Subgroups — #27401521

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277208	C₂	1	1	yes	maximal, characteristic
2	#20275274	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275273	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275271	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274899	C₂	1	1	yes	maximal, characteristic
6	#20274891	C₂	1	1	yes	maximal, characteristic
7	#20274859	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274831	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274785	C₂	1	1	yes	maximal, characteristic
10	#20274575	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274500	C₂	1	1	yes	maximal, characteristic
12	#20274485	C₂	1	1	yes	maximal, characteristic
13	#20274459	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274410	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#11719912	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×#13133174	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133146	C₂²	1	1	yes	characteristic
21	C₂×#13133146	C₂²	1	1	yes	characteristic
22	C₂×#13133129	C₂²	1	1	yes	characteristic
23	C₂×#13133129	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
27	#16810515	C₄	1	1	yes	characteristic
28	#16810515	C₄	1	1	yes	characteristic
29	#16810513	C₄	1	1	yes	characteristic
30	#16810513	C₄	1	1	yes	characteristic
31	#16810510	C₄	1	1	yes	characteristic
32	#16810510	C₄	1	1	yes	characteristic
33	#16810504	C₄	1	1	yes	characteristic
34	#16810504	C₄	1	1	yes	characteristic
35	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
36	C₄×#11719921	C₂²	1	1	yes	characteristic
37	C₄×#11719921	C₂²	1	1	yes	characteristic
38	#16810469	C₂²	1	1	yes	characteristic
39	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
40	#16810378	C₂²	1	1	yes	characteristic
41	#16810378	C₂²	1	1	yes	characteristic
42	#16810378	C₂²	1	1	yes	characteristic
43	#16810378	C₂²	1	1	yes	characteristic
44	#16810377	C₂²	1	1	yes	characteristic
45	#16810377	C₂²	1	1	yes	characteristic
46	#16810377	C₂²	1	1	yes	characteristic
47	#16810377	C₂²	1	1	yes	characteristic
48	#16810377	C₂²	1	1	yes	characteristic
49	#16810377	C₂²	1	1	yes	characteristic
50	#16810377	C₂²	1	1	yes	characteristic
51	#16810377	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₅₃×#336	C₂²	1	1	yes	characteristic
56	C₅₃×#336	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
60	C₂×#11719912	—	2	2	no	—
61	C₂×#11719912	—	2	2	no	—
62	C₂×#11719912	—	4	1	no	—
63	C₂²×#88322	C₂³	1	1	yes	characteristic
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂²×#88322	C₂³	1	1	yes	characteristic
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂×C₈×D₅₃	—	2	2	no	—
68	C₂×C₈×D₅₃	—	2	2	no	—
69	C₂×C₈×D₅₃	—	4	1	no	—
70	#13133174	—	2	2	no	—
71	#13133174	—	2	2	no	—
72	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
75	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
76	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
77	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
78	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
79	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
80	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
