Subgroups — #27353229

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283829	C₂	1	1	yes	maximal, characteristic
2	#20283823	C₂	1	1	yes	maximal, characteristic
3	#20283354	C₂	1	1	yes	maximal, characteristic
4	#20273502	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273500	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273499	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273493	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273492	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273491	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273488	C₂	1	1	yes	maximal, characteristic, complemented
11	#20273466	C₂	1	1	yes	maximal, characteristic, complemented
12	#20273464	C₂	1	1	yes	maximal, characteristic, complemented
13	#20273460	C₂	1	1	yes	maximal, characteristic, complemented
14	#335×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#11719945	C₂²	1	1	yes	characteristic
17	#16810752	—	2	1	no	—
18	C₈×Dic₁₀₆	—	2	1	no	—
19	#16810699	—	2	1	no	—
20	#16810693	—	2	1	no	—
21	#16810692	—	2	1	no	—
22	C₈×D₂₁₂	—	2	1	no	—
23	#16810686	—	2	1	no	—
24	#16810686	—	2	1	no	—
25	#16810683	C₄	1	2	yes	—
26	#16810681	C₄	1	2	yes	—
27	#16810680	C₂²	1	1	yes	characteristic
28	#16810679	C₂²	1	1	yes	characteristic
29	#16810676	C₂²	1	1	yes	characteristic
30	#16810675	C₂²	1	1	yes	characteristic
31	#16810675	C₂²	1	1	yes	characteristic
32	#16810675	C₂²	1	1	yes	characteristic
33	#16810674	C₂²	1	1	yes	characteristic
34	#16810674	C₂²	1	1	yes	characteristic
35	#16810648	C₄	1	2	yes	—
36	#16810645	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
41	#16810588	C₄	1	2	yes	—
42	C₄×#11719912	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
44	#16810376	C₂²	1	1	yes	characteristic
45	#16810376	C₂²	1	1	yes	characteristic
46	#16810375	C₂²	1	1	yes	characteristic
47	#16810375	C₂²	1	1	yes	characteristic
48	#16810375	C₂²	1	1	yes	characteristic
49	#16810375	C₂²	1	1	yes	characteristic
50	#16810375	C₂²	1	1	yes	characteristic
51	#16810375	C₂²	1	1	yes	characteristic
52	#16810374	C₂²	1	1	yes	characteristic
53	#16810374	C₂²	1	1	yes	characteristic
54	#16810373	C₂²	1	1	yes	characteristic
55	#16810373	C₂²	1	1	yes	characteristic
56	C₅₃×#431	C₂²	1	1	yes	characteristic
57	C₅₃×#431	C₂²	1	1	yes	characteristic
58	C₅₃×#423	C₂²	1	1	yes	characteristic
59	C₅₃×#335	C₂²	1	1	yes	characteristic
60	C₅₃×#335	C₂²	1	1	yes	characteristic
61	C₅₃×#334	C₂²	1	1	yes	characteristic
62	C₅₃×#334	C₂²	1	1	yes	characteristic
63	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
64	C₂×#11719912	—	2	1	no	—
65	C₂×#11719912	C₂×C₄	1	2	yes	—
66	C₂²×#88322	—	2	1	no	—
67	C₂×C₈×D₅₃	—	2	1	no	—
68	C₂×C₈×D₅₃	—	2	1	no	—
69	C₂×C₈×D₅₃	—	2	1	no	—
70	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
72	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
73	C₄×D₂₁₂	D₄	1	1	yes	characteristic
74	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
75	C₄×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
