Subgroups — #27349793

Order 13568.

734 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	#20283679	C₂	1	1	yes	maximal, characteristic
2	#20283671	C₂	1	1	yes	maximal, characteristic
3	#20283354	C₂	1	1	yes	maximal, characteristic
4	#20273016	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273011	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273010	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273008	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273006	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	#20272996	C₂	1	1	yes	maximal, characteristic, complemented
12	#20272994	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
14	#322×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#11719945	C₂²	1	1	yes	characteristic
17	#16810752	C₄	1	2	yes	—
18	#16810747	C₄	1	2	yes	—
19	#16810698	C₄	1	2	yes	—
20	#16810697	C₄	1	2	yes	—
21	#16810693	C₄	1	2	yes	—
22	#16810688	C₄	1	2	yes	—
23	#16810686	C₄	1	2	yes	—
24	#16810686	C₄	1	2	yes	—
25	#16810683	C₄	1	2	yes	—
26	#16810681	C₄	1	2	yes	—
27	#16810676	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	#16810674	C₂²	1	1	yes	characteristic
32	#16810674	C₂²	1	1	yes	characteristic
33	#16810674	C₂²	1	1	yes	characteristic
34	#16810648	C₄	1	2	yes	—
35	#16810588	C₄	1	2	yes	—
36	C₄×#11719917	C₂²	1	1	yes	characteristic
37	C₄×#11719912	C₂²	1	1	yes	characteristic
38	C₄×#11719912	C₂²	1	1	yes	characteristic
39	C₂×C₄×#88322	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	#16810344	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	C₅₃×#431	C₂²	1	1	yes	characteristic
58	C₅₃×#431	C₂²	1	1	yes	characteristic
59	C₅₃×#322	C₂²	1	1	yes	characteristic
60	C₅₃×#322	C₂²	1	1	yes	characteristic
61	C₅₃×#322	C₂²	1	1	yes	characteristic
62	C₈²×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
64	C₂×#11719917	C₂×C₄	1	2	yes	—
65	C₂×#11719912	C₂×C₄	1	2	yes	—
66	C₂×#11719912	C₂×C₄	1	2	yes	—
67	C₂²×#88322	C₂×C₄	1	2	yes	—
68	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
72	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
74	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
75	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
76	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
80	#13133176	C₂×C₄	1	1	yes	characteristic
81	#13133174	C₂×C₄	1	1	yes	characteristic
82	#13133174	C₂×C₄	1	1	yes	characteristic
83	C₄²×D₅₃	C₂³	1	1	yes	characteristic
84	C₄²×D₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊#88322	C₂×C₄	1	2	yes	—
86	#13133146	C₂×C₄	1	2	yes	—
87	#13133146	C₂×C₄	1	2	yes	—
88	#13133146	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	C₂²⋊#88322	C₂×C₄	1	2	yes	—
93	#13133129	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	#13133129	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	C₄×#88322	C₂×C₄	1	2	yes	—
103	C₄×#88322	C₂³	1	1	yes	characteristic
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	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
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	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
115	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
116	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
117	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
118	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
119	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
