂²⋊D₅₃	—	8	4	no	—
175	C₂×Dic₅₃	—	4	2	no	—
176	D₂₁₂	—	4	2	no	—
177	D₂₁₂	—	4	2	no	—
178	D₂₁₂	—	4	2	no	—
179	D₂₁₂	—	4	2	no	—
180	D₂₁₂	—	8	2	no	—
181	C₄×D₅₃	—	8	4	no	—
182	Dic₁₀₆	—	8	2	no	—
183	C₈×C₅₃	—	2	1	no	—
184	C₈×C₅₃	—	2	1	no	—
185	C₈×C₅₃	—	2	1	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	—	4	1	no	—
188	C₈×C₅₃	#135	1	2	yes	—
189	#88322	—	4	2	no	—
190	#88322	—	4	2	no	—
191	#19227	—	53	1	no	maximal, Sylow
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	—	4	1	no	—
197	C₂²×C₅₃	#472	1	2	yes	—
198	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
199	C₂×D₅₃	—	8	2	no	—
200	C₂×D₅₃	—	8	2	no	—
201	C₂×D₅₃	—	8	2	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	4	1	no	—
207	C₄×C₅₃	—	4	1	no	—
208	C₄×C₅₃	—	4	1	no	—
209	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
210	C₄×C₅₃	#575	1	1	yes	characteristic
211	C₄×C₅₃	#577	1	2	yes	—
212	Dic₅₃	—	8	2	no	—
213	#3247	—	53	2	no	—
214	#3169	—	53	1	no	—
215	#2966	D₅₃	1	1	yes	characteristic, complemented
216	#1696	—	53	1	no	—
217	#1695	—	53	1	no	—
218	#1688	—	53	1	no	—
219	#1687	—	53	1	no	—
220	#1678	—	53	2	no	—
221	#1677	—	53	2	no	—
222	C₂×#334	—	53	1	no	—
223	#1551	—	53	2	no	—
224	C₂×C₅₃	—	2	2	no	—
225	C₂×C₅₃	—	8	1	no	—
226	C₂×C₅₃	#3035	1	1	yes	characteristic
227	C₂×C₅₃	#3060	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
229	D₅₃	—	16	2	no	—
230	#532	—	53	2	no	—
231	#529	C₂×D₅₃	1	1	yes	characteristic
232	#500	—	53	1	no	—
233	#493	—	53	1	no	—
234	#486	—	53	2	no	—
235	#478	—	106	2	no	—
236	#477	—	53	2	no	—
237	#476	—	53	2	no	—
238	#466	—	106	1	no	—
239	#466	—	106	1	no	—
240	#466	—	106	2	no	—
241	#465	—	106	2	no	—
242	#463	—	106	2	no	—
243	#460	—	53	2	no	—
244	#459	—	53	2	no	—
245	#446	C₂×D₅₃	1	1	yes	characteristic
246	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
247	C₄×D₈	—	106	1	no	—
248	C₄×D₈	—	106	1	no	—
249	#436	C₂×D₅₃	1	1	yes	characteristic
250	#435	—	2	1	no	—
251	C₈×D₄	—	2	1	no	—
252	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
253	#433	—	2	1	no	—
254	#432	C₂×D₅₃	1	1	yes	characteristic
255	C₂×C₄⋊C₈	—	53	2	no	—
256	#405	—	2	1	no	—
257	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
258	#334	—	53	2	no	—
259	#334	—	53	1	no	—
260	#334	—	53	1	no	—
261	#333	—	53	2	no	—
262	#333	—	53	2	no	—
263	#331	—	53	2	no	—
264	#331	—	53	2	no	—
265	C₅₃	#19227	1	1	yes	characteristic, complemented, Sylow
266	C₂×#41	—	106	2	no	—
267	C₂×SD₁₆	—	106	2	no	—
268	C₂×D₈	—	106	2	no	—
269	C₂×D₈	—	106	1	no	—
270	C₂×D₈	—	106	1	no	—
271	C₂×M₁₆	—	2	1	no	—
272	C₂²×C₈	—	2	1	no	—
273	C₂²×C₈	—	106	2	no	—
274	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
276	C₄⋊Q₈	—	53	2	no	—
277	#127	—	53	2	no	—
278	#126	—	2	1	no	—
279	#125	D₂₁₂	1	1	yes	characteristic
280	#124	—	106	2	no	—
281	#124	D₂₁₂	1	1	yes	characteristic
282	#123	—	2	1	no	—
283	C₂²⋊Q₈	D₂₁₂	1	1	yes	characteristic
284	#121	—	106	2	no	—
285	#121	—	106	2	no	—
286	#121	D₂₁₂	1	1	yes	characteristic
287	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₄×D₄	—	2	1	no	—
289	C₄×D₄	—	106	2	no	—
290	C₄×D₄	—	106	2	no	—
291	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
292	#117	—	2	1	no	—
293	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
294	#107	—	106	1	no	—
295	#107	—	106	1	no	—
296	#107	—	106	2	no	—
297	#106	—	106	2	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	—	53	2	no	—
301	C₄⋊C₈	—	53	2	no	—
302	C₄⋊C₈	—	106	2	no	—
303	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
304	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
305	#103	—	106	2	no	—
306	#102	—	106	2	no	—
307	#102	—	106	2	no	—
308	#102	—	106	2	no	—
309	#102	—	106	2	no	—
310	#102	—	106	2	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	2	1	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	#97	—	2	1	no	—
