Subgroups — #27401516

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	#20277211	C₂	1	1	yes	maximal, characteristic
2	#20275277	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275272	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275272	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275270	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274897	C₂	1	1	yes	maximal, characteristic
7	#20274893	C₂	1	1	yes	maximal, characteristic
8	#20274859	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274830	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274784	C₂	1	1	yes	maximal, characteristic
11	#20274573	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274499	C₂	1	1	yes	maximal, characteristic
13	#20274460	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274411	C₂	1	1	yes	maximal, characteristic
15	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂×#13133176	C₂²	1	1	yes	characteristic
18	C₂×#13133146	C₂²	1	1	yes	characteristic
19	C₂×#13133146	C₂²	1	1	yes	characteristic
20	C₂×#13133146	C₂²	1	1	yes	characteristic
21	C₂×#13133146	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×#13133128	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	#16810517	C₄	1	1	yes	characteristic
26	#16810517	C₄	1	1	yes	characteristic
27	#16810511	C₄	1	1	yes	characteristic
28	#16810511	C₄	1	1	yes	characteristic
29	#16810508	C₄	1	1	yes	characteristic
30	#16810508	C₄	1	1	yes	characteristic
31	#16810506	C₄	1	1	yes	characteristic
32	#16810506	C₄	1	1	yes	characteristic
33	C₄×#11719919	C₂²	1	1	yes	characteristic
34	C₄×#11719919	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
36	#16810469	C₂²	1	1	yes	characteristic
37	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
38	#16810378	C₂²	1	1	yes	characteristic
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810377	C₂²	1	1	yes	characteristic
41	#16810377	C₂²	1	1	yes	characteristic
42	#16810377	C₂²	1	1	yes	characteristic
43	#16810377	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	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₂×#11719917	—	2	2	no	—
60	C₂×#11719917	—	2	2	no	—
61	C₂×#11719917	—	4	1	no	—
62	C₂²×#88322	C₂³	1	1	yes	characteristic
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	#13133176	—	2	2	no	—
67	#13133176	—	2	2	no	—
68	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
72	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
73	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
86	#13133146	—	2	2	no	—
87	#13133146	—	2	2	no	—
88	#13133146	—	4	1	no	—
89	#13133146	—	4	1	no	—
90	C₂²⋊#88322	—	2	2	no	—
91	#13133128	—	4	1	no	—
92	C₄×#88322	—	2	2	no	—
93	#13133124	C₂×C₄	1	1	yes	characteristic
94	#13133124	C₂×C₄	1	1	yes	characteristic
95	#13133124	C₂×C₄	1	1	yes	characteristic
96	#13133124	C₂×C₄	1	1	yes	characteristic
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	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	—	2	2	no	—
105	C₂²×C₈×C₅₃	—	2	2	no	—
106	C₂²×C₈×C₅₃	—	4	1	no	—
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×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₂²⋊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	C₂²⋊C₈×C₅₃	—	2	2	no	—
116	C₄×C₈×C₅₃	—	2	2	no	—
117	C₄×C₈×C₅₃	—	4	1	no	—
118	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	#41	1	2	yes	—
131	C₂²×C₄×C₅₃	#41	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
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	—	4	4	no	—
153	C₂×C₈×C₅₃	#41	1	2	yes	—
154	C₂²⋊C₄×C₅₃	—	2	2	no	—
155	C₂²⋊C₄×C₅₃	—	2	2	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	C₂²⋊Dic₅₃	—	2	2	no	—
161	C₂²⋊Dic₅₃	—	2	2	no	—
162	C₂²⋊Dic₅₃	—	2	2	no	—
163	C₂²⋊Dic₅₃	—	4	1	no	—
