Subgroups — #27355862

Order 13568.

654 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	#20285274	C₂	1	1	yes	maximal, characteristic
2	#20283662	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283461	C₂	1	1	yes	maximal, characteristic
4	#20273607	C₂	1	2	yes	maximal, complemented
5	#20273560	C₂	1	2	yes	maximal, complemented
6	#20273553	C₂	1	2	yes	maximal, complemented
7	C₈×#11719913	C₂	1	2	yes	maximal, complemented
8	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273119	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273115	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	—	2	1	no	—
13	#16811180	C₂²	1	1	yes	characteristic
14	#16811173	C₂²	1	1	yes	characteristic
15	#16811021	C₄	1	1	yes	characteristic
16	#16810996	C₄	1	1	yes	characteristic
17	#16810974	C₄	1	1	yes	characteristic
18	#16810970	C₄	1	1	yes	characteristic
19	#16810852	C₄	1	2	yes	—
20	#16810852	C₄	1	2	yes	—
21	#16810753	C₂²	1	2	yes	—
22	#16810751	—	2	2	no	—
23	C₈×Dic₁₀₆	C₂²	1	2	yes	—
24	C₄×#11719913	C₂²	1	2	yes	—
25	C₄×#11719913	C₂²	1	2	yes	—
26	#16810702	C₂²	1	2	yes	—
27	#16810696	—	2	2	no	—
28	C₈×D₂₁₂	C₂²	1	2	yes	—
29	#16810651	—	2	1	no	—
30	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
32	#16810591	—	2	1	no	—
33	#16810581	—	2	1	no	—
34	#16810374	C₂²	1	1	yes	characteristic
35	#16810374	C₂²	1	1	yes	characteristic
36	#16810374	C₂²	1	1	yes	characteristic
37	#16810374	C₂²	1	1	yes	characteristic
38	#16810355	C₂²	1	2	yes	—
39	#16810355	C₂²	1	2	yes	—
40	#16810352	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	C₅₃×#431	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	—
46	C₈²×C₅₃	C₂²	1	2	yes	—
47	C₂×#11719945	—	2	1	no	—
48	C₂×#11719913	C₂×C₄	1	2	yes	—
49	C₂×#11719913	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	—	2	1	no	—
54	C₂×C₈×D₅₃	—	2	2	no	—
55	#13133217	C₂×C₄	1	1	yes	characteristic
56	#13133217	C₂×C₄	1	1	yes	characteristic
57	#13133217	C₂×C₄	1	1	yes	characteristic
58	#13133214	C₂×C₄	1	1	yes	characteristic
59	#13133209	—	2	1	no	—
60	#13133205	—	2	1	no	—
61	#13133196	—	2	1	no	—
62	#13133196	C₂×C₄	1	2	yes	—
63	#13133191	—	2	1	no	—
64	#13133191	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₀₆	—	2	1	no	—
66	C₄×Dic₁₀₆	C₂³	1	2	yes	—
67	C₄×D₂₁₂	—	2	1	no	—
68	C₄×D₂₁₂	C₂³	1	2	yes	—
69	#13133176	—	2	1	no	—
70	#13133149	C₂×C₄	1	2	yes	—
71	#13133149	C₂×C₄	1	2	yes	—
72	C₄⋊#88322	—	2	1	no	—
73	C₄⋊#88322	—	2	1	no	—
74	C₄⋊#88322	C₂³	1	1	yes	characteristic
75	C₄⋊#88322	C₂³	1	1	yes	characteristic
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	1	yes	characteristic
78	#13133146	—	2	2	no	—
79	#13133146	—	2	2	no	—
80	#13133142	C₂×C₄	1	2	yes	—
81	#13133142	C₂×C₄	1	2	yes	—
82	#13133139	C₂×C₄	1	2	yes	—
83	#13133139	C₂×C₄	1	2	yes	—
84	#13133129	—	2	2	no	—
85	#13133129	—	2	2	no	—
86	#13133127	—	2	2	no	—
87	C₈×Dic₅₃	—	2	2	no	—
88	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
89	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	—	2	1	no	—
94	C₂²⋊C₈×C₅₃	—	2	1	no	—
95	#97×C₅₃	D₄	1	2	yes	—
96	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄×C₈×C₅₃	C₂³	1	2	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
105	C₂×C₂²⋊D₅₃	—	2	2	no	—
106	#11719945	—	4	2	no	—
107	#11719945	—	4	2	no	—
108	C₂×D₂₁₂	—	2	1	no	—
109	C₂×D₂₁₂	—	2	1	no	—
110	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
111	C₂×C₄×D₅₃	—	2	2	no	—
112	C₂×Dic₁₀₆	—	2	1	no	—
113	C₂×Dic₁₀₆	—	2	1	no	—
114	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
115	M₁₆×C₅₃	—	2	2	no	—
116	M₁₆×C₅₃	SD₁₆	1	4	yes	complemented
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	4	1	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
126	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅₃	#41	1	1	yes	characteristic
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₅₃	—	2	2	no	—