76	C₄×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
77	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
78	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
79	#13133176	C₂×C₄	1	1	yes	characteristic
80	#13133174	C₂×C₄	1	1	yes	characteristic
81	#13133174	C₂×C₄	1	1	yes	characteristic
82	C₄²×D₅₃	C₂³	1	1	yes	characteristic
83	C₄²×D₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊#88322	—	2	1	no	—
85	C₄⋊#88322	C₂×C₄	1	2	yes	—
86	C₄⋊#88322	C₂³	1	1	yes	characteristic
87	C₄⋊#88322	C₂³	1	1	yes	characteristic
88	C₄⋊#88322	C₂³	1	1	yes	characteristic
89	C₄⋊#88322	C₂³	1	1	yes	characteristic
90	#13133146	—	2	1	no	—
91	#13133146	—	2	1	no	—
92	#13133146	—	2	1	no	—
93	#13133146	—	2	1	no	—
94	#13133146	C₂×C₄	1	2	yes	—
95	#13133146	C₂×C₄	1	2	yes	—
96	C₂²⋊#88322	—	2	1	no	—
97	C₂²⋊#88322	C₂×C₄	1	2	yes	—
98	#13133129	—	2	1	no	—
99	#13133129	—	2	1	no	—
100	#13133129	—	2	1	no	—
101	#13133129	—	2	1	no	—
102	#13133129	C₂×C₄	1	2	yes	—
103	#13133129	C₂×C₄	1	2	yes	—
104	#13133128	—	2	1	no	—
105	#13133127	—	2	1	no	—
106	#13133127	C₂×C₄	1	2	yes	—
107	C₄×#88322	C₂³	1	1	yes	characteristic
108	C₄×#88322	C₂³	1	1	yes	characteristic
109	C₈×Dic₅₃	—	2	1	no	—
110	C₈×Dic₅₃	—	2	1	no	—
111	C₈×Dic₅₃	—	2	1	no	—
112	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
113	C₂×M₁₆×C₅₃	—	2	1	no	—
114	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
115	C₄⋊C₈×C₅₃	—	2	1	no	—
116	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
117	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
121	C₂²⋊C₈×C₅₃	—	2	1	no	—
122	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
123	C₄×C₈×C₅₃	—	2	1	no	—
124	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
131	#11719945	—	2	4	no	—
132	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
139	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
140	M₁₆×C₅₃	—	8	1	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	1	no	—
144	C₂×C₈×C₅₃	—	2	1	no	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
147	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
150	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
152	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
153	#11719921	C₄⋊C₄	1	1	yes	characteristic
154	#11719921	C₄⋊C₄	1	1	yes	characteristic
155	#11719921	C₄⋊C₄	1	1	yes	characteristic
156	#11719921	C₄⋊C₄	1	1	yes	characteristic
157	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
158	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
159	#11719919	C₄⋊C₄	1	1	yes	characteristic
160	#11719919	C₄⋊C₄	1	1	yes	characteristic
161	#11719919	C₄⋊C₄	1	1	yes	characteristic
162	#11719919	C₄⋊C₄	1	1	yes	characteristic
163	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
166	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
167	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
168	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
169	C₂×#88322	—	2	1	no	—
170	C₂×#88322	—	2	1	no	—
