Subgroups — #27353187

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 502 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×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273499	C₂	1	2	yes	maximal, complemented
5	#20273497	C₂	1	2	yes	maximal, complemented
6	#20273488	C₂	1	2	yes	maximal, complemented
7	#20273484	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×C₈×D₅₃	—	2	1	no	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810679	C₂²	1	1	yes	characteristic
15	#16810679	C₂²	1	1	yes	characteristic
16	#16810678	C₄	1	2	yes	—
17	#16810678	C₄	1	2	yes	—
18	#16810677	—	2	1	no	—
19	#16810676	C₂²	1	1	yes	characteristic
20	#16810674	—	2	1	no	—
21	#16810674	C₂²	1	2	yes	—
22	#16810674	C₂²	1	2	yes	—
23	#16810646	C₄	1	1	yes	characteristic
24	#16810646	C₄	1	1	yes	characteristic
25	#16810645	C₂²	1	1	yes	characteristic
26	#16810645	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133146	—	2	1	no	—
29	C₄⋊C₈×D₅₃	—	2	1	no	—
30	C₄⋊C₈×D₅₃	—	2	1	no	—
31	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
32	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
33	#16810586	C₄	1	1	yes	characteristic
34	#16810586	C₄	1	1	yes	characteristic
35	#16810585	—	2	1	no	—
36	C₄×#11719917	C₂²	1	1	yes	characteristic
37	C₄×#11719912	—	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×C₈×D₅₃	—	2	2	no	—
58	C₂×C₈×D₅₃	—	2	2	no	—
59	C₂×C₈×D₅₃	—	4	4	no	—
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	D₄	1	1	yes	characteristic
66	C₄²×D₅₃	D₄	1	1	yes	characteristic
67	C₄²×D₅₃	Q₈	1	1	yes	characteristic
68	C₄²×D₅₃	Q₈	1	1	yes	characteristic
69	C₄²×D₅₃	C₂³	1	1	yes	characteristic
70	C₄²×D₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊#88322	—	2	1	no	—
72	C₄⋊#88322	C₂³	1	1	yes	characteristic
73	C₄⋊#88322	C₂³	1	2	yes	—
74	C₄⋊#88322	C₂³	1	1	yes	characteristic
75	#13133146	—	2	2	no	—
76	#13133146	—	2	2	no	—
77	#13133146	—	2	1	no	—
78	#13133146	—	2	1	no	—
79	#13133146	C₂×C₄	1	2	yes	—
80	#13133146	C₂×C₄	1	2	yes	—
81	#13133146	C₂×C₄	1	2	yes	—
82	#13133146	C₂×C₄	1	2	yes	—
83	C₂²⋊#88322	—	2	1	no	—
84	#13133129	—	2	1	no	—
85	#13133129	—	2	1	no	—
86	#13133129	C₂×C₄	1	2	yes	—
87	#13133129	C₂×C₄	1	2	yes	—
88	#13133129	C₂×C₄	1	2	yes	—
89	#13133129	C₂×C₄	1	2	yes	—
90	#13133128	C₂×C₄	1	2	yes	—
91	#13133127	—	2	1	no	—
92	#13133127	—	2	1	no	—
93	C₄×#88322	—	2	1	no	—
94	C₄×#88322	C₂³	1	2	yes	—
95	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₅₃	—	2	1	no	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	—	2	1	no	—
100	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
102	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²⋊C₈×C₅₃	—	2	1	no	—
104	#97×C₅₃	C₂×C₄	1	2	yes	—
105	C₄×C₈×C₅₃	—	2	1	no	—
106	C₄×C₈×C₅₃	C₂³	1	2	yes	—
107	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	—	2	2	no	—
115	C₂×C₄×D₅₃	—	2	2	no	—
116	C₂×C₄×D₅₃	—	2	2	no	—
117	C₂×C₄×D₅₃	—	2	2	no	—
118	C₂×C₄×D₅₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	4	2	no	—
137	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