131	#11719921	—	2	2	no	—
132	#11719921	—	2	2	no	—
133	C₄⋊Dic₅₃	—	2	1	no	—
134	C₄⋊Dic₅₃	—	2	1	no	—
135	C₄⋊Dic₅₃	—	2	1	no	—
136	C₄⋊Dic₅₃	—	2	1	no	—
137	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
138	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
139	#11719919	—	2	2	no	—
140	#11719919	—	2	2	no	—
141	C₄×Dic₅₃	—	2	2	no	—
142	C₂×#88322	—	2	1	no	—
143	C₂×#88322	—	2	1	no	—
144	C₂×#88322	—	2	1	no	—
145	C₂×#88322	—	2	1	no	—
146	C₂×#88322	—	4	2	no	—
147	C₂×#88322	—	4	2	no	—
148	#11719913	—	2	4	no	—
149	#11719913	—	4	2	no	—
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	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₅₃	#135	1	2	yes	—
162	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂²⋊D₅₃	—	4	4	no	—
166	C₂×Dic₅₃	—	2	2	no	—
167	C₂×Dic₅₃	—	2	2	no	—
168	C₂×Dic₅₃	—	2	2	no	—
169	D₂₁₂	—	2	2	no	—
170	D₂₁₂	—	2	2	no	—
171	D₂₁₂	—	4	2	no	—
172	D₂₁₂	—	8	1	no	—
173	C₄×D₅₃	—	4	2	no	—
174	C₄×D₅₃	—	4	2	no	—
175	Dic₁₀₆	—	2	2	no	—
176	Dic₁₀₆	—	2	2	no	—
177	Dic₁₀₆	—	4	2	no	—
178	Dic₁₀₆	—	8	1	no	—
179	C₈×C₅₃	—	2	1	no	—
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	#131	1	2	yes	complemented
185	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
186	#88322	—	4	1	no	—
187	#88322	—	4	1	no	—
188	#88322	—	4	1	no	—
189	#88322	—	4	1	no	—
190	#18067	—	53	1	no	maximal, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	C₄×SD₁₆	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	C₄×SD₁₆	1	2	yes	—
202	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
203	C₄×C₅₃	#568	1	1	yes	characteristic
204	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
205	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
206	Dic₅₃	—	4	2	no	—
207	Dic₅₃	—	4	2	no	—
208	Dic₅₃	—	4	2	no	—
209	#3167	—	53	1	no	—
210	#2962	—	53	1	no	—
211	#2935	—	53	1	no	—
212	#1590	—	53	2	no	—
213	#1583	—	53	2	no	—
214	#1582	—	53	2	no	—
215	C₈×SD₁₆	—	53	2	no	—
216	C₂×#334	—	53	1	no	—
217	#1481	—	53	1	no	—
218	#1480	—	53	1	no	—
219	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
220	C₂×C₅₃	—	2	2	no	—
221	C₂×C₅₃	#2935	1	1	yes	characteristic
222	C₂×C₅₃	C₂×C₄×SD₁₆	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	—	53	1	no	—
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	—	53	2	no	—
235	#446	—	106	2	no	—
236	C₈×Q₈	—	53	2	no	—
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	C₂×D₅₃	1	1	yes	characteristic
243	#424	—	106	1	no	—
244	C₂×C₄⋊C₈	—	53	1	no	—
245	C₂×C₄⋊C₈	—	53	1	no	—
246	#408	—	106	1	no	—
247	#405	—	106	1	no	—
248	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
249	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
250	#334	—	53	1	no	—
251	#334	—	53	1	no	—
252	#334	—	53	1	no	—
253	#334	—	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	C₂×D₅₃	1	2	yes	—
259	C₈²	C₂×D₅₃	1	2	yes	—
260	C₅₃	#18067	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	D₂₁₂	1	1	yes	characteristic
267	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
268	C₂²×C₈	—	106	1	no	—
269	C₂²×C₈	—	106	1	no	—
270	C₂²×C₈	—	106	2	no	—
271	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
273	C₄⋊Q₈	—	53	1	no	—
274	C₄⋊Q₈	—	53	1	no	—
275	C₄⋊Q₈	—	53	1	no	—
276	#127	—	53	1	no	—
277	#125	—	106	1	no	—
278	#124	—	106	1	no	—
279	C₂²⋊Q₈	—	53	2	no	—
280	C₂²⋊Q₈	—	106	1	no	—
281	#121	—	53	2	no	—
282	#121	—	106	1	no	—
283	C₄×Q₈	—	53	2	no	—
284	C₄×Q₈	—	106	1	no	—
285	C₄×D₄	—	53	2	no	—
286	C₄×D₄	—	106	1	no	—
287	#117	—	106	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	1	no	—
292	C₄⋊C₈	—	53	1	no	—
293	C₄⋊C₈	—	53	1	no	—
294	C₄⋊C₈	—	53	1	no	—
295	C₄⋊C₈	—	53	1	no	—
296	C₄⋊C₈	—	106	2	no	—
297	C₄⋊C₈	—	106	2	no	—
298	C₄⋊C₈	—	106	1	no	—
299	C₄⋊C₈	—	106	1	no	—
300	#103	—	53	2	no	—
301	#103	—	53	2	no	—
302	#102	—	53	2	no	—
303	#102	—	53	2	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	C₂²⋊C₈	—	106	2	no	—
