Subgroups — #27349811

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283679	C₂	1	1	yes	maximal, characteristic
2	#20283677	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
4	#20273019	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273018	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273011	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273010	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	#20272997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133223	C₄	1	2	yes	—
17	C₄×#11719945	C₂²	1	1	yes	characteristic
18	#16810752	C₄	1	2	yes	—
19	#16810751	C₄	1	2	yes	—
20	#16810699	C₄	1	2	yes	—
21	#16810698	C₄	1	2	yes	—
22	#16810697	C₄	1	2	yes	—
23	#16810696	C₄	1	2	yes	—
24	#16810693	C₄	1	2	yes	—
25	#16810692	C₄	1	2	yes	—
26	#16810676	C₂²	1	1	yes	characteristic
27	#16810674	C₂²	1	1	yes	characteristic
28	#16810674	C₂²	1	1	yes	characteristic
29	#16810674	C₂²	1	1	yes	characteristic
30	#16810674	C₂²	1	1	yes	characteristic
31	#16810578	C₄	1	2	yes	—
32	#16810577	C₄	1	2	yes	—
33	#16810577	C₄	1	2	yes	—
34	C₄×#11719917	C₂²	1	1	yes	characteristic
35	C₄×#11719917	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	C₄×#11719912	C₂²	1	1	yes	characteristic
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
45	#16810345	C₂²	1	1	yes	characteristic
46	#16810345	C₂²	1	1	yes	characteristic
47	#16810345	C₂²	1	1	yes	characteristic
48	#16810345	C₂²	1	1	yes	characteristic
49	#16810345	C₂²	1	1	yes	characteristic
50	#16810344	C₂²	1	1	yes	characteristic
51	#16810344	C₂²	1	1	yes	characteristic
52	C₈×#88322	C₂²	1	1	yes	characteristic
53	C₈×#88322	C₂²	1	1	yes	characteristic
54	C₈×#88322	C₂²	1	1	yes	characteristic
55	C₈×#88322	C₂²	1	1	yes	characteristic
56	C₅₃×#431	C₂²	1	1	yes	characteristic
57	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₅₃×#322	C₂²	1	1	yes	characteristic
60	C₅₃×#322	C₂²	1	1	yes	characteristic
61	C₈²×C₅₃	C₂²	1	1	yes	characteristic
62	C₈²×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
64	#13133223	—	2	8	no	—
65	C₂×#11719917	C₂×C₄	1	2	yes	—
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	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
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	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
77	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
78	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
79	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
83	#13133176	C₂×C₄	1	1	yes	characteristic
84	#13133174	C₂×C₄	1	1	yes	characteristic
85	#13133174	C₂×C₄	1	1	yes	characteristic
86	C₄²×D₅₃	C₂³	1	1	yes	characteristic
87	C₄²×D₅₃	C₂³	1	1	yes	characteristic
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	#13133129	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	#13133128	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	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
118	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
119	#97×C₅₃	C₂×C₄	1	2	yes	—
120	C₄×C₈×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	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
132	#11719945	—	2	4	no	—
133	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
134	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
141	M₁₆×C₅₃	—	2	4	no	—
142	C₂×C₈×C₅₃	—	2	4	no	—
143	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
149	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