164	C₄⋊Dic₅₃	—	2	2	no	—
165	C₄⋊Dic₅₃	—	2	2	no	—
166	#11719919	—	2	2	no	—
167	#11719919	—	2	2	no	—
168	#11719919	—	2	2	no	—
169	#11719919	—	4	1	no	—
170	C₄×Dic₅₃	—	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	1	no	—
179	C₄×Dic₅₃	—	2	1	no	—
180	C₄×Dic₅₃	—	2	2	no	—
181	C₄×Dic₅₃	—	4	1	no	—
182	#11719917	—	4	4	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	—	2	1	no	—
193	C₂×#88322	—	2	1	no	—
194	C₂×#88322	#41	1	2	yes	—
195	C₂³×C₅₃	—	2	1	no	—
196	C₂³×C₅₃	—	2	1	no	—
197	C₂³×C₅₃	—	2	1	no	—
198	C₂³×C₅₃	—	2	1	no	—
199	C₂³×C₅₃	—	2	1	no	—
200	C₂³×C₅₃	—	2	1	no	—
201	C₂³×C₅₃	#117	1	2	yes	—
202	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
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	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	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	#117	1	2	yes	—
220	C₂×C₄×C₅₃	#131	1	2	yes	—
221	C₂×C₄×C₅₃	#131	1	2	yes	—
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	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
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	—	2	1	no	—
246	C₂×Dic₅₃	—	2	1	no	—
247	C₂×Dic₅₃	—	2	1	no	—
248	C₂×Dic₅₃	—	2	1	no	—
249	C₂×Dic₅₃	—	2	1	no	—
250	C₂×Dic₅₃	—	2	1	no	—
251	C₂×Dic₅₃	#131	1	2	yes	—
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	—	2	2	no	—
262	C₂²×C₅₃	—	2	2	no	—
263	C₂²×C₅₃	—	2	2	no	—
264	C₂²×C₅₃	—	4	1	no	—
265	C₂²×C₅₃	—	4	1	no	—
266	C₂²×C₅₃	—	4	1	no	—
267	C₂²×C₅₃	—	4	1	no	—
268	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
269	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
270	C₂²×C₅₃	#520	1	1	yes	characteristic
271	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
272	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
273	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
274	C₂²×C₅₃	#568	1	1	yes	characteristic
275	C₄×C₅₃	—	2	2	no	—
276	C₄×C₅₃	—	2	2	no	—
277	C₄×C₅₃	—	2	2	no	—
278	C₄×C₅₃	—	4	1	no	—
279	C₄×C₅₃	—	4	1	no	—
280	C₄×C₅₃	#551	1	1	yes	characteristic
281	C₄×C₅₃	#551	1	1	yes	characteristic
282	C₄×C₅₃	#552	1	1	yes	characteristic
283	C₄×C₅₃	#552	1	1	yes	characteristic
284	Dic₅₃	—	2	2	no	—
285	Dic₅₃	—	4	1	no	—
286	Dic₅₃	—	4	1	no	—
287	Dic₅₃	—	4	1	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	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic
305	#1744	—	53	1	no	—
306	#1729	—	53	1	no	—
307	C₂×C₅₃	—	4	2	no	—
308	C₂×C₅₃	#2343	1	1	yes	characteristic
309	C₂×C₅₃	#2922	1	1	yes	characteristic
310	C₂×C₅₃	#2925	1	1	yes	characteristic
311	C₂×C₅₃	#2966	1	1	yes	characteristic
312	C₂×C₅₃	#2968	1	1	yes	characteristic
313	C₂×C₅₃	#2993	1	1	yes	characteristic
314	C₂×C₅₃	#2994	1	1	yes	characteristic
315	C₂²×M₁₆	—	53	1	no	—
316	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
322	C₂×C₂²⋊C₈	—	53	1	no	—
323	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
324	C₂×#97	—	53	1	no	—
325	C₂×C₄×C₈	—	53	1	no	—
326	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
327	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
338	C₄×C₂²⋊C₄	—	53	1	no	—
339	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
355	#336	—	53	1	no	—
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	C₂²×D₅₃	1	1	yes	characteristic
360	C₂×M₁₆	—	106	2	no	—
361	C₂×M₁₆	—	106	2	no	—
362	C₂×M₁₆	—	212	1	no	—
363	C₂²×C₈	—	2	2	no	—
364	C₂²×C₈	—	2	2	no	—
365	C₂²×C₈	—	4	1	no	—
366	C₂²×C₈	—	53	1	no	—
367	C₂²×C₈	—	53	1	no	—
368	C₂²×C₈	—	53	1	no	—
369	C₂²×C₈	—	53	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	—	53	1	no	—
381	C₂×C₄⋊C₄	—	53	1	no	—
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	C₄×D₅₃	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
396	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
397	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