307	#97	—	106	2	no	—
308	#97	D₂₁₂	1	2	yes	—
309	C₄×C₈	—	106	2	no	—
310	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
311	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
312	C₄×C₈	C₂²×D₅₃	1	2	yes	—
313	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
315	C₄×C₈	C₂²×D₅₃	1	2	yes	—
316	#41	—	212	2	no	—
317	#41	—	212	2	no	—
318	C₂×Q₈	—	53	2	no	—
319	C₂×Q₈	—	106	1	no	—
320	C₂×Q₈	—	106	1	no	—
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	C₂×D₂₁₂	1	1	yes	characteristic
328	C₂²×C₄	#11719945	1	1	yes	characteristic
329	SD₁₆	—	106	4	no	—
330	SD₁₆	—	212	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	212	4	no	—
333	M₁₆	#11719913	1	4	yes	complemented
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	106	1	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	106	1	no	—
340	C₂×C₈	—	106	1	no	—
341	C₂×C₈	—	212	2	no	—
342	C₂×C₈	—	212	2	no	—
343	C₂×C₈	—	212	4	no	—
344	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
346	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
347	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
348	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
349	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
350	C₄⋊C₄	—	53	2	no	—
351	C₄⋊C₄	—	53	2	no	—
352	C₄⋊C₄	—	106	1	no	—
353	C₄⋊C₄	—	106	1	no	—
354	C₄⋊C₄	—	106	1	no	—
355	C₄⋊C₄	—	106	2	no	—
356	C₄⋊C₄	—	106	1	no	—
357	C₄⋊C₄	—	106	2	no	—
358	C₂²⋊C₄	—	106	2	no	—
359	C₂²⋊C₄	—	106	2	no	—
360	C₂²⋊C₄	—	106	2	no	—
361	C₄²	—	106	2	no	—
362	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
363	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
364	C₄²	#11719945	1	1	yes	characteristic
365	C₄²	C₂³×D₅₃	1	1	yes	characteristic
366	C₂³	—	106	2	no	—
367	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
368	Q₈	—	106	2	no	—
369	Q₈	—	106	2	no	—
370	Q₈	—	424	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	—	4	1	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	—	106	2	no	—
383	C₂×C₄	—	212	2	no	—
384	C₂×C₄	—	212	2	no	—
385	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
386	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
387	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
388	C₂×C₄	C₂×#11719913	1	2	yes	—
389	C₂×C₄	#13133239	1	2	yes	—
390	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
392	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	2	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	212	1	no	—
399	C₈	—	212	1	no	—
400	C₈	—	212	1	no	—
401	C₈	—	212	1	no	—
402	C₈	#13133225	1	2	yes	complemented
403	C₈	C₂×#11719913	1	4	yes	—
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	C₄×#11719913	1	2	yes	—
410	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	212	2	no	—
415	C₄	—	212	2	no	—
416	C₄	—	212	2	no	—
417	C₄	C₄×#11719913	1	2	yes	—
418	C₄	C₂×#13133225	1	1	yes	characteristic
419	C₄	#16811389	1	1	yes	characteristic
420	C₄	C₂²×#11719913	1	1	yes	characteristic
421	C₄	C₂×#13133239	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	424	2	no	—
424	C₂	#20283461	1	1	yes	characteristic
425	C₂	C₂×C₄×#11719913	1	1	yes	characteristic
426	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#18067
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20283662	C₂	semidirect
4. #20273607	C₂	semidirect
5. #20273560	C₂	semidirect
6. #20273553	C₂	semidirect
7. C₈×#11719913	C₂	semidirect
8. C₂×#16810374	C₂	semidirect
9. #20273119	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
116. M₁₆×C₅₃	SD₁₆	semidirect
184. C₈×C₅₃	#131	semidirect
219. C₈×M₁₆	D₅₃	semidirect
260. C₅₃	#18067	semidirect
333. M₁₆	#11719913	semidirect
402. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20285274	2
#20283662	2
#20283461	2
#20273607	2
#20273560	2
#20273553	2
C₈×#11719913	2
C₂×#16810374	2
#20273119	2
#20273115	2
C₈×M₁₆×C₅₃	2
#18067	53