120	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
131	#11719945	—	2	4	no	—
132	C₂×D₂₁₂	C₄²	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	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×Dic₁₀₆	C₄²	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
151	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
152	#11719921	C₄²	1	1	yes	characteristic
153	#11719921	C₄²	1	1	yes	characteristic
154	#11719921	C₄²	1	1	yes	characteristic
155	#11719921	C₄²	1	1	yes	characteristic
156	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
157	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
158	#11719919	C₄²	1	1	yes	characteristic
159	#11719919	C₄²	1	1	yes	characteristic
160	#11719919	C₄²	1	1	yes	characteristic
161	#11719919	C₄²	1	1	yes	characteristic
162	C₄×Dic₅₃	C₂²×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	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	#11719917	—	2	4	no	—
169	C₂×#88322	—	4	2	no	—
170	C₂×#88322	C₂²×C₄	1	2	yes	—
171	C₂×#88322	C₂²×C₄	1	2	yes	—
172	C₂×#88322	C₂²×C₄	1	2	yes	—
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	#11719912	—	2	4	no	—
177	#11719912	—	2	4	no	—
178	C₈×D₅₃	—	2	4	no	—
179	C₈×D₅₃	—	2	4	no	—
180	C₈×D₅₃	—	4	2	no	—
181	C₈×D₅₃	—	4	2	no	—
182	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
183	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
184	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
194	C₂²⋊D₅₃	—	2	2	no	—
195	C₂²⋊D₅₃	—	2	2	no	—
196	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
202	D₂₁₂	—	2	2	no	—
203	C₄×D₅₃	—	2	2	no	—
204	C₄×D₅₃	—	2	2	no	—
205	C₄×D₅₃	—	2	2	no	—
206	C₄×D₅₃	—	2	2	no	—
207	C₄×D₅₃	—	2	2	no	—
208	C₄×D₅₃	#131	1	4	yes	—
209	Dic₁₀₆	—	2	2	no	—
210	C₈×C₅₃	—	2	2	no	—
211	C₈×C₅₃	—	2	2	no	—
212	C₈×C₅₃	—	2	2	no	—
213	C₈×C₅₃	—	2	2	no	—
214	C₈×C₅₃	—	2	2	no	—
215	C₈×C₅₃	—	2	2	no	—
216	#88322	—	2	2	no	—
217	#88322	—	2	2	no	—
218	#88322	—	2	2	no	—
219	#88322	—	2	2	no	—
220	#88322	—	2	2	no	—
221	#88322	#131	1	4	yes	—
222	#17331	—	53	1	no	maximal, Sylow
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
227	C₂×D₅₃	—	2	1	no	—
228	C₂×D₅₃	—	2	1	no	—
229	C₂×D₅₃	—	2	1	no	—
230	C₂×D₅₃	—	2	1	no	—
231	C₂×D₅₃	—	2	1	no	—
232	C₂×D₅₃	#405	1	2	yes	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	C₂×#131	1	2	yes	—
237	C₄×C₅₃	C₂×#131	1	2	yes	—
238	C₄×C₅₃	#568	1	2	yes	—
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	#405	1	2	yes	—
245	#2965	—	53	1	no	—
246	C₈×#41	—	53	1	no	—
247	#2922	—	53	1	no	—
248	#1454	—	53	1	no	—
249	#1454	D₅₃	1	1	yes	characteristic, complemented
250	#1453	—	53	1	no	—
251	#1453	—	53	1	no	—
252	#1452	—	53	1	no	—
253	#1452	—	53	1	no	—
254	#1451	—	53	1	no	—
255	#1450	—	53	1	no	—
256	C₈×M₁₆	—	53	1	no	—
257	C₈×M₁₆	—	53	1	no	—
258	C₈×M₁₆	—	53	1	no	—
259	C₂×#322	—	53	1	no	—
260	C₂×C₅₃	—	4	1	no	—
261	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
262	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
263	C₂×C₅₃	#2875	1	1	yes	characteristic
264	D₅₃	—	2	2	no	—
265	D₅₃	—	4	1	no	—
266	C₄×#41	—	53	1	no	—
267	#446	—	53	2	no	—
268	C₈×Q₈	—	53	2	no	—
269	#436	—	53	2	no	—
270	#435	—	53	2	no	—
271	#435	—	53	2	no	—
272	C₈×D₄	—	53	2	no	—
273	C₈×D₄	—	53	2	no	—
274	C₈×D₄	—	53	2	no	—
275	#433	—	53	2	no	—
276	#433	—	53	2	no	—
277	#431	—	53	1	no	—
278	#431	—	53	1	no	—
279	#431	—	53	1	no	—
280	#431	—	53	1	no	—
281	#431	—	53	1	no	—
282	#431	—	53	1	no	—
283	#431	—	53	1	no	—
