Subgroups — #27349746

Order 13568.

836 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283343	C₂	1	1	yes	maximal, characteristic
2	#431×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273012	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273010	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273010	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273003	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273003	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273001	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273001	C₂	1	1	yes	maximal, characteristic, complemented
11	#20272994	C₂	1	1	yes	maximal, characteristic, complemented
12	#20272994	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	#16810679	C₄	1	2	yes	—
19	#16810677	C₄	1	2	yes	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810675	C₄	1	2	yes	—
23	#16810674	C₂²	1	1	yes	characteristic
24	#16810674	C₂²	1	1	yes	characteristic
25	#16810674	C₂²	1	1	yes	characteristic
26	#16810674	C₂²	1	1	yes	characteristic
27	#16810673	C₄	1	2	yes	—
28	#16810645	C₄	1	2	yes	—
29	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
30	#16810585	C₄	1	2	yes	—
31	C₂²⋊C₈×D₅₃	C₄	1	2	yes	—
32	C₄×#11719917	C₂²	1	1	yes	characteristic
33	#16810575	C₄	1	2	yes	—
34	M₁₆×Dic₅₃	C₄	1	2	yes	—
35	C₄×#11719912	C₂²	1	1	yes	characteristic
36	C₄×#11719912	C₂²	1	1	yes	characteristic
37	C₄×#11719912	C₂²	1	1	yes	characteristic
38	C₄×#11719912	C₂²	1	1	yes	characteristic
39	#97×D₅₃	C₄	1	2	yes	—
40	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
44	#16810345	C₂²	1	1	yes	characteristic, complemented
45	#16810345	C₂²	1	1	yes	characteristic, complemented
46	#16810345	C₂²	1	1	yes	characteristic, complemented
47	#16810345	C₂²	1	1	yes	characteristic, complemented
48	#16810345	C₂²	1	1	yes	characteristic, complemented
49	#16810345	C₂²	1	1	yes	characteristic, complemented
50	#16810344	C₂²	1	1	yes	characteristic, complemented
51	#16810344	C₂²	1	1	yes	characteristic, complemented
52	#16810344	C₂²	1	1	yes	characteristic, complemented
53	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
54	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
55	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
56	C₅₃×#431	C₂²	1	1	yes	characteristic
57	C₅₃×#431	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
59	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
60	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
61	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
62	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
63	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×#11719917	C₂×C₄	1	2	yes	—
67	C₂×#11719912	C₂×C₄	1	2	yes	—
68	C₂×#11719912	C₂×C₄	1	2	yes	—
69	C₂×#11719912	C₂×C₄	1	2	yes	—
70	C₂×#11719912	C₂×C₄	1	2	yes	—
71	M₁₆×D₅₃	—	2	8	no	—
72	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
74	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
75	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
76	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄²×D₅₃	C₂³	1	1	yes	characteristic
86	C₄²×D₅₃	C₂³	1	1	yes	characteristic
87	C₄⋊#88322	C₂×C₄	1	2	yes	—
88	C₄⋊#88322	C₂×C₄	1	2	yes	—
89	#13133146	C₂×C₄	1	2	yes	—
90	#13133146	C₂×C₄	1	2	yes	—
91	#13133146	C₂×C₄	1	2	yes	—
92	#13133146	C₂×C₄	1	2	yes	—
93	C₂²⋊#88322	C₂×C₄	1	2	yes	—
94	C₂²⋊#88322	C₂×C₄	1	2	yes	—
95	#13133129	C₂×C₄	1	2	yes	—
96	#13133129	C₂×C₄	1	2	yes	—