171	C₂×#88322	—	2	1	no	—
172	C₂×#88322	—	2	1	no	—
173	C₂×#88322	—	8	1	no	—
174	C₂×#88322	C₂²×C₄	1	2	yes	—
175	C₂×#88322	C₂²×C₄	1	2	yes	—
176	#11719912	—	2	4	no	—
177	#11719912	—	8	1	no	—
178	C₈×D₅₃	—	2	4	no	—
179	C₈×D₅₃	—	8	1	no	—
180	C₈×D₅₃	—	8	1	no	—
181	C₈×D₅₃	—	8	1	no	—
182	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
195	C₂²⋊D₅₃	—	2	2	no	—
196	C₂²⋊D₅₃	—	2	2	no	—
197	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
203	D₂₁₂	—	2	2	no	—
204	C₄×D₅₃	—	2	2	no	—
205	C₄×D₅₃	—	2	1	no	—
206	C₄×D₅₃	—	2	1	no	—
207	C₄×D₅₃	—	2	1	no	—
208	C₄×D₅₃	—	2	1	no	—
209	C₄×D₅₃	—	4	1	no	—
210	C₄×D₅₃	—	4	1	no	—
211	C₄×D₅₃	#131	1	4	yes	—
212	Dic₁₀₆	—	2	2	no	—
213	C₈×C₅₃	—	2	2	no	—
214	C₈×C₅₃	—	2	2	no	—
215	C₈×C₅₃	—	4	1	no	—
216	C₈×C₅₃	—	4	1	no	—
217	C₈×C₅₃	—	4	1	no	—
218	C₈×C₅₃	—	4	1	no	—
219	#88322	—	2	2	no	—
220	#88322	—	2	2	no	—
221	#88322	—	4	1	no	—
222	#88322	—	4	1	no	—
223	#88322	—	4	1	no	—
224	#88322	—	4	1	no	—
225	#17812	—	53	1	no	maximal, Sylow
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	—	2	1	no	—
228	C₂²×C₅₃	—	2	1	no	—
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	#424	1	2	yes	—
236	C₄×C₅₃	—	2	1	no	—
237	C₄×C₅₃	—	2	1	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	C₂×#131	1	2	yes	—
241	C₄×C₅₃	#576	1	1	yes	characteristic
242	C₄×C₅₃	#578	1	1	yes	characteristic
243	Dic₅₃	—	2	1	no	—
244	Dic₅₃	—	2	1	no	—
245	Dic₅₃	—	2	1	no	—
246	Dic₅₃	—	2	1	no	—
247	Dic₅₃	—	2	1	no	—
248	Dic₅₃	#424	1	2	yes	—
249	#2989	—	53	1	no	—
250	#2988	—	53	1	no	—
251	#2922	—	53	1	no	—
252	#1574	—	53	1	no	—
253	#1574	D₅₃	1	1	yes	characteristic, complemented
254	#1573	—	53	1	no	—
255	#1573	—	53	1	no	—
256	#1572	—	53	1	no	—
257	#1572	—	53	1	no	—
258	#1571	—	53	1	no	—
259	#1571	—	53	1	no	—
260	#1565	—	53	1	no	—
261	#1565	—	53	1	no	—
262	#1564	—	53	1	no	—
263	C₂×#335	—	53	1	no	—
264	C₂×C₅₃	—	4	1	no	—
265	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
266	C₂×C₅₃	#2913	1	1	yes	characteristic
267	C₂×C₅₃	#2916	1	1	yes	characteristic
268	D₅₃	—	2	2	no	—
269	D₅₃	—	4	1	no	—
270	C₄×#41	—	53	1	no	—
271	#446	—	106	1	no	—
272	C₈×Q₈	—	106	1	no	—
273	#436	—	106	1	no	—
274	#435	—	106	1	no	—
275	#435	—	106	1	no	—
276	C₈×D₄	—	106	1	no	—
277	C₈×D₄	—	106	1	no	—
278	C₈×D₄	—	106	1	no	—
279	#433	—	53	2	no	—
280	#433	—	53	2	no	—
281	#432	—	53	1	no	—
282	#432	—	53	1	no	—
283	#431	—	53	1	no	—
284	#431	—	53	1	no	—
285	#431	—	53	1	no	—
286	#431	—	53	1	no	—
287	#431	—	53	1	no	—
288	#431	—	53	1	no	—
289	#431	C₂×D₅₃	1	1	yes	characteristic
290	#431	C₂×D₅₃	1	1	yes	characteristic
291	#424	—	53	2	no	—
292	#423	—	53	1	no	—
293	#423	C₂×D₅₃	1	1	yes	characteristic
294	C₂×C₄⋊C₈	—	53	1	no	—
295	C₂×C₄⋊C₈	—	53	1	no	—
296	C₂×C₄⋊C₈	—	53	1	no	—
297	C₂×C₄⋊C₈	—	53	1	no	—
298	#408	—	53	2	no	—
299	C₄×M₁₆	—	53	1	no	—
300	C₂×C₄×C₈	—	53	1	no	—
301	#335	—	53	1	no	—
302	#335	—	53	1	no	—
303	#335	—	53	1	no	—
304	#335	—	53	1	no	—
305	#335	—	53	1	no	—
306	#335	—	53	1	no	—
307	#335	—	53	1	no	—