81	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊#88322	—	4	1	no	—
90	#13133146	—	2	2	no	—
91	#13133146	—	2	2	no	—
92	#13133129	—	2	2	no	—
93	#13133129	—	2	2	no	—
94	C₄×#88322	—	4	1	no	—
95	C₈×Dic₅₃	—	2	2	no	—
96	C₈×Dic₅₃	—	2	2	no	—
97	#13133124	C₂×C₄	1	1	yes	characteristic
98	#13133124	C₂×C₄	1	1	yes	characteristic
99	#13133124	C₂×C₄	1	1	yes	characteristic
100	#13133124	C₂×C₄	1	1	yes	characteristic
101	#13133124	C₂×C₄	1	1	yes	characteristic
102	#13133124	C₂×C₄	1	1	yes	characteristic
103	#13133124	C₂×C₄	1	1	yes	characteristic
104	#13133124	C₂×C₄	1	1	yes	characteristic
105	#13133124	C₂×C₄	1	1	yes	characteristic
106	#13133124	C₂×C₄	1	1	yes	characteristic
107	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₅₃	—	4	1	no	—
117	#97×C₅₃	—	4	1	no	—
118	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	—	2	1	no	—
136	C₂×C₄×D₅₃	—	2	1	no	—
137	C₂×C₄×D₅₃	—	2	1	no	—
138	C₂×C₄×D₅₃	—	2	1	no	—
139	C₂×C₄×D₅₃	#41	1	2	yes	—
140	C₂×C₄×D₅₃	#41	1	2	yes	—
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	—	2	1	no	—
146	C₂×C₈×C₅₃	—	2	1	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	1	no	—
150	C₂×C₈×C₅₃	—	2	1	no	—
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	#41	1	2	yes	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	4	1	no	—
155	C₄²×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	—	2	1	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	#11719921	—	2	2	no	—
162	#11719921	—	2	2	no	—
163	#11719921	—	2	2	no	—
164	#11719921	—	2	2	no	—
165	#11719921	—	2	2	no	—
166	#11719921	—	4	1	no	—
167	C₄⋊Dic₅₃	—	2	2	no	—
168	C₄⋊Dic₅₃	—	2	2	no	—
169	#11719919	—	2	2	no	—
170	#11719919	—	2	2	no	—
171	C₄×Dic₅₃	—	2	2	no	—
172	C₄×Dic₅₃	—	2	1	no	—
173	C₄×Dic₅₃	—	2	1	no	—
174	C₄×Dic₅₃	—	2	1	no	—
175	C₄×Dic₅₃	—	2	1	no	—
176	C₄×Dic₅₃	—	2	1	no	—
177	C₄×Dic₅₃	—	2	1	no	—
178	C₄×Dic₅₃	—	2	2	no	—
179	C₄×Dic₅₃	—	2	2	no	—
180	C₄×Dic₅₃	—	4	1	no	—
181	C₂×#88322	—	2	1	no	—
182	C₂×#88322	—	2	1	no	—
183	C₂×#88322	—	2	1	no	—
184	C₂×#88322	—	2	1	no	—
185	C₂×#88322	—	2	1	no	—
186	C₂×#88322	—	2	1	no	—
187	C₂×#88322	—	2	1	no	—
188	C₂×#88322	—	2	1	no	—
189	C₂×#88322	—	2	1	no	—
190	C₂×#88322	—	2	1	no	—
191	C₂×#88322	—	2	1	no	—
192	C₂×#88322	#41	1	2	yes	—
193	#11719912	—	4	4	no	—
194	C₈×D₅₃	—	4	4	no	—
195	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
196	C₂²×D₅₃	—	2	1	no	—
197	C₂²×D₅₃	—	2	1	no	—
198	C₂²×D₅₃	—	2	1	no	—
199	C₂²×D₅₃	—	2	1	no	—
200	C₂²×D₅₃	—	2	1	no	—
201	C₂²×D₅₃	—	2	1	no	—
202	C₂²×D₅₃	#117	1	2	yes	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	#131	1	2	yes	—
221	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
227	C₂×Dic₅₃	—	2	1	no	—
228	C₂×Dic₅₃	—	2	1	no	—
229	C₂×Dic₅₃	—	2	1	no	—
230	C₂×Dic₅₃	—	2	1	no	—
231	C₂×Dic₅₃	—	2	1	no	—
232	C₂×Dic₅₃	—	2	1	no	—
233	C₂×Dic₅₃	—	2	1	no	—
234	C₂×Dic₅₃	—	2	1	no	—
235	C₂×Dic₅₃	—	2	1	no	—
236	C₂×Dic₅₃	—	2	1	no	—
237	C₂×Dic₅₃	—	2	1	no	—
238	C₂×Dic₅₃	—	2	1	no	—
239	C₂×Dic₅₃	—	2	1	no	—
240	C₂×Dic₅₃	—	2	1	no	—
241	C₂×Dic₅₃	—	2	1	no	—
242	C₂×Dic₅₃	—	2	1	no	—
243	C₂×Dic₅₃	—	2	1	no	—
244	C₂×Dic₅₃	—	2	1	no	—
245	C₂×Dic₅₃	#117	1	2	yes	—
246	C₂×Dic₅₃	#131	1	2	yes	—
