Subgroups — #27353117

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 514 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	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273481	C₂	1	2	yes	maximal, complemented
5	#20273479	C₂	1	2	yes	maximal, complemented
6	#20273472	C₂	1	2	yes	maximal, complemented
7	#20273469	C₂	1	2	yes	maximal, complemented
8	#20273468	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273465	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719912	—	2	1	no	—
13	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
14	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
16	#16810679	C₂²	1	1	yes	characteristic
17	#16810679	C₂²	1	1	yes	characteristic
18	#16810677	—	2	1	no	—
19	#16810676	C₂²	1	1	yes	characteristic
20	#16810674	—	2	1	no	—
21	#16810645	—	2	1	no	—
22	#16810645	—	2	1	no	—
23	#16810645	C₂²	1	1	yes	characteristic
24	#16810645	C₂²	1	2	yes	—
25	#16810645	C₂²	1	1	yes	characteristic
26	#16810645	C₂²	1	2	yes	—
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133146	—	2	1	no	—
29	#16810585	—	2	1	no	—
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	#16810575	C₄	1	2	yes	—
32	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
33	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
34	C₄×#11719912	—	2	1	no	—
35	C₄×#11719912	C₂²	1	2	yes	—
36	#97×D₅₃	C₄	1	2	yes	—
37	C₄×C₈×D₅₃	C₂²	1	2	yes	—
38	#16810376	C₂²	1	2	yes	complemented
39	#16810375	C₂²	1	2	yes	complemented
40	#16810375	C₂²	1	2	yes	complemented
41	#16810374	C₂²	1	2	yes	complemented
42	#16810373	C₂²	1	2	yes	complemented
43	#16810373	C₂²	1	2	yes	complemented
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#335	C₂²	1	2	yes	complemented
48	C₅₃×#334	C₂²	1	2	yes	complemented
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719912	—	2	1	no	—
55	C₂×#11719912	—	2	1	no	—
56	C₂×#11719912	—	2	2	no	—
57	C₂×#11719912	—	2	2	no	—
58	C₂×#11719912	—	4	4	no	—
59	C₂×#11719912	C₂×C₄	1	2	yes	—
60	C₂×#11719912	C₂×C₄	1	2	yes	—
61	M₁₆×D₅₃	—	2	4	no	—
62	M₁₆×D₅₃	—	2	4	no	—
63	C₂²×#88322	—	2	1	no	—
64	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
68	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
69	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×D₅₃	D₄	1	1	yes	characteristic
72	C₄²×D₅₃	D₄	1	1	yes	characteristic
73	C₄²×D₅₃	Q₈	1	1	yes	characteristic
74	C₄²×D₅₃	Q₈	1	1	yes	characteristic
75	C₄²×D₅₃	C₂³	1	1	yes	characteristic
76	C₄²×D₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	—	2	1	no	—
78	C₄⋊#88322	C₂³	1	1	yes	characteristic
79	C₄⋊#88322	C₂³	1	2	yes	—
80	C₄⋊#88322	C₂³	1	1	yes	characteristic
81	#13133146	—	2	2	no	—
82	#13133146	—	2	2	no	—
83	#13133146	—	2	1	no	—
84	#13133146	—	2	1	no	—
85	C₂²⋊#88322	—	2	1	no	—
86	#13133129	—	2	1	no	—
87	#13133129	—	2	1	no	—
88	#13133128	C₂×C₄	1	2	yes	—