315	#97	—	2	1	no	—
316	C₄×C₈	—	2	1	no	—
317	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
318	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
319	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
320	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
321	#41	—	424	4	no	—
322	C₂×Q₈	—	106	2	no	—
323	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	106	2	no	—
326	C₂×D₄	—	106	2	no	—
327	C₂×D₄	—	106	2	no	—
328	C₂×D₄	—	212	2	no	—
329	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	212	2	no	—
332	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
333	C₂²×C₄	#11719945	1	1	yes	characteristic
334	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
335	SD₁₆	—	424	2	no	—
336	D₈	—	424	2	no	—
337	D₈	—	212	2	no	—
338	D₈	—	212	2	no	—
339	M₁₆	—	4	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	4	2	no	—
345	C₂×C₈	—	106	2	no	—
346	C₂×C₈	—	106	2	no	—
347	C₂×C₈	—	424	2	no	—
348	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
349	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
350	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
351	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	106	2	no	—
357	C₄⋊C₄	—	106	2	no	—
358	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	1	no	—
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	2	1	no	—
365	C₂²⋊C₄	—	212	2	no	—
366	C₂²⋊C₄	—	212	2	no	—
367	C₄²	—	2	1	no	—
368	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
369	C₄²	#11719945	1	1	yes	characteristic
370	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
371	C₄²	C₂³×D₅₃	1	1	yes	characteristic
372	C₂³	—	2	1	no	—
373	C₂³	—	212	2	no	—
374	C₂³	#13133191	1	1	yes	characteristic
375	Q₈	—	2	2	no	—
376	Q₈	—	424	2	no	—
377	D₄	—	2	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	424	2	no	—
380	D₄	—	424	4	no	—
381	D₄	—	212	2	no	—
382	D₄	—	212	2	no	—
383	D₄	—	212	2	no	—
384	D₄	—	212	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	424	4	no	—
395	C₂×C₄	—	212	2	no	—
396	C₂×C₄	#13133191	1	1	yes	characteristic
397	C₂×C₄	#13133191	1	1	yes	characteristic
398	C₂×C₄	#13133191	1	1	yes	characteristic
399	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
401	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	—	2	1	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	1	no	—
407	C₈	—	212	2	no	—
408	C₈	—	212	2	no	—
409	C₈	#13133237	1	2	yes	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	424	2	no	—
416	C₂²	—	424	2	no	—
417	C₂²	—	424	2	no	—
418	C₂²	#16810893	1	2	yes	—
419	C₂²	C₂×#13133191	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	2	1	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	1	no	—
427	C₄	—	424	2	no	—
428	C₄	C₂×#13133237	1	1	yes	characteristic
429	C₄	#16811421	1	1	yes	characteristic
430	C₄	#16811435	1	2	yes	—
431	C₂	—	2	2	no	—
432	C₂	—	8	1	no	—
433	C₂	—	848	2	no	—
434	C₂	#20284170	1	1	yes	characteristic
435	C₂	#20284377	1	1	yes	characteristic
436	C₂	C₂×#16810893	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2966	C₂
socle	C₂²×C₅₃	C₂×#121
hypercenter	#2966	D₅₃
hypocenter	C₅₃	#19227
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #20286041	C₂	semidirect
2. #20285294	C₂	semidirect
3. #20274243	C₂	semidirect
4. #20274238	C₂	semidirect
5. #20274199	C₂	semidirect
6. #20274194	C₂	semidirect
7. #20274131	C₂	semidirect
8. #20274124	C₂	semidirect
9. C₂×#16810374	C₂	semidirect
10. #20273411	C₂	semidirect
11. C₅₃×#2966	C₂	semidirect
215. #2966	D₅₃	semidirect
265. C₅₃	#19227	semidirect

Maximal subgroups

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

subgroup	index
#20286041	2
#20285294	2
#20274243	2
#20274238	2
#20274199	2
#20274194	2
#20274131	2
#20274124	2
C₂×#16810374	2
#20273411	2
C₅₃×#2966	2
#19227	53