139	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
142	C₄²×C₅₃	C₂⁴	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	C₄⋊C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
154	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
155	#11719917	—	2	2	no	—
156	#11719917	—	2	2	no	—
157	C₂×#88322	—	2	1	no	—
158	C₂×#88322	—	2	1	no	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	—	4	2	no	—
161	C₂×#88322	C₂²×C₄	1	2	yes	—
162	C₂×#88322	C₂²×C₄	1	2	yes	—
163	#11719912	—	4	2	no	—
164	#11719912	—	4	2	no	—
165	C₈×D₅₃	—	4	4	no	—
166	C₈×D₅₃	—	4	4	no	—
167	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×D₅₃	—	2	2	no	—
169	C₂²×D₅₃	—	2	2	no	—
170	C₂²×D₅₃	—	2	2	no	—
171	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#131	1	2	yes	—
180	C₂×C₄×C₅₃	#131	1	2	yes	—
181	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
184	C₂×Dic₅₃	—	2	2	no	—
185	C₂×Dic₅₃	—	2	2	no	—
186	C₂×Dic₅₃	—	4	1	no	—
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	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄×D₅₃	—	2	2	no	—
194	C₄×D₅₃	—	2	2	no	—
195	C₄×D₅₃	—	2	2	no	—
196	C₄×D₅₃	—	2	1	no	—
197	C₄×D₅₃	—	2	1	no	—
198	C₄×D₅₃	—	2	1	no	—
199	C₄×D₅₃	—	2	1	no	—
200	C₄×D₅₃	—	2	1	no	—
201	C₄×D₅₃	—	2	1	no	—
202	C₄×D₅₃	—	2	2	no	—
203	C₄×D₅₃	—	2	2	no	—
204	C₄×D₅₃	—	2	2	no	—
205	C₄×D₅₃	—	4	1	no	—
206	C₄×D₅₃	—	4	1	no	—
207	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
208	C₄×D₅₃	#137	1	2	yes	—
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	2	2	no	—
211	C₈×C₅₃	—	4	2	no	—
212	C₈×C₅₃	—	4	1	no	—
213	C₈×C₅₃	—	4	1	no	—
214	#88322	—	2	2	no	—
215	#88322	—	2	2	no	—
216	#88322	—	4	2	no	—
217	#88322	—	4	1	no	—
218	#88322	—	4	1	no	—
219	#17807	—	53	1	no	maximal, Sylow
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	#424	1	2	yes	—
223	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₂×D₅₃	—	2	8	no	—
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	2	1	no	—
228	C₂×D₅₃	—	2	1	no	—
229	C₂×D₅₃	#428	1	2	yes	—
230	C₂×D₅₃	#430	1	2	yes	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	#424	1	2	yes	—
235	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
236	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
237	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
239	Dic₅₃	—	2	1	no	—
240	Dic₅₃	—	2	1	no	—
241	Dic₅₃	—	2	1	no	—
242	Dic₅₃	—	2	1	no	—
243	Dic₅₃	—	2	1	no	—
244	Dic₅₃	#428	1	2	yes	—
245	#2920	—	53	1	no	—
246	#2920	—	53	1	no	—
247	C₂²×C₄⋊C₈	—	53	1	no	—
248	#1573	—	53	2	no	—
249	#1573	—	53	2	no	—
250	#1571	—	53	2	no	—
251	#1570	—	53	2	no	—
252	#1565	—	53	1	no	—
253	#1565	—	53	1	no	—
254	#1565	—	53	1	no	—
255	#1565	D₅₃	1	1	yes	characteristic, complemented
256	C₂×C₅₃	—	2	2	no	—
257	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
258	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
260	D₅₃	—	2	2	no	—
261	D₅₃	—	2	2	no	—
262	C₂³×C₈	—	106	1	no	—
263	C₂²×C₄²	—	53	1	no	—
264	#432	—	53	2	no	—
265	#432	—	53	2	no	—
266	#432	—	53	1	no	—