89	#13133127	—	2	1	no	—
90	#13133127	—	2	1	no	—
91	#13133127	C₂×C₄	1	2	yes	—
92	#13133127	C₂×C₄	1	2	yes	—
93	C₄×#88322	—	2	1	no	—
94	C₄×#88322	C₂³	1	2	yes	—
95	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
96	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
97	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	—	2	1	no	—
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	—	2	1	no	—
102	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
104	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₈×C₅₃	—	2	1	no	—
106	#97×C₅₃	C₂×C₄	1	2	yes	—
107	C₄×C₈×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	—	2	2	no	—
117	C₂×C₄×D₅₃	—	2	2	no	—
118	C₂×C₄×D₅₃	—	2	2	no	—
119	C₂×C₄×D₅₃	—	2	2	no	—
120	C₂×C₄×D₅₃	—	4	2	no	—
121	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
132	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
133	M₁₆×C₅₃	—	2	2	no	—
134	M₁₆×C₅₃	M₁₆	1	4	yes	complemented
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	4	2	no	—
139	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
141	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
144	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
145	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
157	#11719917	—	2	2	no	—
158	#11719917	M₁₆	1	4	yes	complemented
159	C₂×#88322	—	2	1	no	—
160	C₂×#88322	—	2	1	no	—
161	C₂×#88322	—	2	2	no	—
162	C₂×#88322	—	4	2	no	—
163	C₂×#88322	C₂²×C₄	1	2	yes	—
164	C₂×#88322	C₂²×C₄	1	2	yes	—
165	#11719912	—	2	4	no	—
166	#11719912	—	2	4	no	—
167	#11719912	—	4	4	no	—
168	#11719912	—	4	2	no	—
169	#11719912	—	4	4	no	—
170	#11719912	—	4	2	no	—
171	C₈×D₅₃	—	2	4	no	—
172	C₈×D₅₃	—	2	4	no	—
173	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×D₅₃	—	2	2	no	—
175	C₂²×D₅₃	—	2	2	no	—
176	C₂²×D₅₃	—	4	1	no	—
177	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
187	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
190	C₂×Dic₅₃	—	2	2	no	—
191	C₂×Dic₅₃	—	2	2	no	—
192	C₂×Dic₅₃	—	2	2	no	—
193	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₄×D₅₃	—	2	2	no	—
200	C₄×D₅₃	—	2	1	no	—
201	C₄×D₅₃	—	2	1	no	—
202	C₄×D₅₃	—	2	1	no	—
203	C₄×D₅₃	—	2	1	no	—
204	C₄×D₅₃	—	2	1	no	—
205	C₄×D₅₃	—	2	1	no	—
206	C₄×D₅₃	—	2	2	no	—
207	C₄×D₅₃	—	2	2	no	—
208	C₄×D₅₃	—	2	2	no	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	4	1	no	—
212	C₄×D₅₃	—	4	1	no	—
213	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
214	C₄×D₅₃	#137	1	2	yes	—
215	C₈×C₅₃	—	2	2	no	—
216	C₈×C₅₃	—	4	2	no	—
217	C₈×C₅₃	—	4	1	no	—
218	C₈×C₅₃	—	4	1	no	—
219	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
220	#88322	—	2	2	no	—
221	#88322	—	4	2	no	—
222	#88322	—	4	1	no	—
223	#88322	—	4	1	no	—
224	#88322	C₂×M₁₆	1	4	yes	complemented
225	#17796	—	53	1	no	maximal, Sylow
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	—	2	1	no	—
228	C₂²×C₅₃	#423	1	2	yes	—
