Subgroups — #27353194

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	C₂	1	1	yes	maximal, characteristic
2	#20283343	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
4	#20273505	C₂	1	2	yes	maximal, complemented
5	#20273502	C₂	1	2	yes	maximal, complemented
6	#20273494	C₂	1	2	yes	maximal, complemented
7	#20273492	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	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	#16810679	C₂²	1	1	yes	characteristic
14	#16810679	C₂²	1	1	yes	characteristic
15	#16810678	C₄	1	2	yes	—
16	#16810678	C₄	1	2	yes	—
17	#16810677	—	2	1	no	—
18	#16810676	C₂²	1	1	yes	characteristic
19	#16810675	—	2	1	no	—
20	#16810675	—	2	1	no	—
21	#16810675	C₂²	1	2	yes	—
22	#16810675	C₂²	1	2	yes	—
23	#16810674	—	2	1	no	—
24	#16810674	C₂²	1	2	yes	—
25	#16810674	C₂²	1	2	yes	—
26	#16810646	C₄	1	1	yes	characteristic
27	#16810646	C₄	1	1	yes	characteristic
28	#16810645	C₂²	1	1	yes	characteristic
29	#16810645	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
31	#16810586	C₄	1	1	yes	characteristic
32	#16810586	C₄	1	1	yes	characteristic
33	#16810585	—	2	1	no	—
34	C₂×#13133129	—	2	1	no	—
35	C₄×#11719917	C₂²	1	1	yes	characteristic
36	C₄×#11719912	—	2	1	no	—
37	C₂×C₈×Dic₅₃	—	2	1	no	—
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	C₂²×#88322	—	2	1	no	—
57	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
60	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	D₄	1	1	yes	characteristic
63	C₄²×D₅₃	D₄	1	1	yes	characteristic
64	C₄²×D₅₃	Q₈	1	1	yes	characteristic
65	C₄²×D₅₃	Q₈	1	1	yes	characteristic
66	C₄²×D₅₃	C₂³	1	1	yes	characteristic
67	C₄²×D₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	C₂³	1	1	yes	characteristic
70	C₄⋊#88322	C₂³	1	2	yes	—
71	C₄⋊#88322	C₂³	1	1	yes	characteristic
72	#13133146	—	2	1	no	—
73	#13133146	—	2	1	no	—
74	#13133146	C₂×C₄	1	2	yes	—
75	#13133146	C₂×C₄	1	2	yes	—
76	#13133146	C₂×C₄	1	2	yes	—
77	#13133146	C₂×C₄	1	2	yes	—
78	C₂²⋊#88322	—	2	1	no	—
79	#13133129	—	2	2	no	—
80	#13133129	—	2	2	no	—
81	#13133129	—	2	1	no	—
82	#13133129	—	2	1	no	—
83	#13133129	C₂×C₄	1	2	yes	—
84	#13133129	C₂×C₄	1	2	yes	—
85	#13133129	C₂×C₄	1	2	yes	—
86	#13133129	C₂×C₄	1	2	yes	—
87	#13133128	C₂×C₄	1	2	yes	—
88	#13133127	—	2	1	no	—
89	#13133127	—	2	1	no	—
90	C₄×#88322	—	2	1	no	—
91	C₄×#88322	C₂³	1	2	yes	—
92	C₈×Dic₅₃	—	2	2	no	—
93	C₈×Dic₅₃	—	2	2	no	—
94	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	—	2	1	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	—	2	1	no	—
99	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
101	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²⋊C₈×C₅₃	—	2	1	no	—
103	#97×C₅₃	C₂×C₄	1	2	yes	—
104	C₄×C₈×C₅₃	—	2	1	no	—
105	C₄×C₈×C₅₃	C₂³	1	2	yes	—
106	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	—	2	2	no	—
114	C₂×C₄×D₅₃	—	2	2	no	—
115	C₂×C₄×D₅₃	—	4	2	no	—
116	C₂×C₄×D₅₃	—	4	1	no	—
117	C₂×C₄×D₅₃	—	4	1	no	—
118	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	4	2	no	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