267	#432	—	53	1	no	—
268	#432	—	106	1	no	—
269	#431	—	53	2	no	—
270	#431	—	53	2	no	—
271	#431	—	53	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	—	53	2	no	—
280	C₂×C₄⋊C₈	—	53	2	no	—
281	C₂×C₄⋊C₈	—	106	1	no	—
282	C₂×C₄⋊C₈	—	106	1	no	—
283	C₂×C₄⋊C₈	—	106	1	no	—
284	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
285	#407	—	53	1	no	—
286	#407	—	53	1	no	—
287	#407	—	106	1	no	—
288	C₄×M₁₆	—	53	1	no	—
289	C₄×M₁₆	—	106	1	no	—
290	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
291	#335	—	53	2	no	—
292	#335	—	53	2	no	—
293	#335	—	53	2	no	—
294	#335	C₂×D₅₃	1	2	yes	complemented
295	#334	—	53	2	no	—
296	#334	—	53	2	no	—
297	#334	—	53	2	no	—
298	#334	C₂×D₅₃	1	2	yes	complemented
299	C₅₃	#17807	1	1	yes	characteristic, complemented, Sylow
300	C₂³×C₄	—	53	1	no	—
301	C₂³×C₄	—	53	1	no	—
302	C₂³×C₄	—	53	1	no	—
303	C₂×M₁₆	—	53	1	no	—
304	C₂×M₁₆	—	53	1	no	—
305	C₂×M₁₆	—	106	1	no	—
306	C₂×M₁₆	—	106	1	no	—
307	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
308	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	—	106	2	no	—
311	C₂²×C₈	—	106	2	no	—
312	C₂²×C₈	—	106	1	no	—
313	C₂²×C₈	—	212	4	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	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	1	no	—
323	C₂×C₄²	—	53	1	no	—
324	C₂×C₄²	—	53	1	no	—
325	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	53	2	no	—
328	C₄⋊C₈	—	53	2	no	—
329	C₄⋊C₈	—	53	2	no	—
330	C₄⋊C₈	—	53	2	no	—
331	C₄⋊C₈	—	53	1	no	—
332	C₄⋊C₈	—	53	2	no	—
333	C₄⋊C₈	—	53	1	no	—
334	C₄⋊C₈	—	106	2	no	—
335	C₄⋊C₈	—	106	2	no	—
336	C₄⋊C₈	—	106	1	no	—
337	C₄⋊C₈	—	106	1	no	—
338	C₄⋊C₈	—	106	1	no	—
339	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
340	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
341	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	53	2	no	—
344	C₂²⋊C₈	—	53	2	no	—
345	C₂²⋊C₈	—	53	2	no	—
346	C₂²⋊C₈	—	53	2	no	—
347	C₂²⋊C₈	—	106	1	no	—
348	C₂²⋊C₈	—	106	1	no	—
349	C₂²⋊C₈	—	106	1	no	—
350	#97	—	53	2	no	—
351	#97	—	106	1	no	—
352	#97	—	106	1	no	—
353	#97	C₄×D₅₃	1	2	yes	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	53	2	no	—
356	C₄×C₈	—	106	1	no	—
357	C₄×C₈	C₂²×D₅₃	1	2	yes	—
358	C₂⁴	—	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	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	106	2	no	—
375	C₂²×C₄	—	106	2	no	—
376	C₂²×C₄	—	106	2	no	—
377	C₂²×C₄	—	106	2	no	—
378	C₂²×C₄	—	212	2	no	—
379	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
380	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
381	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
382	M₁₆	—	2	2	no	—
383	M₁₆	—	2	2	no	—
384	M₁₆	—	106	2	no	—
385	M₁₆	—	106	2	no	—
386	M₁₆	—	212	2	no	—
387	M₁₆	—	212	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	4	2	no	—
392	C₂×C₈	—	53	2	no	—
393	C₂×C₈	—	53	2	no	—
394	C₂×C₈	—	106	1	no	—
395	C₂×C₈	—	106	1	no	—
396	C₂×C₈	—	106	2	no	—
397	C₂×C₈	—	212	4	no	—
398	C₂×C₈	—	212	4	no	—
399	C₂×C₈	—	212	2	no	—
400	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