154	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
155	#11719921	C₄²	1	1	yes	characteristic
156	#11719921	C₄²	1	1	yes	characteristic
157	#11719921	C₄²	1	1	yes	characteristic
158	#11719921	C₄²	1	1	yes	characteristic
159	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
160	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
161	#11719919	C₄²	1	1	yes	characteristic
162	#11719919	C₄²	1	1	yes	characteristic
163	#11719919	C₄²	1	1	yes	characteristic
164	#11719919	C₄²	1	1	yes	characteristic
165	C₄×Dic₅₃	C₂²×C₄	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	#11719917	—	2	4	no	—
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	—	2	4	no	—
183	C₈×D₅₃	—	2	4	no	—
184	C₈×D₅₃	—	2	4	no	—
185	C₈×D₅₃	—	2	4	no	—
186	C₈×D₅₃	M₁₆	1	8	yes	complemented
187	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
188	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
189	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
199	C₂²⋊D₅₃	—	2	2	no	—
200	C₂²⋊D₅₃	—	2	2	no	—
201	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
203	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
204	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
205	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
206	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
207	D₂₁₂	—	2	2	no	—
208	C₄×D₅₃	—	2	2	no	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	2	2	no	—
212	C₄×D₅₃	—	2	2	no	—
213	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
214	Dic₁₀₆	—	2	2	no	—
215	C₈×C₅₃	—	2	2	no	—
216	C₈×C₅₃	—	2	2	no	—
217	C₈×C₅₃	—	2	2	no	—
218	C₈×C₅₃	—	2	2	no	—
219	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
220	C₈×C₅₃	#131	1	4	yes	complemented
221	#88322	—	2	2	no	—
222	#88322	—	2	2	no	—
223	#88322	—	2	2	no	—
224	#88322	—	2	2	no	—
225	#88322	C₂×M₁₆	1	4	yes	—
226	#88322	#131	1	4	yes	complemented
227	#17335	—	53	1	no	maximal, Sylow
228	C₂²×C₅₃	—	2	1	no	—
229	C₂²×C₅₃	—	2	1	no	—
230	C₂²×C₅₃	—	2	1	no	—
231	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
232	C₂×D₅₃	—	2	1	no	—
233	C₂×D₅₃	—	2	1	no	—
234	C₂×D₅₃	—	2	1	no	—
235	C₂×D₅₃	—	2	1	no	—
236	C₂×D₅₃	—	2	1	no	—
237	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	—	2	1	no	—
241	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
242	C₄×C₅₃	C₂×#131	1	2	yes	—
243	C₄×C₅₃	C₂×#131	1	2	yes	—
244	Dic₅₃	—	2	1	no	—
245	Dic₅₃	—	2	1	no	—
246	Dic₅₃	—	2	1	no	—
247	Dic₅₃	—	2	1	no	—
248	Dic₅₃	—	2	1	no	—
249	Dic₅₃	C₄×M₁₆	1	2	yes	—
250	#2965	—	53	1	no	—
251	#2965	—	53	1	no	—
252	C₄×#131	—	53	1	no	—
253	#1455	—	53	1	no	—
254	#1455	—	53	1	no	—
255	#1453	—	53	1	no	—
256	#1453	—	53	1	no	—
257	#1451	—	53	1	no	—
258	#1451	—	53	1	no	—
259	#1450	—	53	1	no	—
260	C₈×M₁₆	—	53	1	no	—
261	C₈×M₁₆	—	53	1	no	—
262	C₈×M₁₆	—	53	1	no	—
263	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
264	C₂×C₈²	—	53	1	no	—
265	C₂×C₅₃	—	4	1	no	—
266	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
267	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
268	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
269	D₅₃	—	2	2	no	—
270	D₅₃	—	4	1	no	—
271	C₂×#131	—	53	2	no	—
272	C₄×#41	—	53	1	no	—
273	#446	—	53	2	no	—
274	#446	—	53	2	no	—
275	#436	—	53	2	no	—
276	#436	—	53	2	no	—
277	#435	—	53	2	no	—
278	#435	—	53	2	no	—
279	#435	—	53	2	no	—
280	#435	—	53	2	no	—
281	#431	—	53	1	no	—
282	#431	—	53	1	no	—
283	#431	—	53	1	no	—
284	#431	—	53	1	no	—
285	#431	—	53	1	no	—
286	#431	C₂×D₅₃	1	1	yes	characteristic
