Subgroups — #27354118

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284793	C₂	1	1	yes	maximal, characteristic
2	#20283687	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273564	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273562	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273561	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273559	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273535	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273532	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273527	C₂	1	1	yes	maximal, characteristic, complemented
11	#20273499	C₂	1	1	yes	maximal, characteristic, complemented
12	#20273160	C₂	1	1	yes	maximal, characteristic, complemented
13	#20273125	C₂	1	1	yes	maximal, characteristic
14	#20273003	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811155	C₂²	1	1	yes	characteristic
17	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
18	#16810955	C₄	1	1	yes	characteristic
19	#16810935	C₄	1	1	yes	characteristic
20	#16810933	C₄	1	1	yes	characteristic
21	#16810932	C₄	1	1	yes	characteristic
22	#16810844	C₄	1	1	yes	characteristic
23	#16810843	C₄	1	1	yes	characteristic
24	#16810780	C₄	1	1	yes	characteristic
25	#16810760	C₄	1	1	yes	characteristic
26	#16810750	C₂²	1	1	yes	characteristic
27	Q₈×#88322	C₂²	1	1	yes	characteristic
28	C₈×Dic₁₀₆	—	2	1	no	—
29	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
30	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
31	#16810747	—	2	1	no	—
32	#16810747	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
34	C₄×#11719925	C₂²	1	1	yes	characteristic
35	C₄×#11719923	C₂²	1	1	yes	characteristic
36	#16810701	C₂²	1	1	yes	characteristic
37	#16810697	—	2	1	no	—
38	C₈×C₂²⋊D₅₃	—	2	1	no	—
39	#16810681	—	2	1	no	—
40	#16810679	C₂²	1	1	yes	characteristic
41	#16810674	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×D₅₃	—	2	1	no	—
43	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
44	#16810578	—	2	1	no	—
45	C₄×#11719912	C₂²	1	1	yes	characteristic
46	C₄×C₈×D₅₃	—	2	1	no	—
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	#16810375	C₂²	1	1	yes	characteristic
49	#16810374	C₂²	1	1	yes	characteristic
50	#16810373	C₂²	1	1	yes	characteristic
51	#16810355	C₂²	1	1	yes	characteristic
52	#16810355	C₂²	1	1	yes	characteristic
53	Q₈⋊#88322	C₂²	1	1	yes	characteristic
54	#16810353	C₂²	1	1	yes	characteristic
55	#16810353	C₂²	1	1	yes	characteristic
56	D₄⋊#88322	C₂²	1	1	yes	characteristic
57	#16810345	C₂²	1	1	yes	characteristic
58	#16810345	C₂²	1	1	yes	characteristic
59	#16810345	C₂²	1	1	yes	characteristic
60	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
62	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
63	C₅₃×#335	C₂²	1	1	yes	characteristic
64	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅₃×#322	C₂²	1	1	yes	characteristic
67	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719917	—	2	1	no	—