401	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
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	—	53	1	no	—
409	C₄²	—	53	1	no	—
410	C₄²	—	53	1	no	—
411	C₄²	—	53	1	no	—
412	C₄²	—	53	1	no	—
413	C₄²	—	53	1	no	—
414	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
415	C₄²	D₄×D₅₃	1	1	yes	characteristic
416	C₄²	Q₈×D₅₃	1	1	yes	characteristic
417	C₄²	C₂³×D₅₃	1	1	yes	characteristic
418	C₂³	—	53	1	no	—
419	C₂³	—	53	1	no	—
420	C₂³	—	106	2	no	—
421	C₂³	—	106	2	no	—
422	C₂³	—	106	2	no	—
423	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	53	1	no	—
428	C₂×C₄	—	53	2	no	—
429	C₂×C₄	—	53	2	no	—
430	C₂×C₄	—	53	1	no	—
431	C₂×C₄	—	53	1	no	—
432	C₂×C₄	—	53	1	no	—
433	C₂×C₄	—	53	1	no	—
434	C₂×C₄	—	53	1	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	106	2	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	106	2	no	—
440	C₂×C₄	—	106	2	no	—
441	C₂×C₄	—	106	1	no	—
442	C₂×C₄	—	106	1	no	—
443	C₂×C₄	—	106	1	no	—
444	C₂×C₄	—	106	1	no	—
445	C₂×C₄	—	106	1	no	—
446	C₂×C₄	—	106	1	no	—
447	C₂×C₄	—	106	2	no	—
448	C₂×C₄	—	106	2	no	—
449	C₂×C₄	—	212	1	no	—
450	C₂×C₄	—	212	1	no	—
451	C₂×C₄	—	212	1	no	—
452	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
454	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
455	C₂×C₄	#13133223	1	2	yes	—
456	C₂×C₄	#13133225	1	2	yes	—
457	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
458	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
460	C₈	—	2	2	no	—
461	C₈	—	2	2	no	—
462	C₈	—	4	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	1	no	—
465	C₈	—	106	2	no	—
466	C₈	—	106	2	no	—
467	C₈	—	212	2	no	—
468	C₈	—	212	1	no	—
469	C₈	—	212	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	53	2	no	—
473	C₂²	—	53	2	no	—
474	C₂²	—	106	8	no	—
475	C₂²	—	106	1	no	—
476	C₂²	—	106	1	no	—
477	C₂²	—	106	1	no	—
478	C₂²	—	106	1	no	—
479	C₂²	#16810648	1	2	yes	—
480	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	2	1	no	—
484	C₄	—	53	2	no	—
485	C₄	—	106	1	no	—
486	C₄	—	106	1	no	—
487	C₄	—	106	1	no	—
488	C₄	—	106	1	no	—
489	C₄	—	106	1	no	—
490	C₄	#16810648	1	2	yes	—
491	C₄	C₂×#13133223	1	1	yes	characteristic
492	C₄	C₂×#13133225	1	1	yes	characteristic
493	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
494	C₄	#137×D₅₃	1	1	yes	characteristic
495	C₂	—	2	2	no	—
496	C₂	—	106	2	no	—
497	C₂	—	106	2	no	—
498	C₂	C₂×#16810648	1	1	yes	characteristic
499	C₂	#428×D₅₃	1	1	yes	characteristic
500	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	#17807
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. #20273499	C₂	semidirect
5. #20273497	C₂	semidirect
6. #20273488	C₂	semidirect
7. #20273484	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
255. #1565	D₅₃	semidirect
294. #335	C₂×D₅₃	semidirect
298. #334	C₂×D₅₃	semidirect
299. C₅₃	#17807	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×C₄⋊C₈×D₅₃	2
#20273499	2
#20273497	2
#20273488	2
#20273484	2
#20273468	2
#20273467	2
#20273465	2
C₅₃×#1565	2
#17807	53