308	#335	—	53	1	no	—
309	#335	C₂×D₅₃	1	1	yes	characteristic
310	#335	C₂×D₅₃	1	1	yes	characteristic
311	#334	—	53	1	no	—
312	#334	—	53	1	no	—
313	#334	—	53	1	no	—
314	#334	—	53	1	no	—
315	#334	C₂×D₅₃	1	1	yes	characteristic
316	#334	C₂×D₅₃	1	1	yes	characteristic
317	C₅₃	#17812	1	1	yes	characteristic, complemented, Sylow
318	C₂×#41	—	53	1	no	—
319	C₂×M₁₆	—	2	1	no	—
320	C₂×M₁₆	—	53	2	no	—
321	C₂×M₁₆	—	106	1	no	—
322	C₂²×C₈	—	53	2	no	—
323	C₂²×C₈	—	106	1	no	—
324	C₂²×C₈	—	106	1	no	—
325	C₂²×C₈	—	106	1	no	—
326	C₂²×C₈	—	106	1	no	—
327	C₄×Q₈	—	53	1	no	—
328	C₄×Q₈	—	53	1	no	—
329	C₄×D₄	—	53	1	no	—
330	C₄×D₄	—	53	1	no	—
331	C₄×D₄	—	53	1	no	—
332	C₄×D₄	—	53	1	no	—
333	C₄×D₄	—	53	1	no	—
334	C₄×D₄	—	53	1	no	—
335	#117	—	53	1	no	—
336	#117	—	53	1	no	—
337	#117	—	53	1	no	—
338	C₂×C₄²	—	53	1	no	—
339	C₂×C₄²	—	53	1	no	—
340	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	53	2	no	—
343	C₄⋊C₈	—	53	2	no	—
344	C₄⋊C₈	—	53	2	no	—
345	C₄⋊C₈	—	53	1	no	—
346	C₄⋊C₈	—	53	1	no	—
347	C₄⋊C₈	—	53	1	no	—
348	C₄⋊C₈	—	53	1	no	—
349	C₄⋊C₈	—	106	1	no	—
350	C₄⋊C₈	—	106	1	no	—
351	C₄⋊C₈	—	106	1	no	—
352	C₄⋊C₈	—	106	1	no	—
353	C₄⋊C₈	—	106	1	no	—
354	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
355	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
356	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
357	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
358	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
359	C₂²⋊C₈	—	2	1	no	—
360	C₂²⋊C₈	—	53	2	no	—
361	C₂²⋊C₈	—	53	2	no	—
362	C₂²⋊C₈	—	53	2	no	—
363	C₂²⋊C₈	—	106	1	no	—
364	C₂²⋊C₈	—	106	1	no	—
365	C₂²⋊C₈	—	106	1	no	—
366	C₂²⋊C₈	—	106	1	no	—
367	C₂²⋊C₈	—	106	1	no	—
368	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
369	#97	—	53	2	no	—
370	#97	—	106	1	no	—
371	#97	—	106	1	no	—
372	C₄×C₈	—	2	1	no	—
373	C₄×C₈	—	53	2	no	—
374	C₄×C₈	—	53	1	no	—
375	C₄×C₈	—	53	1	no	—
376	C₄×C₈	—	106	1	no	—
377	C₄×C₈	—	106	1	no	—
378	C₄×C₈	—	106	1	no	—
379	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
380	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
381	#41	—	106	4	no	—
382	C₂×Q₈	—	53	1	no	—
383	C₂×D₄	—	53	1	no	—
384	C₂×D₄	—	53	1	no	—
385	C₂×D₄	—	53	1	no	—
386	C₂²×C₄	—	53	1	no	—
387	C₂²×C₄	—	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	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
394	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
395	M₁₆	—	8	1	no	—
396	M₁₆	—	106	4	no	—
397	M₁₆	—	424	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	53	2	no	—
403	C₂×C₈	—	53	2	no	—
404	C₂×C₈	—	106	4	no	—
405	C₂×C₈	—	106	1	no	—
406	C₂×C₈	—	106	1	no	—
407	C₂×C₈	—	106	1	no	—
408	C₂×C₈	—	106	1	no	—
409	C₂×C₈	—	424	1	no	—
410	C₂×C₈	—	424	1	no	—
411	C₂×C₈	—	424	1	no	—
412	C₂×C₈	—	424	1	no	—
413	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
414	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
415	C₄⋊C₄	—	53	1	no	—
416	C₄⋊C₄	—	53	1	no	—
417	C₄⋊C₄	—	53	1	no	—