72	C₂×#11719912	—	2	1	no	—
73	C₂×C₈×D₅₃	—	2	1	no	—
74	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
75	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
76	#13133210	—	2	1	no	—
77	#13133208	C₂×C₄	1	1	yes	characteristic
78	#13133204	C₂×C₄	1	1	yes	characteristic
79	#13133199	—	2	1	no	—
80	#13133197	C₂×C₄	1	1	yes	characteristic
81	#13133188	C₂×C₄	1	1	yes	characteristic
82	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Dic₁₀₆	—	2	1	no	—
84	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
86	#13133182	—	2	1	no	—
87	#13133182	C₂×C₄	1	1	yes	characteristic
88	C₄×C₂²⋊D₅₃	—	2	1	no	—
89	#13133176	—	2	1	no	—
90	C₄⋊C₄×D₅₃	—	2	1	no	—
91	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄²×D₅₃	—	2	1	no	—
93	C₄²×D₅₃	C₂³	1	1	yes	characteristic
94	#13133151	C₂×C₄	1	1	yes	characteristic
95	#13133150	C₂×C₄	1	1	yes	characteristic
96	#13133148	C₂×C₄	1	1	yes	characteristic
97	C₄⋊#88322	—	2	1	no	—
98	C₄⋊#88322	C₂³	1	1	yes	characteristic
99	C₄⋊#88322	C₂³	1	1	yes	characteristic
100	#13133146	—	2	1	no	—
101	#13133146	—	2	1	no	—
102	#13133146	—	2	1	no	—
103	#13133146	—	2	1	no	—
104	#13133146	—	2	1	no	—
105	#13133146	C₂×C₄	1	1	yes	characteristic
106	#13133146	C₂×C₄	1	1	yes	characteristic
107	#13133142	C₂×C₄	1	1	yes	characteristic
108	#13133142	C₂×C₄	1	1	yes	characteristic
109	#13133141	C₂×C₄	1	1	yes	characteristic
110	#13133140	C₂×C₄	1	1	yes	characteristic
111	#13133140	C₂×C₄	1	1	yes	characteristic
112	#13133138	C₂×C₄	1	1	yes	characteristic
113	C₂²⋊#88322	—	2	1	no	—
114	#13133129	—	2	1	no	—
115	#13133129	C₂×C₄	1	1	yes	characteristic
116	#13133129	C₂×C₄	1	1	yes	characteristic
117	#13133127	—	2	1	no	—
118	#13133127	—	2	1	no	—
119	#13133127	—	2	1	no	—
120	C₄×#88322	—	2	1	no	—
121	C₄×#88322	C₂³	1	1	yes	characteristic
122	C₄×#88322	C₂³	1	1	yes	characteristic
123	C₄×#88322	C₂³	1	1	yes	characteristic
124	C₈×Dic₅₃	—	2	1	no	—
125	C₈×Dic₅₃	—	2	1	no	—
126	C₈×Dic₅₃	—	2	1	no	—
127	C₈×Dic₅₃	D₄	1	1	yes	characteristic
128	C₈×Dic₅₃	D₄	1	1	yes	characteristic
129	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
130	C₂²×C₈×C₅₃	—	2	1	no	—
131	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
132	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
133	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
134	C₄⋊C₈×C₅₃	—	2	1	no	—
135	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
136	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
137	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
138	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
139	C₂²⋊C₈×C₅₃	—	2	1	no	—
140	C₄×C₈×C₅₃	—	2	1	no	—
141	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	—	2	1	no	—
147	C₂×C₂²⋊D₅₃	—	2	1	no	—
148	Q₈×D₅₃	—	2	2	no	—
149	Q₈×D₅₃	—	2	2	no	—
150	C₂×C₄×D₅₃	—	2	1	no	—
151	C₂×C₄×D₅₃	—	2	1	no	—
152	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
155	C₂×Dic₁₀₆	—	2	1	no	—
156	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