229	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
230	C₂×D₅₃	—	2	1	no	—
231	C₂×D₅₃	—	2	1	no	—
232	C₂×D₅₃	—	2	1	no	—
233	C₂×D₅₃	—	2	1	no	—
234	C₂×D₅₃	—	2	1	no	—
235	C₂×D₅₃	—	4	4	no	—
236	C₂×D₅₃	#428	1	2	yes	—
237	C₄×C₅₃	—	2	1	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	#423	1	2	yes	—
241	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
242	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
243	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
245	Dic₅₃	—	2	1	no	—
246	Dic₅₃	—	2	1	no	—
247	Dic₅₃	—	2	1	no	—
248	Dic₅₃	—	2	1	no	—
249	Dic₅₃	#428	1	2	yes	—
250	Dic₅₃	#430	1	2	yes	—
251	#2920	—	53	1	no	—
252	C₂×#423	—	53	1	no	—
253	C₂×C₄×M₁₆	—	53	1	no	—
254	#1569	—	53	2	no	—
255	#1569	—	53	2	no	—
256	#1567	—	53	2	no	—
257	#1566	—	53	2	no	—
258	#1565	—	53	1	no	—
259	#1565	—	53	1	no	—
260	#1565	—	53	1	no	—
261	#1565	D₅₃	1	1	yes	characteristic, complemented
262	C₂×C₅₃	—	2	2	no	—
263	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
264	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
265	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
266	D₅₃	—	2	2	no	—
267	D₅₃	—	4	1	no	—
268	C₂²×M₁₆	—	53	1	no	—
269	C₂²×M₁₆	—	53	1	no	—
270	C₂²×M₁₆	—	106	1	no	—
271	C₂²×C₄²	—	53	1	no	—
272	#432	—	53	1	no	—
273	#432	—	53	1	no	—
274	#432	—	106	1	no	—
275	#431	—	53	1	no	—
276	#431	—	106	1	no	—
277	#431	C₂×D₅₃	1	1	yes	characteristic
278	#423	—	53	1	no	—
279	#423	—	53	2	no	—
280	#423	—	53	1	no	—
281	#423	—	53	2	no	—
282	#423	—	106	1	no	—
283	#423	—	106	1	no	—
284	C₂×C₄⋊C₈	—	53	1	no	—
285	C₂×C₄⋊C₈	—	106	1	no	—
286	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
287	#407	—	106	1	no	—
288	C₄×M₁₆	—	53	1	no	—
289	C₄×M₁₆	—	53	1	no	—
290	C₄×M₁₆	—	53	1	no	—
291	C₄×M₁₆	—	53	2	no	—
292	C₄×M₁₆	—	53	2	no	—
293	C₄×M₁₆	—	106	1	no	—
294	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
295	C₂×#97	—	53	2	no	—
296	C₂×C₄×C₈	—	53	2	no	—
297	#335	—	53	2	no	—
298	#335	—	53	2	no	—
299	#335	—	53	2	no	—
300	#335	C₂×D₅₃	1	2	yes	complemented
301	#334	—	53	2	no	—
302	#334	—	53	2	no	—
303	#334	—	53	2	no	—
304	#334	C₂×D₅₃	1	2	yes	complemented
305	C₅₃	#17796	1	1	yes	characteristic, complemented, Sylow
306	C₂³×C₄	—	53	1	no	—
307	C₂³×C₄	—	53	1	no	—
308	C₂³×C₄	—	53	1	no	—
309	C₂×M₁₆	—	53	1	no	—
310	C₂×M₁₆	—	53	1	no	—
311	C₂×M₁₆	—	53	2	no	—
312	C₂×M₁₆	—	53	2	no	—
313	C₂×M₁₆	—	106	2	no	—
314	C₂×M₁₆	—	106	2	no	—
315	C₂×M₁₆	—	106	1	no	—
316	C₂×M₁₆	—	106	1	no	—
317	C₂×M₁₆	—	106	4	no	—
318	C₂×M₁₆	—	106	4	no	—
319	C₂×M₁₆	—	212	4	no	—
320	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
321	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	53	2	no	—
324	C₂²×C₈	—	53	2	no	—
325	C₂²×C₈	—	106	1	no	—
326	C₂×C₄²	—	53	1	no	—
327	C₂×C₄²	—	53	1	no	—
328	C₂×C₄²	—	53	1	no	—
329	C₂×C₄²	—	53	1	no	—
330	C₂×C₄²	—	53	1	no	—