141	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
144	C₄×Dic₅₃	C₄⋊C₄	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	C₂²×C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
154	#11719917	—	2	2	no	—
155	#11719917	—	2	2	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	2	1	no	—
158	C₂×#88322	—	2	1	no	—
159	C₂×#88322	—	4	2	no	—
160	C₂×#88322	C₂²×C₄	1	2	yes	—
161	C₂×#88322	C₂²×C₄	1	2	yes	—
162	#11719912	—	4	2	no	—
163	#11719912	—	4	2	no	—
164	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×D₅₃	—	2	2	no	—
166	C₂²×D₅₃	—	4	1	no	—
167	C₂²×D₅₃	—	4	1	no	—
168	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	#131	1	2	yes	—
177	C₂×C₄×C₅₃	#131	1	2	yes	—
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
181	C₂×Dic₅₃	—	2	2	no	—
182	C₂×Dic₅₃	—	2	2	no	—
183	C₂×Dic₅₃	—	4	1	no	—
184	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄×D₅₃	—	2	1	no	—
191	C₄×D₅₃	—	2	1	no	—
192	C₄×D₅₃	—	2	1	no	—
193	C₄×D₅₃	—	2	1	no	—
194	C₄×D₅₃	—	2	1	no	—
195	C₄×D₅₃	—	2	1	no	—
196	C₄×D₅₃	—	2	1	no	—
197	C₄×D₅₃	—	2	1	no	—
198	C₄×D₅₃	—	2	2	no	—
199	C₄×D₅₃	—	2	2	no	—
200	C₄×D₅₃	—	2	2	no	—
201	C₄×D₅₃	—	2	2	no	—
202	C₄×D₅₃	—	4	1	no	—
203	C₄×D₅₃	—	4	1	no	—
204	C₄×D₅₃	—	4	1	no	—
205	C₄×D₅₃	—	4	1	no	—
206	C₈×C₅₃	—	2	2	no	—
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	—	4	1	no	—
209	C₈×C₅₃	—	4	1	no	—
210	C₈×C₅₃	—	4	2	no	—
211	#88322	—	2	2	no	—
212	#88322	—	2	2	no	—
213	#88322	—	4	1	no	—
214	#88322	—	4	1	no	—
215	#88322	—	4	2	no	—
216	#17808	—	53	1	no	maximal, Sylow
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	#424	1	2	yes	—
220	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₂×D₅₃	—	2	1	no	—
222	C₂×D₅₃	—	2	1	no	—
223	C₂×D₅₃	—	2	1	no	—
224	C₂×D₅₃	—	2	1	no	—
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	4	4	no	—
228	C₄×C₅₃	—	2	1	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	#424	1	2	yes	—
232	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
233	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
234	C₄×C₅₃	#575	1	1	yes	characteristic
235	C₄×C₅₃	#575	1	1	yes	characteristic
236	Dic₅₃	—	2	1	no	—
237	Dic₅₃	—	2	1	no	—
238	Dic₅₃	—	2	1	no	—
239	Dic₅₃	—	2	1	no	—
240	Dic₅₃	—	2	1	no	—
241	Dic₅₃	#430	1	2	yes	—
242	#2920	—	53	1	no	—
243	#2920	—	53	1	no	—
244	C₂×#431	—	53	1	no	—
245	#1574	—	53	2	no	—
246	#1574	—	53	2	no	—
247	#1572	—	53	2	no	—
248	#1572	—	53	2	no	—
249	#1565	—	53	1	no	—
250	#1565	—	53	1	no	—
251	#1565	—	53	1	no	—
252	#1565	D₅₃	1	1	yes	characteristic, complemented
253	C₂×C₅₃	—	2	2	no	—
254	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
255	C₂×C₅₃	#2911	1	1	yes	characteristic
256	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
257	D₅₃	—	4	1	no	—
258	D₅₃	—	4	1	no	—
259	C₂²×C₄²	—	53	1	no	—
260	#432	—	53	2	no	—
261	#432	—	53	2	no	—
262	#432	—	53	1	no	—
263	#432	—	53	1	no	—
264	#432	—	106	1	no	—
265	#431	—	53	2	no	—
266	#431	—	53	2	no	—
267	#431	—	53	1	no	—
268	#431	—	53	2	no	—
269	#431	—	53	2	no	—
270	#431	—	106	1	no	—
271	#431	—	106	1	no	—
272	#431	—	106	1	no	—
273	#431	C₂×D₅₃	1	1	yes	characteristic