287	#405	—	53	2	no	—
288	#405	—	53	2	no	—
289	#405	—	53	2	no	—
290	C₄×M₁₆	—	53	1	no	—
291	C₄×M₁₆	—	53	1	no	—
292	C₄×M₁₆	—	53	1	no	—
293	C₄×M₁₆	—	53	1	no	—
294	C₄×M₁₆	—	53	1	no	—
295	C₄×M₁₆	—	53	1	no	—
296	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
297	C₂×C₄×C₈	—	53	1	no	—
298	C₂×C₄×C₈	—	53	1	no	—
299	C₂×C₄×C₈	—	53	1	no	—
300	C₂×C₄×C₈	—	53	1	no	—
301	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
302	#322	—	53	1	no	—
303	#322	—	53	1	no	—
304	#322	—	53	1	no	—
305	#322	—	53	1	no	—
306	#322	—	53	1	no	—
307	#322	—	53	1	no	—
308	#322	—	53	1	no	—
309	#322	—	53	1	no	—
310	#322	C₂×D₅₃	1	1	yes	characteristic
311	#322	C₂×D₅₃	1	1	yes	characteristic
312	C₈²	—	53	1	no	—
313	C₈²	—	53	1	no	—
314	C₈²	—	53	1	no	—
315	C₈²	—	53	1	no	—
316	C₈²	C₂×D₅₃	1	1	yes	characteristic
317	C₈²	C₂×D₅₃	1	1	yes	characteristic
318	C₅₃	#17335	1	1	yes	characteristic, complemented, Sylow
319	C₂×#41	—	53	1	no	—
320	#131	—	106	8	no	—
321	C₂×M₁₆	—	53	2	no	—
322	C₂×M₁₆	—	53	2	no	—
323	C₂×M₁₆	—	53	2	no	—
324	C₂×M₁₆	—	53	2	no	—
325	C₂×M₁₆	—	53	2	no	—
326	C₂×M₁₆	—	53	2	no	—
327	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
328	C₂²×C₈	—	53	2	no	—
329	C₂²×C₈	—	53	2	no	—
330	C₂²×C₈	—	53	2	no	—
331	C₂²×C₈	—	53	2	no	—
332	C₂²×C₈	C₄×D₅₃	1	2	yes	—
333	C₄×Q₈	—	53	1	no	—
334	C₄×Q₈	—	53	1	no	—
335	C₄×D₄	—	53	1	no	—
336	C₄×D₄	—	53	1	no	—
337	C₄×D₄	—	53	1	no	—
338	C₄×D₄	—	53	1	no	—
339	C₄×D₄	—	53	1	no	—
340	C₄×D₄	—	53	1	no	—
341	#117	—	53	1	no	—
342	#117	—	53	1	no	—
343	#117	—	53	1	no	—
344	C₂×C₄²	—	53	1	no	—
345	C₂×C₄²	—	53	1	no	—
346	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
347	C₄⋊C₈	—	53	2	no	—
348	C₄⋊C₈	—	53	2	no	—
349	C₄⋊C₈	—	53	2	no	—
350	C₄⋊C₈	—	53	2	no	—
351	C₄⋊C₈	—	53	2	no	—
352	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
353	C₂²⋊C₈	—	53	2	no	—
354	C₂²⋊C₈	—	53	2	no	—
355	C₂²⋊C₈	—	53	2	no	—
356	C₂²⋊C₈	—	53	2	no	—
357	C₂²⋊C₈	—	53	2	no	—
358	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
359	#97	—	53	2	no	—
360	#97	—	53	2	no	—
361	#97	—	53	2	no	—
362	#97	—	53	2	no	—
363	#97	—	53	2	no	—
364	#97	—	53	2	no	—
365	#97	C₄×D₅₃	1	2	yes	—
366	C₄×C₈	—	53	2	no	—
367	C₄×C₈	—	53	2	no	—
368	C₄×C₈	—	53	2	no	—
369	C₄×C₈	—	53	2	no	—
370	C₄×C₈	—	53	1	no	—
371	C₄×C₈	—	53	1	no	—
372	C₄×C₈	—	53	1	no	—
373	C₄×C₈	—	53	1	no	—
374	C₄×C₈	—	53	1	no	—
375	C₄×C₈	—	53	1	no	—
376	C₄×C₈	C₄×D₅₃	1	2	yes	—
377	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
378	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
379	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
380	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
381	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
382	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
383	#41	—	106	4	no	—
384	C₂×Q₈	—	53	1	no	—
385	C₂×D₄	—	53	1	no	—
386	C₂×D₄	—	53	1	no	—
387	C₂×D₄	—	53	1	no	—
388	C₂²×C₄	—	53	1	no	—
389	C₂²×C₄	—	53	1	no	—
390	C₂²×C₄	—	53	1	no	—
391	C₂²×C₄	—	53	1	no	—
392	C₂²×C₄	—	53	1	no	—
393	C₂²×C₄	—	53	1	no	—
394	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
397	M₁₆	—	2	4	no	—
398	M₁₆	—	106	4	no	—
399	M₁₆	—	106	4	no	—
400	M₁₆	—	106	4	no	—
401	M₁₆	—	106	4	no	—
402	M₁₆	—	106	4	no	—
403	M₁₆	—	106	4	no	—
404	C₂×C₈	—	2	4	no	—
405	C₂×C₈	—	53	2	no	—
406	C₂×C₈	—	53	2	no	—
407	C₂×C₈	—	53	8	no	—
408	C₂×C₈	—	53	2	no	—
409	C₂×C₈	—	53	2	no	—
410	C₂×C₈	—	53	2	no	—
411	C₂×C₈	—	53	2	no	—
412	C₂×C₈	—	106	4	no	—
413	C₂×C₈	—	106	4	no	—
414	C₂×C₈	—	106	4	no	—