331	C₂×C₄²	—	53	1	no	—
332	C₂×C₄²	—	53	1	no	—
333	C₂×C₄²	—	53	1	no	—
334	C₂×C₄²	—	53	1	no	—
335	C₂×C₄²	—	53	1	no	—
336	C₂×C₄²	—	53	1	no	—
337	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
338	C₄⋊C₈	—	2	1	no	—
339	C₄⋊C₈	—	53	1	no	—
340	C₄⋊C₈	—	53	2	no	—
341	C₄⋊C₈	—	53	1	no	—
342	C₄⋊C₈	—	106	2	no	—
343	C₄⋊C₈	—	106	2	no	—
344	C₄⋊C₈	—	106	1	no	—
345	C₄⋊C₈	—	106	1	no	—
346	C₄⋊C₈	—	106	1	no	—
347	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
348	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
349	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	106	1	no	—
352	C₂²⋊C₈	—	106	1	no	—
353	C₂²⋊C₈	—	106	1	no	—
354	#97	—	53	2	no	—
355	#97	—	53	2	no	—
356	#97	—	53	2	no	—
357	#97	—	106	1	no	—
358	#97	—	106	1	no	—
359	#97	C₄×D₅₃	1	2	yes	—
360	C₄×C₈	—	2	1	no	—
361	C₄×C₈	—	53	2	no	—
362	C₄×C₈	—	53	2	no	—
363	C₄×C₈	—	53	2	no	—
364	C₄×C₈	—	106	1	no	—
365	C₄×C₈	C₂²×D₅₃	1	2	yes	—
366	C₂⁴	—	53	1	no	—
367	C₂²×C₄	—	53	1	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	53	1	no	—
375	C₂²×C₄	—	53	1	no	—
376	C₂²×C₄	—	53	1	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	53	1	no	—
379	C₂²×C₄	—	53	1	no	—
380	C₂²×C₄	—	53	1	no	—
381	C₂²×C₄	—	53	1	no	—
382	C₂²×C₄	—	106	2	no	—
383	C₂²×C₄	—	106	2	no	—
384	C₂²×C₄	—	106	2	no	—
385	C₂²×C₄	—	106	2	no	—
386	C₂²×C₄	—	212	2	no	—
387	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
388	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
390	M₁₆	—	2	2	no	—
391	M₁₆	—	53	4	no	—
392	M₁₆	—	106	4	no	—
393	M₁₆	—	106	2	no	—
394	M₁₆	—	106	4	no	—
395	M₁₆	—	212	4	no	—
396	M₁₆	—	212	2	no	—
397	M₁₆	—	212	2	no	—
398	M₁₆	—	212	4	no	—
399	M₁₆	#11719912	1	4	yes	complemented
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	4	2	no	—
404	C₂×C₈	—	53	2	no	—
405	C₂×C₈	—	53	2	no	—
406	C₂×C₈	—	106	4	no	—
407	C₂×C₈	—	106	4	no	—
408	C₂×C₈	—	106	1	no	—
409	C₂×C₈	—	106	1	no	—
410	C₂×C₈	—	106	2	no	—
411	C₂×C₈	—	212	2	no	—
412	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
413	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
414	C₄²	—	53	1	no	—
415	C₄²	—	53	1	no	—
416	C₄²	—	53	1	no	—
417	C₄²	—	53	1	no	—
418	C₄²	—	53	1	no	—
419	C₄²	—	53	1	no	—
420	C₄²	—	53	1	no	—
421	C₄²	—	53	1	no	—
422	C₄²	—	53	1	no	—
423	C₄²	—	53	1	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₄²	D₄×D₅₃	1	1	yes	characteristic
428	C₄²	Q₈×D₅₃	1	1	yes	characteristic
429	C₄²	C₂³×D₅₃	1	1	yes	characteristic
430	C₂³	—	53	1	no	—
431	C₂³	—	53	1	no	—
432	C₂³	—	106	2	no	—
433	C₂³	—	106	2	no	—
434	C₂³	—	212	1	no	—
435	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	53	1	no	—
440	C₂×C₄	—	53	1	no	—
441	C₂×C₄	—	53	2	no	—
442	C₂×C₄	—	53	2	no	—
443	C₂×C₄	—	53	1	no	—
444	C₂×C₄	—	53	1	no	—
445	C₂×C₄	—	53	1	no	—
446	C₂×C₄	—	53	1	no	—
447	C₂×C₄	—	106	2	no	—
448	C₂×C₄	—	106	2	no	—
449	C₂×C₄	—	106	2	no	—
450	C₂×C₄	—	106	2	no	—