97	#13133129	C₂×C₄	1	2	yes	—
98	#13133129	C₂×C₄	1	2	yes	—
99	#13133128	C₂×C₄	1	2	yes	—
100	#13133127	C₂×C₄	1	2	yes	—
101	#13133127	C₂×C₄	1	2	yes	—
102	#13133127	C₂×C₄	1	2	yes	—
103	#13133127	C₂×C₄	1	2	yes	—
104	C₄×#88322	C₂³	1	1	yes	characteristic
105	C₄×#88322	C₂³	1	1	yes	characteristic
106	C₄×#88322	C₂³	1	1	yes	characteristic
107	C₄×#88322	C₂³	1	1	yes	characteristic
108	C₄×#88322	C₂³	1	1	yes	characteristic
109	C₄×#88322	C₂³	1	1	yes	characteristic
110	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
111	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
112	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
113	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
114	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
117	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
118	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
119	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
120	#97×C₅₃	C₂×C₄	1	2	yes	—
121	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
122	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
123	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂³×D₅₃	C₄²	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
132	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
134	C₂×C₄×D₅₃	—	2	4	no	—
135	C₂×C₄×D₅₃	—	2	4	no	—
136	C₂×C₄×D₅₃	—	4	2	no	—
137	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
138	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
139	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
140	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
141	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
142	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
143	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
149	M₁₆×C₅₃	—	2	4	no	—
150	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
151	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
153	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
156	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
161	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
162	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
163	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
164	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
165	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
166	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
172	#11719917	—	2	4	no	—
173	C₂×#88322	C₂²×C₄	1	2	yes	—
174	C₂×#88322	C₂²×C₄	1	2	yes	—
175	C₂×#88322	C₂²×C₄	1	2	yes	—
176	C₂×#88322	C₂²×C₄	1	2	yes	—
177	C₂×#88322	C₂²×C₄	1	2	yes	—
178	C₂×#88322	C₂²×C₄	1	2	yes	—
179	#11719912	—	2	4	no	—
180	#11719912	—	2	4	no	—
181	#11719912	—	2	4	no	—
182	#11719912	M₁₆	1	8	yes	complemented
183	C₈×D₅₃	—	2	4	no	—
184	C₈×D₅₃	—	2	4	no	—
185	C₈×D₅₃	—	4	2	no	—
186	C₈×D₅₃	—	4	2	no	—
187	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
188	C₂²×D₅₃	—	2	2	no	—
189	C₂²×D₅₃	—	2	2	no	—
190	C₂²×D₅₃	—	4	1	no	—
191	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
192	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
202	C₂×Dic₅₃	—	2	2	no	—
203	C₂×Dic₅₃	—	2	2	no	—
204	C₂×Dic₅₃	—	4	1	no	—
205	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
206	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
207	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
208	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
209	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