157	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
158	SD₁₆×C₅₃	—	2	2	no	—
159	C₂×C₈×C₅₃	—	2	1	no	—
160	C₂×C₈×C₅₃	—	2	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	4	2	no	—
163	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₄⋊C₄×C₅₃	—	2	1	no	—
168	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²⋊C₄×C₅₃	—	2	1	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂²⋊Dic₅₃	—	2	1	no	—
174	C₂²⋊Dic₅₃	—	2	1	no	—
175	C₂²⋊Dic₅₃	—	2	1	no	—
176	#11719925	—	2	2	no	—
177	#11719923	—	2	2	no	—
178	#11719921	—	2	1	no	—
179	C₄⋊Dic₅₃	—	2	1	no	—
180	C₄⋊Dic₅₃	—	2	1	no	—
181	C₄⋊Dic₅₃	—	2	1	no	—
182	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
183	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
184	#11719919	—	2	1	no	—
185	#11719919	—	2	1	no	—
186	#11719919	—	2	1	no	—
187	#11719919	—	2	1	no	—
188	#11719919	—	2	1	no	—
189	C₄×Dic₅₃	—	2	1	no	—
190	C₄×Dic₅₃	—	2	1	no	—
191	C₄×Dic₅₃	—	2	1	no	—
192	C₄×Dic₅₃	—	2	1	no	—
193	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
195	C₄×Dic₅₃	#41	1	1	yes	characteristic
196	#11719917	—	4	2	no	—
197	C₂×#88322	—	2	1	no	—
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
201	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
202	C₂×#88322	C₂×D₄	1	1	yes	characteristic
203	C₂×#88322	C₂×D₄	1	1	yes	characteristic
204	Dic₂₁₂	—	2	2	no	—
205	#11719912	—	4	2	no	—
206	C₈×D₅₃	—	2	2	no	—
207	C₈×D₅₃	—	4	1	no	—
208	C₈×D₅₃	—	4	2	no	—
209	C₂³×C₅₃	—	2	1	no	—
210	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
211	Q₈×C₅₃	—	2	1	no	—
212	Q₈×C₅₃	#131	1	2	yes	—
213	D₄×C₅₃	—	2	1	no	—
214	D₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	4	1	no	—
219	C₂×C₄×C₅₃	—	4	1	no	—
220	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
223	C₂²⋊D₅₃	—	4	2	no	—
224	C₂×Dic₅₃	—	2	1	no	—
225	C₂×Dic₅₃	—	2	1	no	—
226	C₂×Dic₅₃	—	2	1	no	—
227	C₂×Dic₅₃	—	2	1	no	—
228	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
229	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
230	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
231	C₄×D₅₃	—	2	4	no	—
232	C₄×D₅₃	—	2	2	no	—
233	C₄×D₅₃	—	2	1	no	—
234	C₄×D₅₃	—	2	1	no	—
235	C₄×D₅₃	—	2	1	no	—
236	C₄×D₅₃	—	2	1	no	—
237	C₄×D₅₃	—	4	2	no	—
238	Dic₁₀₆	—	2	4	no	—
239	Dic₁₀₆	—	2	1	no	—
240	Dic₁₀₆	—	2	1	no	—
241	Dic₁₀₆	—	2	1	no	—
242	Dic₁₀₆	#131	1	2	yes	—
243	C₈×C₅₃	—	2	1	no	—
244	C₈×C₅₃	—	2	1	no	—
245	C₈×C₅₃	—	2	1	no	—
246	C₈×C₅₃	—	2	1	no	—
247	C₈×C₅₃	—	4	1	no	—
248	C₈×C₅₃	—	4	1	no	—
249	C₈×C₅₃	—	4	1	no	—
250	#88322	—	2	1	no	—
251	#88322	—	2	1	no	—
252	#88322	—	2	1	no	—
253	#88322	—	4	1	no	—
254	#88322	—	4	1	no	—
255	#88322	—	4	1	no	—
256	#88322	#137	1	2	yes	—
257	#17915	—	53	1	no	maximal, Sylow
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	4	1	no	—
260	C₂²×C₅₃	—	4	1	no	—
261	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
262	C₂×D₅₃	—	2	1	no	—
263	C₂×D₅₃	—	2	1	no	—
264	C₂×D₅₃	#444	1	2	yes	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	4	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