247	C₂×Dic₅₃	#131	1	2	yes	—
248	C₄×D₅₃	—	2	2	no	—
249	C₄×D₅₃	—	2	2	no	—
250	C₄×D₅₃	—	2	2	no	—
251	C₄×D₅₃	—	2	2	no	—
252	C₈×C₅₃	—	2	2	no	—
253	C₈×C₅₃	—	4	1	no	—
254	C₈×C₅₃	—	4	1	no	—
255	C₈×C₅₃	—	4	1	no	—
256	#88322	—	2	2	no	—
257	#88322	—	4	1	no	—
258	#88322	—	4	1	no	—
259	#88322	—	4	1	no	—
260	#22593	—	53	1	no	maximal, Sylow
261	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
262	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
263	C₂²×C₅₃	#520	1	1	yes	characteristic
264	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
265	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
266	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
267	C₂²×C₅₃	#568	1	1	yes	characteristic
268	C₂×D₅₃	—	2	2	no	—
269	C₂×D₅₃	—	2	2	no	—
270	C₂×D₅₃	—	2	2	no	—
271	C₂×D₅₃	—	4	1	no	—
272	C₂×D₅₃	—	4	1	no	—
273	C₂×D₅₃	—	4	1	no	—
274	C₂×D₅₃	—	4	1	no	—
275	C₄×C₅₃	—	2	2	no	—
276	C₄×C₅₃	—	4	1	no	—
277	C₄×C₅₃	—	4	1	no	—
278	C₄×C₅₃	—	4	1	no	—
279	C₄×C₅₃	—	4	1	no	—
280	C₄×C₅₃	—	4	1	no	—
281	C₄×C₅₃	#551	1	1	yes	characteristic
282	C₄×C₅₃	#551	1	1	yes	characteristic
283	C₄×C₅₃	#552	1	1	yes	characteristic
284	C₄×C₅₃	#552	1	1	yes	characteristic
285	Dic₅₃	—	2	2	no	—
286	Dic₅₃	—	2	2	no	—
287	Dic₅₃	—	2	2	no	—
288	Dic₅₃	—	4	1	no	—
289	Dic₅₃	—	4	1	no	—
290	Dic₅₃	—	4	1	no	—
291	Dic₅₃	—	4	1	no	—
292	#2298	—	53	1	no	—
293	#1917	D₅₃	1	1	yes	characteristic, complemented
294	#1916	—	53	1	no	—
295	#1916	—	53	1	no	—
296	#1916	—	53	1	no	—
297	#1851	—	53	1	no	—
298	#1850	—	53	1	no	—
299	#1840	—	53	1	no	—
300	#1831	—	53	1	no	—
301	#1818	—	53	1	no	—
302	#1768	—	53	1	no	—
303	#1753	—	53	1	no	—
304	C₈×C₂²⋊C₄	—	53	1	no	—
305	#1744	—	53	1	no	—
306	#1729	—	53	1	no	—
307	C₂×C₅₃	#2343	1	1	yes	characteristic
308	C₂×C₅₃	#2922	1	1	yes	characteristic
309	C₂×C₅₃	#2925	1	1	yes	characteristic
310	C₂×C₅₃	#2966	1	1	yes	characteristic
311	C₂×C₅₃	#2968	1	1	yes	characteristic
312	C₂×C₅₃	#2993	1	1	yes	characteristic
313	C₂×C₅₃	#2994	1	1	yes	characteristic
314	D₅₃	—	4	2	no	—
315	C₂²×M₁₆	—	53	1	no	—
316	C₂³×C₈	—	53	1	no	—
317	C₂×#117	—	53	1	no	—
318	C₂×C₄⋊C₈	—	53	1	no	—
319	C₂×C₄⋊C₈	—	53	1	no	—
320	C₂×C₄⋊C₈	—	53	1	no	—
321	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
322	C₂×C₂²⋊C₈	—	53	1	no	—
323	C₂×C₂²⋊C₈	—	53	1	no	—
324	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
325	C₂×C₄×C₈	—	53	1	no	—
326	C₂×C₄×C₈	—	53	1	no	—
327	C₂×C₄×C₈	—	53	1	no	—
328	#388	—	53	1	no	—
329	#388	—	53	1	no	—
330	#388	—	53	1	no	—
331	#388	—	53	1	no	—
332	#387	—	53	1	no	—
333	#387	—	53	1	no	—
334	#387	—	53	1	no	—
335	#387	—	53	1	no	—
336	C₄×C₄⋊C₄	—	53	1	no	—
337	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
338	C₄×C₂²⋊C₄	—	53	1	no	—
339	C₄×C₂²⋊C₄	—	53	1	no	—
340	#376	—	53	1	no	—
341	C₄³	—	53	1	no	—
342	#336	—	53	1	no	—
343	#336	—	53	1	no	—
344	#336	—	53	1	no	—
345	#336	—	53	1	no	—
346	#336	—	53	1	no	—
347	#336	—	53	1	no	—
348	#336	—	53	1	no	—
349	#336	—	53	1	no	—
350	#336	—	53	1	no	—
351	#336	—	53	1	no	—
352	#336	—	53	1	no	—
353	#336	—	53	1	no	—
354	#336	C₂×D₅₃	1	1	yes	characteristic
355	#336	C₂×D₅₃	1	1	yes	characteristic
356	#336	C₂×D₅₃	1	1	yes	characteristic
357	#336	C₂×D₅₃	1	1	yes	characteristic