284	#431	C₂×D₅₃	1	1	yes	characteristic
285	#431	C₂×D₅₃	1	1	yes	characteristic
286	#431	C₂×D₅₃	1	1	yes	characteristic
287	#424	—	53	2	no	—
288	#408	—	53	2	no	—
289	C₄×M₁₆	—	53	1	no	—
290	C₄×M₁₆	—	53	1	no	—
291	C₄×M₁₆	—	53	1	no	—
292	C₂×C₄×C₈	—	53	1	no	—
293	C₂×C₄×C₈	—	53	1	no	—
294	C₂×C₄×C₈	—	53	1	no	—
295	C₂×C₄×C₈	—	53	1	no	—
296	C₂×C₄×C₈	—	53	1	no	—
297	#322	—	53	1	no	—
298	#322	—	53	1	no	—
299	#322	—	53	1	no	—
300	#322	—	53	1	no	—
301	#322	—	53	1	no	—
302	#322	—	53	1	no	—
303	#322	—	53	1	no	—
304	#322	—	53	1	no	—
305	#322	—	53	1	no	—
306	#322	C₂×D₅₃	1	1	yes	characteristic
307	#322	C₂×D₅₃	1	1	yes	characteristic
308	#322	C₂×D₅₃	1	1	yes	characteristic
309	C₈²	—	53	1	no	—
310	C₈²	—	53	1	no	—
311	C₈²	—	53	1	no	—
312	C₈²	C₂×D₅₃	1	1	yes	characteristic
313	C₅₃	#17331	1	1	yes	characteristic, complemented, Sylow
314	C₂×#41	—	53	1	no	—
315	C₂×M₁₆	—	53	2	no	—
316	C₂×M₁₆	—	53	2	no	—
317	C₂×M₁₆	—	53	2	no	—
318	C₂²×C₈	—	53	2	no	—
319	C₂²×C₈	—	53	2	no	—
320	C₂²×C₈	—	53	2	no	—
321	C₂²×C₈	—	53	2	no	—
322	C₂²×C₈	—	53	2	no	—
323	C₄×Q₈	—	53	1	no	—
324	C₄×Q₈	—	53	1	no	—
325	C₄×D₄	—	53	1	no	—
326	C₄×D₄	—	53	1	no	—
327	C₄×D₄	—	53	1	no	—
328	C₄×D₄	—	53	1	no	—
329	C₄×D₄	—	53	1	no	—
330	C₄×D₄	—	53	1	no	—
331	#117	—	53	1	no	—
332	#117	—	53	1	no	—
333	#117	—	53	1	no	—
334	C₂×C₄²	—	53	1	no	—
335	C₂×C₄²	—	53	1	no	—
336	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
337	C₄⋊C₈	—	53	2	no	—
338	C₄⋊C₈	—	53	2	no	—
339	C₄⋊C₈	—	53	2	no	—
340	C₄⋊C₈	—	53	2	no	—
341	C₄⋊C₈	—	53	2	no	—
342	C₄⋊C₈	—	53	2	no	—
343	C₄⋊C₈	—	53	2	no	—
344	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
345	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
346	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
347	C₂²⋊C₈	—	53	2	no	—
348	C₂²⋊C₈	—	53	2	no	—
349	C₂²⋊C₈	—	53	2	no	—
350	C₂²⋊C₈	—	53	2	no	—
351	C₂²⋊C₈	—	53	2	no	—
352	C₂²⋊C₈	—	53	2	no	—
353	C₂²⋊C₈	—	53	2	no	—
354	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
355	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
356	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
357	#97	—	53	2	no	—
358	#97	—	53	2	no	—
359	#97	—	53	2	no	—
360	C₄×C₈	—	53	2	no	—
361	C₄×C₈	—	53	2	no	—
362	C₄×C₈	—	53	2	no	—
363	C₄×C₈	—	53	2	no	—
364	C₄×C₈	—	53	2	no	—
365	C₄×C₈	—	53	1	no	—
366	C₄×C₈	—	53	1	no	—
367	C₄×C₈	—	53	1	no	—
368	C₄×C₈	—	53	1	no	—
369	C₄×C₈	—	53	1	no	—
370	C₄×C₈	—	53	1	no	—
371	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
372	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
373	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
374	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
375	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
376	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
377	#41	—	106	4	no	—
378	C₂×Q₈	—	53	1	no	—
379	C₂×D₄	—	53	1	no	—
380	C₂×D₄	—	53	1	no	—
381	C₂×D₄	—	53	1	no	—
382	C₂²×C₄	—	53	1	no	—
383	C₂²×C₄	—	53	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	53	1	no	—
386	C₂²×C₄	—	53	1	no	—
387	C₂²×C₄	—	53	1	no	—
388	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
391	M₁₆	—	106	4	no	—
392	M₁₆	—	106	4	no	—
393	M₁₆	—	106	4	no	—
394	C₂×C₈	—	53	2	no	—
395	C₂×C₈	—	53	2	no	—
396	C₂×C₈	—	53	2	no	—
397	C₂×C₈	—	53	2	no	—
398	C₂×C₈	—	53	2	no	—
399	C₂×C₈	—	53	2	no	—
400	C₂×C₈	—	106	4	no	—
401	C₂×C₈	—	106	4	no	—
402	C₂×C₈	—	212	2	no	—
403	C₂×C₈	—	212	2	no	—
404	C₂×C₈	—	212	2	no	—