418	C₄⋊C₄	—	53	1	no	—
419	C₄⋊C₄	—	53	1	no	—
420	C₄⋊C₄	—	53	1	no	—
421	C₂²⋊C₄	—	53	1	no	—
422	C₂²⋊C₄	—	53	1	no	—
423	C₂²⋊C₄	—	53	1	no	—
424	C₂²⋊C₄	—	53	1	no	—
425	C₂²⋊C₄	—	53	1	no	—
426	C₂²⋊C₄	—	53	1	no	—
427	C₄²	—	53	1	no	—
428	C₄²	—	53	1	no	—
429	C₄²	—	53	1	no	—
430	C₄²	—	53	1	no	—
431	C₄²	—	53	1	no	—
432	C₄²	—	53	1	no	—
433	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
434	C₄²	D₄×D₅₃	1	1	yes	characteristic
435	C₄²	Q₈×D₅₃	1	1	yes	characteristic
436	C₄²	C₂³×D₅₃	1	1	yes	characteristic
437	C₂³	—	53	1	no	—
438	C₂³	—	53	1	no	—
439	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
440	Q₈	—	106	2	no	—
441	D₄	—	106	2	no	—
442	D₄	—	106	2	no	—
443	D₄	—	106	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	53	1	no	—
449	C₂×C₄	—	53	4	no	—
450	C₂×C₄	—	53	1	no	—
451	C₂×C₄	—	53	1	no	—
452	C₂×C₄	—	53	1	no	—
453	C₂×C₄	—	53	1	no	—
454	C₂×C₄	—	53	1	no	—
455	C₂×C₄	—	106	2	no	—
456	C₂×C₄	—	106	1	no	—
457	C₂×C₄	—	106	1	no	—
458	C₂×C₄	—	106	1	no	—
459	C₂×C₄	—	106	1	no	—
460	C₂×C₄	—	212	1	no	—
461	C₂×C₄	—	212	1	no	—
462	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
463	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
464	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
467	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	4	1	no	—
471	C₈	—	4	1	no	—
472	C₈	—	4	1	no	—
473	C₈	—	4	1	no	—
474	C₈	—	106	2	no	—
475	C₈	—	106	2	no	—
476	C₈	—	212	1	no	—
477	C₈	—	212	1	no	—
478	C₈	—	212	1	no	—
479	C₈	—	212	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	53	2	no	—
484	C₂²	—	106	1	no	—
485	C₂²	—	106	1	no	—
486	C₂²	—	106	1	no	—
487	C₂²	—	106	1	no	—
488	C₂²	—	106	1	no	—
489	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	53	2	no	—
495	C₄	—	106	1	no	—
496	C₄	—	106	1	no	—
497	C₄	—	106	1	no	—
498	C₄	—	106	1	no	—
499	C₄	—	106	1	no	—
500	C₄	#131×D₅₃	1	2	yes	—
501	C₄	#16811427	1	1	yes	characteristic
502	C₄	#16811439	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	106	2	no	—
505	C₂	—	212	1	no	—
506	C₂	#424×D₅₃	1	1	yes	characteristic
507	C₂	#20283314	1	1	yes	characteristic
508	C₂	#20283323	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×#1574	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1574	D₅₃
hypocenter	C₅₃	#17812
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
4. #20273502	C₂	semidirect
5. #20273500	C₂	semidirect
6. #20273499	C₂	semidirect
7. #20273493	C₂	semidirect
8. #20273492	C₂	semidirect
9. #20273491	C₂	semidirect
10. #20273488	C₂	semidirect
11. #20273466	C₂	semidirect
12. #20273464	C₂	semidirect
13. #20273460	C₂	semidirect
14. #335×D₅₃	C₂	semidirect
15. C₅₃×#1574	C₂	semidirect
253. #1574	D₅₃	semidirect
317. C₅₃	#17812	semidirect

Maximal subgroups

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

subgroup	index
#20283829	2
#20283823	2
#20283354	2
#20273502	2
#20273500	2
#20273499	2
#20273493	2
#20273492	2
#20273491	2
#20273488	2
#20273466	2
#20273464	2
#20273460	2
#335×D₅₃	2
C₅₃×#1574	2
#17812	53