274	#423	—	53	1	no	—
275	#423	—	53	1	no	—
276	#423	—	53	1	no	—
277	#423	—	53	1	no	—
278	C₂×C₄⋊C₈	—	53	1	no	—
279	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
280	#407	—	53	1	no	—
281	#407	—	53	1	no	—
282	#407	—	106	1	no	—
283	C₂×C₂²⋊C₈	—	106	1	no	—
284	C₄×M₁₆	—	53	1	no	—
285	C₄×M₁₆	—	106	1	no	—
286	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
287	C₂×C₄×C₈	—	106	1	no	—
288	#335	—	53	2	no	—
289	#335	—	53	2	no	—
290	#335	—	53	2	no	—
291	#335	C₂×D₅₃	1	2	yes	complemented
292	#334	—	53	2	no	—
293	#334	—	53	2	no	—
294	#334	—	53	2	no	—
295	#334	C₂×D₅₃	1	2	yes	complemented
296	C₅₃	#17808	1	1	yes	characteristic, complemented, Sylow
297	C₂³×C₄	—	53	1	no	—
298	C₂³×C₄	—	53	1	no	—
299	C₂³×C₄	—	53	1	no	—
300	C₂×M₁₆	—	53	1	no	—
301	C₂×M₁₆	—	53	1	no	—
302	C₂×M₁₆	—	106	1	no	—
303	C₂×M₁₆	—	106	1	no	—
304	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
305	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
306	C₂²×C₈	—	2	1	no	—
307	C₂²×C₈	—	106	1	no	—
308	C₂×C₄²	—	53	1	no	—
309	C₂×C₄²	—	53	1	no	—
310	C₂×C₄²	—	53	1	no	—
311	C₂×C₄²	—	53	1	no	—
312	C₂×C₄²	—	53	1	no	—
313	C₂×C₄²	—	53	1	no	—
314	C₂×C₄²	—	53	1	no	—
315	C₂×C₄²	—	53	1	no	—
316	C₂×C₄²	—	53	1	no	—
317	C₂×C₄²	—	53	1	no	—
318	C₂×C₄²	—	53	1	no	—
319	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
320	C₄⋊C₈	—	2	1	no	—
321	C₄⋊C₈	—	53	2	no	—
322	C₄⋊C₈	—	53	2	no	—
323	C₄⋊C₈	—	53	2	no	—
324	C₄⋊C₈	—	53	2	no	—
325	C₄⋊C₈	—	53	2	no	—
326	C₄⋊C₈	—	53	1	no	—
327	C₄⋊C₈	—	53	1	no	—
328	C₄⋊C₈	—	106	1	no	—
329	C₄⋊C₈	—	106	1	no	—
330	C₄⋊C₈	—	106	1	no	—
331	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
332	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
333	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
334	C₂²⋊C₈	—	2	1	no	—
335	C₂²⋊C₈	—	53	2	no	—
336	C₂²⋊C₈	—	53	2	no	—
337	C₂²⋊C₈	—	53	2	no	—
338	C₂²⋊C₈	—	53	2	no	—
339	C₂²⋊C₈	—	106	2	no	—
340	C₂²⋊C₈	—	106	2	no	—
341	C₂²⋊C₈	—	106	1	no	—
342	C₂²⋊C₈	—	106	1	no	—
343	C₂²⋊C₈	—	106	1	no	—
344	#97	—	53	2	no	—
345	#97	—	106	1	no	—
346	#97	—	106	1	no	—
347	#97	C₄×D₅₃	1	2	yes	—
348	C₄×C₈	—	2	1	no	—
349	C₄×C₈	—	53	2	no	—
350	C₄×C₈	—	106	2	no	—
351	C₄×C₈	—	106	2	no	—
352	C₄×C₈	—	106	1	no	—
353	C₄×C₈	C₂²×D₅₃	1	2	yes	—
354	C₂⁴	—	53	1	no	—
355	C₂²×C₄	—	53	1	no	—
356	C₂²×C₄	—	53	1	no	—
357	C₂²×C₄	—	53	1	no	—
358	C₂²×C₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	—	53	1	no	—
364	C₂²×C₄	—	53	1	no	—
365	C₂²×C₄	—	53	1	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	1	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	106	2	no	—
371	C₂²×C₄	—	106	2	no	—
372	C₂²×C₄	—	212	1	no	—
373	C₂²×C₄	—	212	1	no	—
374	C₂²×C₄	—	212	2	no	—
375	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
376	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
377	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	106	2	no	—
381	M₁₆	—	106	2	no	—
382	M₁₆	—	212	2	no	—
383	M₁₆	—	212	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	4	2	no	—
388	C₂×C₈	—	53	2	no	—
389	C₂×C₈	—	53	2	no	—
390	C₂×C₈	—	106	2	no	—