270	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
271	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
272	C₄×C₅₃	#575	1	1	yes	characteristic
273	Dic₅₃	—	2	1	no	—
274	Dic₅₃	—	2	2	no	—
275	Dic₅₃	—	2	1	no	—
276	Dic₅₃	—	2	1	no	—
277	Dic₅₃	—	4	1	no	—
278	Dic₅₃	—	4	1	no	—
279	Dic₅₃	—	4	1	no	—
280	#3108	—	53	1	no	—
281	#2966	—	53	1	no	—
282	C₂×C₈×Q₈	—	53	1	no	—
283	#1584	—	53	1	no	—
284	#1584	—	53	1	no	—
285	#1583	—	53	1	no	—
286	#1583	—	53	1	no	—
287	#1579	—	53	1	no	—
288	#1579	—	53	1	no	—
289	SD₁₆⋊C₈	—	53	1	no	—
290	SD₁₆⋊C₈	D₅₃	1	1	yes	characteristic, complemented
291	#1573	—	53	1	no	—
292	#1491	—	53	1	no	—
293	#1483	—	53	1	no	—
294	#1451	—	53	1	no	—
295	C₂×C₅₃	—	8	1	no	—
296	C₂×C₅₃	#2929	1	1	yes	characteristic
297	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
298	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
299	D₅₃	—	2	2	no	—
300	#529	—	53	1	no	—
301	C₂×C₄×Q₈	—	53	1	no	—
302	#483	—	53	1	no	—
303	#480	—	53	1	no	—
304	#479	—	53	1	no	—
305	#479	—	53	1	no	—
306	#464	—	53	1	no	—
307	#464	—	53	1	no	—
308	#451	—	53	1	no	—
309	#448	—	53	1	no	—
310	#446	—	53	1	no	—
311	C₈×Q₈	—	53	1	no	—
312	C₈×Q₈	—	53	1	no	—
313	C₈×Q₈	—	53	1	no	—
314	C₈×Q₈	—	53	1	no	—
315	C₈×Q₈	—	106	1	no	—
316	C₈×Q₈	—	106	1	no	—
317	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
318	C₄×Q₁₆	—	53	1	no	—
319	C₄×Q₁₆	—	53	1	no	—
320	C₄×SD₁₆	—	53	1	no	—
321	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
322	#436	—	53	1	no	—
323	#435	—	106	1	no	—
324	C₈×D₄	—	106	1	no	—
325	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
326	#433	—	106	1	no	—
327	#432	—	53	1	no	—
328	#431	—	53	1	no	—
329	C₂×C₄⋊C₈	—	53	1	no	—
330	C₂×C₄⋊C₈	—	106	1	no	—
331	#405	—	106	1	no	—
332	C₄×M₁₆	—	53	1	no	—
333	C₂×C₄×C₈	—	53	1	no	—
334	C₂×C₄×C₈	—	106	1	no	—
335	#335	—	53	1	no	—
336	#335	C₂×D₅₃	1	1	yes	characteristic
337	#334	—	53	1	no	—
338	#334	—	53	1	no	—
339	Q₈⋊C₈	—	53	1	no	—
340	Q₈⋊C₈	—	53	1	no	—
341	Q₈⋊C₈	—	53	1	no	—
342	Q₈⋊C₈	—	53	1	no	—
343	Q₈⋊C₈	—	53	1	no	—
344	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
345	D₄⋊C₈	—	53	1	no	—
346	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
347	#322	—	53	1	no	—
348	#322	—	53	1	no	—
349	#322	—	53	1	no	—
350	#322	C₂×D₅₃	1	1	yes	characteristic
351	C₅₃	#17915	1	1	yes	characteristic, complemented, Sylow
352	C₂²×Q₈	—	53	1	no	—
353	C₂×Q₁₆	—	53	1	no	—
354	C₂×Q₁₆	—	53	1	no	—
355	C₂×SD₁₆	—	53	1	no	—
356	C₂×SD₁₆	C₄×D₅₃	1	1	yes	characteristic
357	C₂×M₁₆	—	106	1	no	—
358	C₂×M₁₆	—	106	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	—	53	1	no	—
361	C₂²×C₈	—	53	1	no	—
362	C₂²×C₈	—	106	1	no	—
363	#126	—	106	1	no	—
364	#125	—	53	1	no	—
365	#124	—	53	1	no	—
366	#123	—	106	1	no	—
367	C₂²⋊Q₈	—	53	1	no	—
368	#121	—	53	1	no	—
369	C₄×Q₈	—	53	1	no	—
370	C₄×Q₈	—	53	1	no	—
371	C₄×Q₈	—	53	1	no	—
372	C₄×Q₈	—	53	1	no	—
373	C₄×Q₈	—	106	1	no	—
374	C₄×Q₈	—	106	1	no	—