210	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
211	C₄×D₅₃	—	2	2	no	—
212	C₄×D₅₃	—	2	2	no	—
213	C₄×D₅₃	—	2	2	no	—
214	C₄×D₅₃	—	2	2	no	—
215	C₄×D₅₃	—	2	2	no	—
216	C₄×D₅₃	—	2	2	no	—
217	C₄×D₅₃	—	2	2	no	—
218	C₄×D₅₃	—	2	2	no	—
219	C₄×D₅₃	—	4	1	no	—
220	C₄×D₅₃	—	4	1	no	—
221	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
222	C₄×D₅₃	#131	1	4	yes	—
223	C₈×C₅₃	—	2	2	no	—
224	C₈×C₅₃	—	2	2	no	—
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	—	2	2	no	—
227	C₈×C₅₃	—	2	2	no	—
228	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
229	#88322	—	2	2	no	—
230	#88322	—	2	2	no	—
231	#88322	—	2	2	no	—
232	#88322	—	2	2	no	—
233	#88322	—	2	2	no	—
234	#88322	C₂×M₁₆	1	4	yes	—
235	#17327	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	1	no	—
239	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
240	C₂×D₅₃	—	2	1	no	—
241	C₂×D₅₃	—	2	1	no	—
242	C₂×D₅₃	—	2	1	no	—
243	C₂×D₅₃	—	2	1	no	—
244	C₂×D₅₃	—	4	4	no	—
245	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
246	C₂×D₅₃	#405	1	2	yes	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
251	C₄×C₅₃	C₂×#131	1	2	yes	—
252	C₄×C₅₃	#568	1	2	yes	—
253	Dic₅₃	—	2	1	no	—
254	Dic₅₃	—	2	1	no	—
255	Dic₅₃	—	2	1	no	—
256	Dic₅₃	—	2	1	no	—
257	Dic₅₃	C₄×M₁₆	1	2	yes	—
258	Dic₅₃	#405	1	2	yes	—
259	#2920	—	53	1	no	—
260	C₂×#431	—	53	1	no	—
261	C₂×C₄×M₁₆	—	53	1	no	—
262	#1453	—	53	1	no	—
263	#1453	—	53	1	no	—
264	#1453	—	53	1	no	—
265	#1453	D₅₃	1	1	yes	characteristic, complemented
266	#1451	—	53	1	no	—
267	#1451	—	53	1	no	—
268	#1451	—	53	1	no	—
269	#1451	—	53	1	no	—
270	C₈×M₁₆	—	53	1	no	—
271	C₈×M₁₆	—	53	1	no	—
272	C₈×M₁₆	—	53	1	no	—
273	C₈×M₁₆	—	53	1	no	—
274	C₂×C₅₃	—	4	1	no	—
275	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
276	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
277	C₂×C₅₃	#2873	1	1	yes	characteristic
278	D₅₃	—	2	2	no	—
279	D₅₃	—	2	2	no	—
280	C₂²×M₁₆	—	53	2	no	—
281	C₂²×C₄²	—	53	1	no	—
282	#432	—	53	2	no	—
283	#432	—	53	2	no	—
284	#431	—	53	1	no	—
285	#431	—	53	1	no	—
286	#431	—	53	2	no	—
287	#431	—	53	2	no	—
288	#431	—	53	1	no	—
289	#431	—	53	1	no	—
290	#431	—	53	1	no	—
291	#431	—	53	1	no	—
292	#431	C₂×D₅₃	1	1	yes	characteristic
293	#431	C₂×D₅₃	1	1	yes	characteristic
294	#423	—	53	2	no	—
295	C₂×C₄⋊C₈	—	53	2	no	—
296	#407	—	53	2	no	—
297	C₂×C₂²⋊C₈	—	53	2	no	—
298	C₄×M₁₆	—	53	2	no	—
299	C₄×M₁₆	—	53	1	no	—
300	C₄×M₁₆	—	53	2	no	—
301	C₄×M₁₆	—	53	1	no	—
302	C₄×M₁₆	—	53	1	no	—
303	C₄×M₁₆	—	53	1	no	—
304	C₄×M₁₆	—	53	1	no	—
305	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
306	C₂×#97	—	53	2	no	—
307	C₂×C₄×C₈	—	53	1	no	—
308	C₂×C₄×C₈	—	53	1	no	—
309	C₂×C₄×C₈	—	53	1	no	—
310	C₂×C₄×C₈	—	53	1	no	—
311	#322	—	53	1	no	—
312	#322	—	53	1	no	—
313	#322	—	53	1	no	—
314	#322	—	53	1	no	—
315	#322	—	53	1	no	—
316	#322	—	53	1	no	—
317	#322	—	53	1	no	—
318	#322	—	53	1	no	—
319	#322	—	53	1	no	—
320	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
321	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
322	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
323	C₈²	—	53	1	no	—
324	C₈²	—	53	1	no	—
325	C₈²	—	53	1	no	—
326	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
327	C₅₃	#17327	1	1	yes	characteristic, complemented, Sylow
328	C₂³×C₄	—	53	1	no	—
329	C₂³×C₄	—	53	1	no	—
330	C₂³×C₄	—	53	1	no	—
331	C₂×M₁₆	—	53	2	no	—
332	C₂×M₁₆	—	53	2	no	—
333	C₂×M₁₆	—	53	2	no	—
334	C₂×M₁₆	—	53	2	no	—
335	C₂×M₁₆	—	53	2	no	—