451	C₂×C₄	—	106	2	no	—
452	C₂×C₄	—	106	2	no	—
453	C₂×C₄	—	106	2	no	—
454	C₂×C₄	—	106	1	no	—
455	C₂×C₄	—	106	1	no	—
456	C₂×C₄	—	106	1	no	—
457	C₂×C₄	—	106	1	no	—
458	C₂×C₄	—	106	1	no	—
459	C₂×C₄	—	106	2	no	—
460	C₂×C₄	—	106	2	no	—
461	C₂×C₄	—	106	1	no	—
462	C₂×C₄	—	212	1	no	—
463	C₂×C₄	—	212	1	no	—
464	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
465	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
467	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
468	C₂×C₄	C₂×#11719912	1	2	yes	—
469	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
470	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
471	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
472	C₈	—	2	2	no	—
473	C₈	—	4	2	no	—
474	C₈	—	4	1	no	—
475	C₈	—	4	1	no	—
476	C₈	—	53	4	no	—
477	C₈	—	106	2	no	—
478	C₈	—	212	2	no	—
479	C₈	—	212	1	no	—
480	C₈	—	212	1	no	—
481	C₈	C₂×#11719912	1	4	yes	complemented
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	53	2	no	—
485	C₂²	—	106	1	no	—
486	C₂²	—	106	1	no	—
487	C₂²	—	106	1	no	—
488	C₂²	—	106	1	no	—
489	C₂²	—	106	1	no	—
490	C₂²	—	212	4	no	—
491	C₂²	#16810645	1	2	yes	—
492	C₂²	C₂×C₄⋊C₄×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	—	53	2	no	—
498	C₄	—	106	1	no	—
499	C₄	—	106	1	no	—
500	C₄	—	106	1	no	—
501	C₄	—	106	1	no	—
502	C₄	#16810645	1	2	yes	—
503	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
504	C₄	C₂²×#11719912	1	1	yes	characteristic
505	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
506	C₄	#137×D₅₃	1	1	yes	characteristic
507	C₂	—	2	2	no	—
508	C₂	—	106	2	no	—
509	C₂	—	212	1	no	—
510	C₂	C₂×#16810645	1	1	yes	characteristic
511	C₂	#428×D₅₃	1	1	yes	characteristic
512	C₂	#430×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×#1565	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1565	D₅₃
hypocenter	C₅₃	#17796
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #20273481	C₂	semidirect
5. #20273479	C₂	semidirect
6. #20273472	C₂	semidirect
7. #20273469	C₂	semidirect
8. #20273468	C₂	semidirect
9. #20273467	C₂	semidirect
10. #20273465	C₂	semidirect
11. C₅₃×#1565	C₂	semidirect
38. #16810376	C₂²	semidirect
39. #16810375	C₂²	semidirect
40. #16810375	C₂²	semidirect
41. #16810374	C₂²	semidirect
42. #16810373	C₂²	semidirect
43. #16810373	C₂²	semidirect
47. C₅₃×#335	C₂²	semidirect
48. C₅₃×#334	C₂²	semidirect
134. M₁₆×C₅₃	M₁₆	semidirect
158. #11719917	M₁₆	semidirect
219. C₈×C₅₃	C₂×M₁₆	semidirect
224. #88322	C₂×M₁₆	semidirect
261. #1565	D₅₃	semidirect
300. #335	C₂×D₅₃	semidirect
304. #334	C₂×D₅₃	semidirect
305. C₅₃	#17796	semidirect
399. M₁₆	#11719912	semidirect
481. C₈	C₂×#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283343	2
C₂×#16810645	2
C₄×M₁₆×D₅₃	2
#20273481	2
#20273479	2
#20273472	2
#20273469	2
#20273468	2
#20273467	2
#20273465	2
C₅₃×#1565	2
#17796	53