358	C₅₃	#22593	1	1	yes	characteristic, complemented, Sylow
359	C₂³×C₄	—	53	1	no	—
360	C₂×M₁₆	—	106	2	no	—
361	C₂×M₁₆	—	106	2	no	—
362	C₂×M₁₆	—	212	1	no	—
363	C₂²×C₈	—	53	1	no	—
364	C₂²×C₈	—	53	1	no	—
365	C₂²×C₈	—	53	1	no	—
366	C₂²×C₈	—	53	1	no	—
367	C₂²×C₈	—	106	2	no	—
368	C₂²×C₈	—	106	2	no	—
369	C₂²×C₈	—	212	1	no	—
370	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
371	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
372	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
373	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
374	#117	—	106	2	no	—
375	#117	—	106	2	no	—
376	C₂×C₄⋊C₄	—	53	1	no	—
377	C₂×C₄⋊C₄	—	53	1	no	—
378	C₂×C₄⋊C₄	—	53	1	no	—
379	C₂×C₄⋊C₄	—	53	1	no	—
380	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	—	53	1	no	—
383	C₂×C₂²⋊C₄	—	53	1	no	—
384	C₂×C₂²⋊C₄	—	53	1	no	—
385	C₂×C₂²⋊C₄	—	53	1	no	—
386	C₂×C₂²⋊C₄	—	53	1	no	—
387	C₂×C₂²⋊C₄	—	53	1	no	—
388	C₂×C₄²	—	53	1	no	—
389	C₂×C₄²	—	53	1	no	—
390	C₂×C₄²	—	53	1	no	—
391	C₂×C₄²	—	53	1	no	—
392	C₂×C₄²	—	53	1	no	—
393	C₂×C₄²	—	53	1	no	—
394	C₂×C₄²	—	53	1	no	—
395	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
396	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
397	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
398	C₄⋊C₈	—	4	1	no	—
399	C₄⋊C₈	—	106	2	no	—
400	C₄⋊C₈	—	106	2	no	—
401	C₄⋊C₈	—	212	1	no	—
402	C₂²⋊C₈	—	106	2	no	—
403	C₂²⋊C₈	—	106	2	no	—
404	#97	—	4	1	no	—
405	C₄×C₈	—	106	2	no	—
406	C₄×C₈	—	106	2	no	—
407	C₄×C₈	—	212	1	no	—
408	#95	—	53	1	no	—
409	#95	—	53	1	no	—
410	#95	—	53	1	no	—
411	#95	—	53	1	no	—
412	#95	—	53	1	no	—
413	#95	—	53	1	no	—
414	#95	—	53	1	no	—
415	#95	—	53	1	no	—
416	#95	—	53	1	no	—
417	#95	—	53	1	no	—
418	#95	C₄×D₅₃	1	1	yes	characteristic
419	#95	C₄×D₅₃	1	1	yes	characteristic
420	C₂⁴	—	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	2	no	—
425	C₂²×C₄	—	53	2	no	—
426	C₂²×C₄	—	53	1	no	—
427	C₂²×C₄	—	53	1	no	—
428	C₂²×C₄	—	53	1	no	—
429	C₂²×C₄	—	53	1	no	—
430	C₂²×C₄	—	106	1	no	—
431	C₂²×C₄	—	106	1	no	—
432	C₂²×C₄	—	106	1	no	—
433	C₂²×C₄	—	106	1	no	—
434	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
440	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
441	M₁₆	—	212	4	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	53	2	no	—
454	C₂×C₈	—	106	1	no	—
455	C₂×C₈	—	106	1	no	—
456	C₂×C₈	—	106	1	no	—
457	C₂×C₈	—	106	1	no	—
458	C₂×C₈	—	106	1	no	—
459	C₂×C₈	—	106	1	no	—
460	C₂×C₈	—	106	1	no	—
461	C₂×C₈	—	106	1	no	—
462	C₂×C₈	—	106	1	no	—
463	C₂×C₈	—	106	1	no	—
464	C₂×C₈	—	106	1	no	—
465	C₂×C₈	—	212	4	no	—
466	C₂×C₈	#11719949	1	2	yes	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	—	106	2	no	—
470	C₄⋊C₄	—	106	2	no	—
471	C₄⋊C₄	—	106	2	no	—
472	C₄⋊C₄	—	106	2	no	—
473	C₂²⋊C₄	—	106	2	no	—
474	C₂²⋊C₄	—	106	2	no	—
475	C₂²⋊C₄	—	106	2	no	—
476	C₂²⋊C₄	—	106	2	no	—
477	C₂²⋊C₄	—	106	2	no	—
478	C₂²⋊C₄	—	212	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	106	2	no	—
486	C₄²	—	106	2	no	—
487	C₄²	—	106	2	no	—
488	C₄²	—	106	1	no	—
489	C₄²	—	106	1	no	—
490	C₄²	—	106	1	no	—
491	C₄²	—	106	1	no	—
492	C₄²	—	106	1	no	—
493	C₄²	—	106	1	no	—
494	C₄²	—	212	1	no	—
495	C₂³	—	53	2	no	—
496	C₂³	—	106	1	no	—