398	C₄⋊C₈	—	106	2	no	—
399	C₄⋊C₈	—	106	2	no	—
400	C₄⋊C₈	—	212	1	no	—
401	C₄⋊C₈	—	212	1	no	—
402	C₂²⋊C₈	—	2	2	no	—
403	C₂²⋊C₈	—	106	2	no	—
404	#97	—	212	1	no	—
405	C₄×C₈	—	2	2	no	—
406	C₄×C₈	—	4	1	no	—
407	C₄×C₈	—	106	2	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	C₂×C₄×D₅₃	1	1	yes	characteristic
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	53	1	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	—	53	1	no	—
431	C₂²×C₄	—	53	1	no	—
432	C₂²×C₄	—	53	1	no	—
433	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
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	#11719945	1	2	yes	—
439	C₂²×C₄	#11719945	1	2	yes	—
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	—	4	4	no	—
454	C₂×C₈	—	53	2	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	—	106	1	no	—
466	C₂×C₈	#11719947	1	2	yes	—
467	C₄⋊C₄	—	106	2	no	—
468	C₄⋊C₄	—	106	2	no	—
469	C₄⋊C₄	—	106	2	no	—
470	C₄⋊C₄	—	106	2	no	—
471	C₄⋊C₄	—	106	2	no	—
472	C₄⋊C₄	—	212	1	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	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	—	106	2	no	—
484	C₄²	—	106	2	no	—
485	C₄²	—	106	2	no	—
486	C₄²	—	106	1	no	—
487	C₄²	—	106	1	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	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	#13133174	1	2	yes	—
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	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	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	—	53	2	no	—
520	C₂×C₄	—	106	1	no	—
521	C₂×C₄	—	106	1	no	—
522	C₂×C₄	—	106	1	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	1	no	—
530	C₂×C₄	—	106	1	no	—
531	C₂×C₄	—	106	1	no	—
532	C₂×C₄	—	106	1	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	#13133174	1	2	yes	—
544	C₂×C₄	#13133223	1	2	yes	—
545	C₂×C₄	#13133223	1	2	yes	—
546	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
547	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
548	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
549	C₂×C₄	C₂×#11719945	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	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	C₂×#13133174	1	1	yes	characteristic
568	C₂²	C₄×#11719947	1	1	yes	characteristic
569	C₂²	#16811102	1	1	yes	characteristic
570	C₂²	#131×D₅₃	1	1	yes	characteristic
571	C₂²	C₂×#13133223	1	1	yes	characteristic
572	C₂²	C₂×#13133223	1	1	yes	characteristic
573	C₂²	#16811388	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	106	2	no	—
580	C₄	—	212	1	no	—
581	C₄	—	212	1	no	—
582	C₄	—	212	1	no	—
583	C₄	—	212	1	no	—
584	C₄	—	212	1	no	—
585	C₄	—	212	1	no	—
586	C₄	—	212	1	no	—
587	C₄	#16811299	1	1	yes	characteristic
588	C₄	#16811302	1	1	yes	characteristic
589	C₄	#16811306	1	1	yes	characteristic
590	C₄	#16811313	1	1	yes	characteristic
591	C₂	—	4	2	no	—
592	C₂	#20277569	1	1	yes	characteristic
593	C₂	#20283357	1	1	yes	characteristic
594	C₂	#20283379	1	1	yes	characteristic
595	C₂	#20283683	1	1	yes	characteristic
596	C₂	#20283703	1	1	yes	characteristic
597	C₂	#20283865	1	1	yes	characteristic
598	C₂	#20283870	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₈×C₂²⋊C₄×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#22593
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20275277	C₂	semidirect
3. #20275272	C₂	semidirect
4. #20275272	C₂	semidirect
5. #20275270	C₂	semidirect
8. #20274859	C₂	semidirect
9. #20274830	C₂	semidirect
11. #20274573	C₂	semidirect
13. #20274460	C₂	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
#20277211	2
#20275277	2
#20275272	2
#20275270	2
#20274897	2
#20274893	2
#20274859	2
#20274830	2
#20274784	2
#20274573	2
#20274499	2
#20274460	2
#20274411	2
C₈×C₂²⋊C₄×C₅₃	2
#22593	53