375	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
376	C₄×D₄	—	106	1	no	—
377	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
378	#117	—	106	1	no	—
379	C₂×C₄⋊C₄	—	53	1	no	—
380	C₂×C₄⋊C₄	—	106	1	no	—
381	C₂×C₄²	—	53	1	no	—
382	C₂×C₄²	—	106	1	no	—
383	#107	—	53	1	no	—
384	#107	—	53	1	no	—
385	#106	—	53	1	no	—
386	#106	C₄×D₅₃	1	1	yes	characteristic
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	53	1	no	—
389	C₄⋊C₈	—	53	1	no	—
390	C₄⋊C₈	—	53	1	no	—
391	C₄⋊C₈	—	53	1	no	—
392	C₄⋊C₈	—	106	1	no	—
393	C₄⋊C₈	—	106	1	no	—
394	C₄⋊C₈	—	106	1	no	—
395	C₄⋊C₈	—	106	1	no	—
396	C₄⋊C₈	—	106	1	no	—
397	C₄⋊C₈	—	106	1	no	—
398	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
399	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
400	#103	—	53	1	no	—
401	#103	—	53	1	no	—
402	#103	—	53	1	no	—
403	#103	—	53	1	no	—
404	#103	—	53	1	no	—
405	#103	C₄×D₅₃	1	1	yes	characteristic
406	#102	—	53	1	no	—
407	#102	C₄×D₅₃	1	1	yes	characteristic
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	53	1	no	—
410	C₂²⋊C₈	—	53	1	no	—
411	C₂²⋊C₈	—	106	1	no	—
412	C₂²⋊C₈	—	106	1	no	—
413	#97	—	106	1	no	—
414	#97	—	106	1	no	—
415	#97	—	106	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	—	53	1	no	—
418	C₄×C₈	—	53	1	no	—
419	C₄×C₈	—	53	1	no	—
420	C₄×C₈	—	53	1	no	—
421	C₄×C₈	—	53	1	no	—
422	C₄×C₈	—	106	1	no	—
423	C₄×C₈	—	106	1	no	—
424	C₄×C₈	—	106	1	no	—
425	C₄×C₈	—	106	1	no	—
426	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
427	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
428	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
429	C₂×Q₈	—	53	1	no	—
430	C₂×Q₈	—	53	1	no	—
431	C₂×Q₈	—	106	1	no	—
432	C₂×Q₈	—	106	2	no	—
433	C₂×Q₈	—	106	2	no	—
434	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
435	C₂×D₄	—	106	1	no	—
436	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	53	1	no	—
439	C₂²×C₄	—	53	1	no	—
440	C₂²×C₄	—	53	1	no	—
441	C₂²×C₄	—	106	1	no	—
442	C₂²×C₄	—	106	1	no	—
443	Q₁₆	—	106	2	no	—
444	Q₁₆	—	106	2	no	—
445	SD₁₆	—	2	2	no	—
446	SD₁₆	—	106	2	no	—
447	M₁₆	—	212	2	no	—
448	M₁₆	—	212	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	4	2	no	—
453	C₂×C₈	—	53	1	no	—
454	C₂×C₈	—	53	1	no	—
455	C₂×C₈	—	53	1	no	—
456	C₂×C₈	—	53	1	no	—
457	C₂×C₈	—	106	2	no	—
458	C₂×C₈	—	106	1	no	—
459	C₂×C₈	—	106	1	no	—
460	C₂×C₈	—	106	1	no	—
461	C₂×C₈	—	212	1	no	—
462	C₂×C₈	—	212	2	no	—
463	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
464	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
465	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
466	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	53	1	no	—
469	C₄⋊C₄	—	53	1	no	—
470	C₄⋊C₄	—	106	1	no	—
471	C₄⋊C₄	—	106	1	no	—
472	C₄⋊C₄	—	106	1	no	—
473	C₄⋊C₄	—	106	1	no	—
474	C₄⋊C₄	—	106	1	no	—
475	C₄⋊C₄	—	106	1	no	—
476	C₄⋊C₄	—	106	1	no	—
477	C₄⋊C₄	—	106	1	no	—
478	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
479	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	106	1	no	—