497	C₂³	—	106	1	no	—
498	C₂³	—	106	1	no	—
499	C₂³	—	106	1	no	—
500	C₂³	—	106	1	no	—
501	C₂³	—	106	1	no	—
502	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	53	2	no	—
521	C₂×C₄	—	53	2	no	—
522	C₂×C₄	—	53	2	no	—
523	C₂×C₄	—	106	1	no	—
524	C₂×C₄	—	106	1	no	—
525	C₂×C₄	—	106	1	no	—
526	C₂×C₄	—	106	1	no	—
527	C₂×C₄	—	106	1	no	—
528	C₂×C₄	—	106	1	no	—
529	C₂×C₄	—	106	2	no	—
530	C₂×C₄	—	106	2	no	—
531	C₂×C₄	—	106	2	no	—
532	C₂×C₄	—	106	2	no	—
533	C₂×C₄	—	106	1	no	—
534	C₂×C₄	—	106	1	no	—
535	C₂×C₄	—	106	1	no	—
536	C₂×C₄	—	106	1	no	—
537	C₂×C₄	—	106	1	no	—
538	C₂×C₄	—	106	1	no	—
539	C₂×C₄	—	106	1	no	—
540	C₂×C₄	—	106	1	no	—
541	C₂×C₄	—	106	1	no	—
542	C₂×C₄	—	106	1	no	—
543	C₂×C₄	—	106	1	no	—
544	C₂×C₄	—	106	1	no	—
545	C₂×C₄	#13133225	1	2	yes	—
546	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
547	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
548	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
549	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
550	C₂×C₄	#13133262	1	1	yes	characteristic
551	C₂×C₄	#13133264	1	1	yes	characteristic
552	C₈	—	2	2	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	106	2	no	—
557	C₈	—	212	1	no	—
558	C₈	—	212	1	no	—
559	C₈	—	212	1	no	—
560	C₂²	—	106	2	no	—
561	C₂²	—	106	2	no	—
562	C₂²	—	106	2	no	—
563	C₂²	—	212	1	no	—
564	C₂²	—	212	1	no	—
565	C₂²	—	212	1	no	—
566	C₂²	—	212	1	no	—
567	C₂²	#117×D₅₃	1	1	yes	characteristic
568	C₂²	C₄×#11719949	1	1	yes	characteristic
569	C₂²	#16811102	1	1	yes	characteristic
570	C₂²	#131×D₅₃	1	1	yes	characteristic
571	C₂²	#131×D₅₃	1	1	yes	characteristic
572	C₂²	C₂×#13133225	1	1	yes	characteristic
573	C₂²	#16811389	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	106	2	no	—
581	C₄	—	106	2	no	—
582	C₄	—	106	2	no	—
583	C₄	—	212	1	no	—
584	C₄	—	212	1	no	—
585	C₄	—	212	1	no	—
586	C₄	—	212	1	no	—
587	C₄	#16811300	1	1	yes	characteristic
588	C₄	#16811301	1	1	yes	characteristic
589	C₄	#16811309	1	1	yes	characteristic
590	C₄	#16811311	1	1	yes	characteristic
591	C₂	—	212	2	no	—
592	C₂	#20277574	1	1	yes	characteristic
593	C₂	#20283356	1	1	yes	characteristic
594	C₂	#20283382	1	1	yes	characteristic
595	C₂	#20283689	1	1	yes	characteristic
596	C₂	#20283708	1	1	yes	characteristic
597	C₂	#20283861	1	1	yes	characteristic
598	C₂	#20283868	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1917	C₂
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1917	D₅₃
hypocenter	C₅₃	#22593
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1917

Complements

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

N	H ≅ G/N	type
2. #20275274	C₂	semidirect
3. #20275273	C₂	semidirect
4. #20275271	C₂	semidirect
7. #20274859	C₂	semidirect
8. #20274831	C₂	semidirect
10. #20274575	C₂	semidirect
13. #20274459	C₂	semidirect
15. C₅₃×#1917	C₂	semidirect
293. #1917	D₅₃	semidirect
358. C₅₃	#22593	semidirect

Maximal subgroups

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

subgroup	index
#20277208	2
#20275274	2
#20275273	2
#20275271	2
#20274899	2
#20274891	2
#20274859	2
#20274831	2
#20274785	2
#20274575	2
#20274500	2
#20274485	2
#20274459	2
#20274410	2
C₅₃×#1917	2
#22593	53