336	C₂×M₁₆	—	106	8	no	—
337	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
338	C₂²×C₈	—	53	2	no	—
339	C₂²×C₈	—	53	2	no	—
340	C₂²×C₈	—	53	2	no	—
341	C₂²×C₈	—	53	2	no	—
342	C₂×C₄²	—	53	1	no	—
343	C₂×C₄²	—	53	1	no	—
344	C₂×C₄²	—	53	1	no	—
345	C₂×C₄²	—	53	1	no	—
346	C₂×C₄²	—	53	1	no	—
347	C₂×C₄²	—	53	1	no	—
348	C₂×C₄²	—	53	1	no	—
349	C₂×C₄²	—	53	1	no	—
350	C₂×C₄²	—	53	1	no	—
351	C₂×C₄²	—	53	1	no	—
352	C₂×C₄²	—	53	1	no	—
353	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
354	C₄⋊C₈	—	53	2	no	—
355	C₄⋊C₈	—	53	2	no	—
356	C₄⋊C₈	—	53	2	no	—
357	C₄⋊C₈	—	53	2	no	—
358	C₄⋊C₈	—	53	2	no	—
359	C₄⋊C₈	—	53	2	no	—
360	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
361	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
362	C₂²⋊C₈	—	53	2	no	—
363	C₂²⋊C₈	—	53	2	no	—
364	C₂²⋊C₈	—	53	2	no	—
365	C₂²⋊C₈	—	53	2	no	—
366	C₂²⋊C₈	—	53	2	no	—
367	C₂²⋊C₈	—	53	2	no	—
368	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
369	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
370	#97	—	53	2	no	—
371	#97	—	53	2	no	—
372	#97	—	53	2	no	—
373	#97	—	53	2	no	—
374	#97	—	53	2	no	—
375	#97	C₄×D₅₃	1	2	yes	—
376	C₄×C₈	—	53	2	no	—
377	C₄×C₈	—	53	2	no	—
378	C₄×C₈	—	53	2	no	—
379	C₄×C₈	—	53	2	no	—
380	C₄×C₈	—	53	1	no	—
381	C₄×C₈	—	53	1	no	—
382	C₄×C₈	—	53	1	no	—
383	C₄×C₈	—	53	1	no	—
384	C₄×C₈	—	53	1	no	—
385	C₄×C₈	—	53	1	no	—
386	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
387	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
388	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
389	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
390	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
391	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
392	C₂⁴	—	53	1	no	—
393	C₂²×C₄	—	53	1	no	—
394	C₂²×C₄	—	53	1	no	—
395	C₂²×C₄	—	53	1	no	—
396	C₂²×C₄	—	53	1	no	—
397	C₂²×C₄	—	53	1	no	—
398	C₂²×C₄	—	53	1	no	—
399	C₂²×C₄	—	53	1	no	—
400	C₂²×C₄	—	53	1	no	—
401	C₂²×C₄	—	53	1	no	—
402	C₂²×C₄	—	53	1	no	—
403	C₂²×C₄	—	53	1	no	—
404	C₂²×C₄	—	53	1	no	—
405	C₂²×C₄	—	53	1	no	—
406	C₂²×C₄	—	53	1	no	—
407	C₂²×C₄	—	53	1	no	—
408	C₂²×C₄	—	106	4	no	—
409	C₂²×C₄	—	106	4	no	—
410	C₂²×C₄	—	212	2	no	—
411	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
412	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
413	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
414	M₁₆	—	2	4	no	—
415	M₁₆	—	53	8	no	—
416	M₁₆	—	106	4	no	—
417	M₁₆	—	106	4	no	—
418	M₁₆	—	106	4	no	—
419	M₁₆	—	106	4	no	—
420	C₂×C₈	—	53	2	no	—
421	C₂×C₈	—	53	2	no	—
422	C₂×C₈	—	53	2	no	—
423	C₂×C₈	—	53	2	no	—
424	C₂×C₈	—	53	2	no	—
425	C₂×C₈	—	53	2	no	—
426	C₂×C₈	—	106	4	no	—
427	C₂×C₈	—	106	4	no	—
428	C₂×C₈	—	212	2	no	—
429	C₂×C₈	—	212	2	no	—
430	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
431	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
432	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
433	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
434	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
435	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
436	C₄²	—	53	1	no	—
437	C₄²	—	53	1	no	—
438	C₄²	—	53	1	no	—
439	C₄²	—	53	1	no	—
440	C₄²	—	53	1	no	—
441	C₄²	—	53	1	no	—
442	C₄²	—	53	1	no	—
443	C₄²	—	53	1	no	—
444	C₄²	—	53	1	no	—
445	C₄²	—	53	1	no	—
446	C₄²	—	53	1	no	—
447	C₄²	—	53	1	no	—
448	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
449	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