482	C₂²⋊C₄	—	106	1	no	—
483	C₂²⋊C₄	—	106	1	no	—
484	C₂²⋊C₄	—	106	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	53	1	no	—
487	C₄²	—	53	1	no	—
488	C₄²	—	53	1	no	—
489	C₄²	—	106	1	no	—
490	C₄²	—	106	1	no	—
491	C₄²	—	106	1	no	—
492	C₄²	—	106	1	no	—
493	C₄²	C₂³×D₅₃	1	1	yes	characteristic
494	C₂³	—	2	1	no	—
495	C₂³	—	53	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	53	2	no	—
498	Q₈	—	106	4	no	—
499	Q₈	—	106	1	no	—
500	Q₈	—	106	1	no	—
501	Q₈	—	106	1	no	—
502	Q₈	#13133223	1	2	yes	—
503	D₄	—	2	1	no	—
504	D₄	—	2	1	no	—
505	D₄	—	212	2	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	53	1	no	—
512	C₂×C₄	—	53	1	no	—
513	C₂×C₄	—	53	1	no	—
514	C₂×C₄	—	106	2	no	—
515	C₂×C₄	—	106	4	no	—
516	C₂×C₄	—	106	1	no	—
517	C₂×C₄	—	106	1	no	—
518	C₂×C₄	—	106	1	no	—
519	C₂×C₄	—	106	1	no	—
520	C₂×C₄	—	106	1	no	—
521	C₂×C₄	—	106	1	no	—
522	C₂×C₄	—	106	1	no	—
523	C₂×C₄	—	106	1	no	—
524	C₂×C₄	—	212	2	no	—
525	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
526	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
527	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	53	2	no	—
536	C₈	—	106	1	no	—
537	C₈	—	106	1	no	—
538	C₈	—	106	1	no	—
539	C₈	—	212	1	no	—
540	C₈	—	212	1	no	—
541	C₈	—	212	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	53	2	no	—
546	C₂²	—	106	1	no	—
547	C₂²	—	106	1	no	—
548	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
549	C₄	—	2	2	no	—
550	C₄	—	2	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	106	1	no	—
554	C₄	—	106	2	no	—
555	C₄	—	106	1	no	—
556	C₄	—	106	1	no	—
557	C₄	—	212	1	no	—
558	C₄	—	212	1	no	—
559	C₄	—	212	1	no	—
560	C₄	#131×D₅₃	1	1	yes	characteristic
561	C₄	C₂×#13133223	1	1	yes	characteristic
562	C₄	#137×D₅₃	1	1	yes	characteristic
563	C₄	#16811419	1	1	yes	characteristic
564	C₂	—	8	1	no	—
565	C₂	—	106	2	no	—
566	C₂	#20283404	1	1	yes	characteristic
567	C₂	#20283537	1	1	yes	characteristic
568	C₂	#444×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×SD₁₆⋊C₈	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#13133188	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	SD₁₆⋊C₈	D₅₃
hypocenter	C₅₃	#17915
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ SD₁₆⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₈×Q₈×D₅₃	C₂	semidirect
4. #20273564	C₂	semidirect
5. #20273562	C₂	semidirect
6. #20273561	C₂	semidirect
7. #20273559	C₂	semidirect
8. #20273535	C₂	semidirect
9. #20273532	C₂	semidirect
10. #20273527	C₂	semidirect
11. #20273499	C₂	semidirect
12. #20273160	C₂	semidirect
14. #20273003	C₂	semidirect
15. C₅₃×SD₁₆⋊C₈	C₂	semidirect
290. SD₁₆⋊C₈	D₅₃	semidirect
351. C₅₃	#17915	semidirect

Maximal subgroups

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

subgroup	index
#20284793	2
#20283687	2
C₈×Q₈×D₅₃	2
#20273564	2
#20273562	2
#20273561	2
#20273559	2
#20273535	2
#20273532	2
#20273527	2
#20273499	2
#20273160	2
#20273125	2
#20273003	2
C₅₃×SD₁₆⋊C₈	2
#17915	53