405	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
406	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
407	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
408	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
409	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
410	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
411	C₄⋊C₄	—	53	1	no	—
412	C₄⋊C₄	—	53	1	no	—
413	C₄⋊C₄	—	53	1	no	—
414	C₄⋊C₄	—	53	1	no	—
415	C₄⋊C₄	—	53	1	no	—
416	C₄⋊C₄	—	53	1	no	—
417	C₂²⋊C₄	—	53	1	no	—
418	C₂²⋊C₄	—	53	1	no	—
419	C₂²⋊C₄	—	53	1	no	—
420	C₂²⋊C₄	—	53	1	no	—
421	C₂²⋊C₄	—	53	1	no	—
422	C₂²⋊C₄	—	53	1	no	—
423	C₄²	—	53	1	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	—	53	1	no	—
427	C₄²	—	53	1	no	—
428	C₄²	—	53	1	no	—
429	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
430	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
432	C₄²	C₂³×D₅₃	1	1	yes	characteristic
433	C₂³	—	53	1	no	—
434	C₂³	—	53	1	no	—
435	C₂³	C₄²×D₅₃	1	1	yes	characteristic
436	Q₈	—	106	2	no	—
437	D₄	—	106	2	no	—
438	D₄	—	106	2	no	—
439	D₄	—	106	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	53	1	no	—
444	C₂×C₄	—	53	4	no	—
445	C₂×C₄	—	53	1	no	—
446	C₂×C₄	—	53	1	no	—
447	C₂×C₄	—	53	1	no	—
448	C₂×C₄	—	53	1	no	—
449	C₂×C₄	—	53	1	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	2	no	—
455	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
456	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
457	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
458	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	2	2	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	2	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	—	53	4	no	—
468	C₈	—	106	2	no	—
469	C₈	—	106	2	no	—
470	C₈	—	106	2	no	—
471	C₈	—	106	2	no	—
472	C₈	—	106	2	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	53	2	no	—
477	C₂²	—	106	1	no	—
478	C₂²	—	106	1	no	—
479	C₂²	—	106	1	no	—
480	C₂²	—	106	1	no	—
481	C₂²	—	106	1	no	—
482	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
483	C₄	—	2	1	no	—
484	C₄	—	2	1	no	—
485	C₄	—	2	1	no	—
486	C₄	—	53	2	no	—
487	C₄	—	106	1	no	—
488	C₄	—	106	1	no	—
489	C₄	—	106	1	no	—
490	C₄	—	106	1	no	—
491	C₄	—	106	1	no	—
492	C₄	#131×D₅₃	1	2	yes	—
493	C₄	#131×D₅₃	1	2	yes	—
494	C₄	#16811388	1	2	yes	—
495	C₂	—	4	1	no	—
496	C₂	—	106	2	no	—
497	C₂	—	212	1	no	—
498	C₂	#405×D₅₃	1	1	yes	characteristic
499	C₂	#20283137	1	1	yes	characteristic
500	C₂	#20283142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1454	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1454	D₅₃
hypocenter	C₅₃	#17331
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1454

Complements

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

N	H ≅ G/N	type
4. #20273016	C₂	semidirect
5. #20273011	C₂	semidirect
6. #20273010	C₂	semidirect
7. #20273008	C₂	semidirect
8. #20273006	C₂	semidirect
9. #20273001	C₂	semidirect
10. #20272997	C₂	semidirect
11. #20272996	C₂	semidirect
12. #20272994	C₂	semidirect
13. C₈×#11719912	C₂	semidirect
14. #322×D₅₃	C₂	semidirect
15. C₅₃×#1454	C₂	semidirect
249. #1454	D₅₃	semidirect
313. C₅₃	#17331	semidirect

Maximal subgroups

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

subgroup	index
#20283679	2
#20283671	2
#20283354	2
#20273016	2
#20273011	2
#20273010	2
#20273008	2
#20273006	2
#20273001	2
#20272997	2
#20272996	2
#20272994	2
C₈×#11719912	2
#322×D₅₃	2
C₅₃×#1454	2
#17331	53