415	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
416	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
417	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
418	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
419	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
420	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
421	C₄⋊C₄	—	53	1	no	—
422	C₄⋊C₄	—	53	1	no	—
423	C₄⋊C₄	—	53	1	no	—
424	C₄⋊C₄	—	53	1	no	—
425	C₄⋊C₄	—	53	1	no	—
426	C₄⋊C₄	—	53	1	no	—
427	C₂²⋊C₄	—	53	1	no	—
428	C₂²⋊C₄	—	53	1	no	—
429	C₂²⋊C₄	—	53	1	no	—
430	C₂²⋊C₄	—	53	1	no	—
431	C₂²⋊C₄	—	53	1	no	—
432	C₂²⋊C₄	—	53	1	no	—
433	C₄²	—	53	1	no	—
434	C₄²	—	53	1	no	—
435	C₄²	—	53	1	no	—
436	C₄²	—	53	1	no	—
437	C₄²	—	53	1	no	—
438	C₄²	—	53	1	no	—
439	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
440	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
441	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
442	C₄²	C₂³×D₅₃	1	1	yes	characteristic
443	C₂³	—	53	1	no	—
444	C₂³	—	53	1	no	—
445	C₂³	C₄²×D₅₃	1	1	yes	characteristic
446	Q₈	—	106	2	no	—
447	D₄	—	106	2	no	—
448	D₄	—	106	2	no	—
449	D₄	—	106	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	53	1	no	—
454	C₂×C₄	—	53	4	no	—
455	C₂×C₄	—	53	1	no	—
456	C₂×C₄	—	53	1	no	—
457	C₂×C₄	—	53	1	no	—
458	C₂×C₄	—	53	1	no	—
459	C₂×C₄	—	53	1	no	—
460	C₂×C₄	—	106	2	no	—
461	C₂×C₄	—	106	2	no	—
462	C₂×C₄	—	106	2	no	—
463	C₂×C₄	—	106	2	no	—
464	C₂×C₄	—	106	2	no	—
465	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
467	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
468	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
471	C₈	—	2	2	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	2	2	no	—
475	C₈	—	53	4	no	—
476	C₈	—	53	4	no	—
477	C₈	—	106	2	no	—
478	C₈	—	106	2	no	—
479	C₈	—	106	2	no	—
480	C₈	—	106	2	no	—
481	C₈	M₁₆×D₅₃	1	4	yes	—
482	C₈	#13133223	1	4	yes	complemented
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	53	2	no	—
487	C₂²	—	106	1	no	—
488	C₂²	—	106	1	no	—
489	C₂²	—	106	1	no	—
490	C₂²	—	106	1	no	—
491	C₂²	—	106	1	no	—
492	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	53	2	no	—
497	C₄	—	106	1	no	—
498	C₄	—	106	1	no	—
499	C₄	—	106	1	no	—
500	C₄	—	106	1	no	—
501	C₄	—	106	1	no	—
502	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
503	C₄	#131×D₅₃	1	2	yes	—
504	C₄	C₂×#13133223	1	2	yes	—
505	C₂	—	4	1	no	—
506	C₂	—	106	2	no	—
507	C₂	—	212	1	no	—
508	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
509	C₂	C₄×#13133223	1	1	yes	characteristic
510	C₂	#20283137	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#17335
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20273019	C₂	semidirect
5. #20273018	C₂	semidirect
6. #20273011	C₂	semidirect
7. #20273010	C₂	semidirect
8. #20273003	C₂	semidirect
9. #20273001	C₂	semidirect
10. #20272997	C₂	semidirect
11. C₈×#11719917	C₂	semidirect
12. C₈×#11719912	C₂	semidirect
13. C₈×#11719912	C₂	semidirect
14. C₈²×D₅₃	C₂	semidirect
15. C₈×M₁₆×C₅₃	C₂	semidirect
186. C₈×D₅₃	M₁₆	semidirect
220. C₈×C₅₃	#131	semidirect
226. #88322	#131	semidirect
263. C₈×M₁₆	D₅₃	semidirect
318. C₅₃	#17335	semidirect
482. C₈	#13133223	semidirect

Maximal subgroups

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

subgroup	index
#20283679	2
#20283677	2
C₄×#13133223	2
#20273019	2
#20273018	2
#20273011	2
#20273010	2
#20273003	2
#20273001	2
#20272997	2
C₈×#11719917	2
C₈×#11719912	2
C₈²×D₅₃	2
C₈×M₁₆×C₅₃	2
#17335	53