391	C₂×C₈	—	106	1	no	—
392	C₂×C₈	—	106	1	no	—
393	C₂×C₈	—	212	2	no	—
394	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
395	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
396	C₄²	—	53	1	no	—
397	C₄²	—	53	1	no	—
398	C₄²	—	53	1	no	—
399	C₄²	—	53	1	no	—
400	C₄²	—	53	1	no	—
401	C₄²	—	53	1	no	—
402	C₄²	—	53	1	no	—
403	C₄²	—	53	1	no	—
404	C₄²	—	53	1	no	—
405	C₄²	—	53	1	no	—
406	C₄²	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
409	C₄²	D₄×D₅₃	1	1	yes	characteristic
410	C₄²	Q₈×D₅₃	1	1	yes	characteristic
411	C₄²	C₂³×D₅₃	1	1	yes	characteristic
412	C₂³	—	53	1	no	—
413	C₂³	—	53	1	no	—
414	C₂³	—	106	2	no	—
415	C₂³	—	212	1	no	—
416	C₂³	—	212	1	no	—
417	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	53	1	no	—
422	C₂×C₄	—	53	1	no	—
423	C₂×C₄	—	53	1	no	—
424	C₂×C₄	—	53	1	no	—
425	C₂×C₄	—	53	1	no	—
426	C₂×C₄	—	53	1	no	—
427	C₂×C₄	—	106	2	no	—
428	C₂×C₄	—	106	2	no	—
429	C₂×C₄	—	106	2	no	—
430	C₂×C₄	—	106	2	no	—
431	C₂×C₄	—	106	1	no	—
432	C₂×C₄	—	106	1	no	—
433	C₂×C₄	—	106	1	no	—
434	C₂×C₄	—	106	1	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	106	1	no	—
438	C₂×C₄	—	106	1	no	—
439	C₂×C₄	—	106	1	no	—
440	C₂×C₄	—	106	1	no	—
441	C₂×C₄	—	212	1	no	—
442	C₂×C₄	—	212	1	no	—
443	C₂×C₄	—	212	1	no	—
444	C₂×C₄	—	212	1	no	—
445	C₂×C₄	—	212	1	no	—
446	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
447	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
448	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
449	C₂×C₄	#13133223	1	2	yes	—
450	C₂×C₄	#13133225	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	4	2	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	106	2	no	—
460	C₈	—	106	2	no	—
461	C₈	—	212	1	no	—
462	C₈	—	212	2	no	—
463	C₈	—	212	1	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	106	1	no	—
467	C₂²	—	106	1	no	—
468	C₂²	—	106	1	no	—
469	C₂²	—	106	1	no	—
470	C₂²	—	106	1	no	—
471	C₂²	—	106	1	no	—
472	C₂²	—	212	4	no	—
473	C₂²	#16810648	1	2	yes	—
474	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	53	2	no	—
479	C₄	—	106	1	no	—
480	C₄	—	106	1	no	—
481	C₄	—	106	1	no	—
482	C₄	—	106	1	no	—
483	C₄	—	106	1	no	—
484	C₄	#16810648	1	2	yes	—
485	C₄	C₂×#13133223	1	1	yes	characteristic
486	C₄	C₂×#13133225	1	1	yes	characteristic
487	C₄	#16811419	1	1	yes	characteristic
488	C₄	#16811422	1	1	yes	characteristic
489	C₂	—	2	2	no	—
490	C₂	—	212	1	no	—
491	C₂	—	212	1	no	—
492	C₂	C₂×#16810648	1	1	yes	characteristic
493	C₂	#20283302	1	1	yes	characteristic
494	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	#17808
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. #20273505	C₂	semidirect
5. #20273502	C₂	semidirect
6. #20273494	C₂	semidirect
7. #20273492	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
252. #1565	D₅₃	semidirect
291. #335	C₂×D₅₃	semidirect
295. #334	C₂×D₅₃	semidirect
296. C₅₃	#17808	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
#20283343	2
C₂×#16810675	2
#20273505	2
#20273502	2
#20273494	2
#20273492	2
#20273468	2
#20273467	2
#20273465	2
C₅₃×#1565	2
#17808	53