450	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
451	C₄²	C₂³×D₅₃	1	1	yes	characteristic
452	C₂³	—	53	1	no	—
453	C₂³	—	53	1	no	—
454	C₂³	—	106	2	no	—
455	C₂³	—	106	2	no	—
456	C₂³	—	212	1	no	—
457	C₂³	C₄²×D₅₃	1	1	yes	characteristic
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	53	1	no	—
462	C₂×C₄	—	53	1	no	—
463	C₂×C₄	—	53	4	no	—
464	C₂×C₄	—	53	4	no	—
465	C₂×C₄	—	53	1	no	—
466	C₂×C₄	—	53	1	no	—
467	C₂×C₄	—	53	1	no	—
468	C₂×C₄	—	53	1	no	—
469	C₂×C₄	—	106	2	no	—
470	C₂×C₄	—	106	2	no	—
471	C₂×C₄	—	106	2	no	—
472	C₂×C₄	—	106	2	no	—
473	C₂×C₄	—	106	2	no	—
474	C₂×C₄	—	106	2	no	—
475	C₂×C₄	—	106	2	no	—
476	C₂×C₄	—	106	2	no	—
477	C₂×C₄	—	106	2	no	—
478	C₂×C₄	—	106	2	no	—
479	C₂×C₄	—	212	1	no	—
480	C₂×C₄	—	212	1	no	—
481	C₂×C₄	—	212	1	no	—
482	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
483	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
484	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
487	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
488	C₈	—	2	2	no	—
489	C₈	—	2	2	no	—
490	C₈	—	2	2	no	—
491	C₈	—	2	2	no	—
492	C₈	—	2	2	no	—
493	C₈	—	53	4	no	—
494	C₈	—	106	2	no	—
495	C₈	—	106	2	no	—
496	C₈	—	106	2	no	—
497	C₈	—	106	2	no	—
498	C₈	—	106	2	no	—
499	C₈	M₁₆×D₅₃	1	4	yes	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	53	2	no	—
504	C₂²	—	53	2	no	—
505	C₂²	—	106	1	no	—
506	C₂²	—	106	1	no	—
507	C₂²	—	106	1	no	—
508	C₂²	—	106	1	no	—
509	C₂²	—	212	4	no	—
510	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	53	2	no	—
515	C₄	—	53	2	no	—
516	C₄	—	106	1	no	—
517	C₄	—	106	1	no	—
518	C₄	—	106	1	no	—
519	C₄	—	106	1	no	—
520	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
521	C₄	#131×D₅₃	1	2	yes	—
522	C₄	#16811388	1	2	yes	—
523	C₂	—	4	1	no	—
524	C₂	—	106	2	no	—
525	C₂	—	106	2	no	—
526	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
527	C₂	#405×D₅₃	1	1	yes	characteristic
528	C₂	#20283131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1453	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1453	D₅₃
hypocenter	C₅₃	#17327
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1453

Complements

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

N	H ≅ G/N	type
4. #20273012	C₂	semidirect
5. #20273010	C₂	semidirect
6. #20273010	C₂	semidirect
7. #20273003	C₂	semidirect
8. #20273003	C₂	semidirect
9. #20273001	C₂	semidirect
10. #20273001	C₂	semidirect
11. #20272994	C₂	semidirect
12. #20272994	C₂	semidirect
13. C₈×#11719912	C₂	semidirect
14. C₈×#11719912	C₂	semidirect
15. C₅₃×#1453	C₂	semidirect
44. #16810345	C₂²	semidirect
45. #16810345	C₂²	semidirect
46. #16810345	C₂²	semidirect
47. #16810345	C₂²	semidirect
48. #16810345	C₂²	semidirect
49. #16810345	C₂²	semidirect
50. #16810344	C₂²	semidirect
51. #16810344	C₂²	semidirect
52. #16810344	C₂²	semidirect
53. C₈×#88322	C₂²	semidirect
54. C₈×#88322	C₂²	semidirect
55. C₈×#88322	C₂²	semidirect
59. C₅₃×#322	C₂²	semidirect
60. C₅₃×#322	C₂²	semidirect
61. C₅₃×#322	C₂²	semidirect
62. C₈²×C₅₃	C₂²	semidirect
182. #11719912	M₁₆	semidirect
265. #1453	D₅₃	semidirect
320. #322	C₂×D₅₃	semidirect
321. #322	C₂×D₅₃	semidirect
322. #322	C₂×D₅₃	semidirect
326. C₈²	C₂×D₅₃	semidirect
327. C₅₃	#17327	semidirect

Maximal subgroups

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

subgroup	index
#20283343	2
#431×D₅₃	2
C₄×M₁₆×D₅₃	2
#20273012	2
#20273010	2
#20273003	2
#20273001	2
#20272994	2
C₈×#11719912	2
C₅₃×#1453	2
#17327	53