Subgroups — #27392348

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20278016	C₂	1	1	yes	maximal, characteristic
2	#20275329	C₂	1	1	yes	maximal, characteristic
3	#20275328	C₂	1	1	yes	maximal, characteristic
4	#20275327	C₂	1	1	yes	maximal, characteristic
5	#20275282	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275280	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274859	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274580	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274576	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274485	C₂	1	1	yes	maximal, characteristic
12	#20274485	C₂	1	1	yes	maximal, characteristic
13	#20274485	C₂	1	1	yes	maximal, characteristic
14	#20274403	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×#13133146	C₂²	1	1	yes	characteristic
19	C₂×#13133129	C₂²	1	1	yes	characteristic
20	C₂×#13133129	C₂²	1	1	yes	characteristic
21	C₂×#13133129	C₂²	1	1	yes	characteristic
22	C₂×#13133127	C₂²	1	1	yes	characteristic
23	C₂×#13133127	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₅₃⋊#401	C₄	1	1	yes	characteristic
31	#16810565	C₄	1	1	yes	characteristic
32	#16810562	C₄	1	1	yes	characteristic
33	#16810561	C₄	1	1	yes	characteristic
34	#16810552	C₄	1	1	yes	characteristic
35	#16810550	C₄	1	1	yes	characteristic
36	#16810549	C₄	1	1	yes	characteristic
37	#16810545	C₄	1	1	yes	characteristic
38	C₄×#11719919	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
40	C₄×#11719921	C₂²	1	1	yes	characteristic
41	C₄×#11719921	C₂²	1	1	yes	characteristic
42	C₄×#11719921	C₂²	1	1	yes	characteristic
43	#16810469	C₂²	1	1	yes	characteristic
44	#16810378	C₂²	1	1	yes	characteristic
45	#16810378	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	C₂²×#88322	C₂³	1	1	yes	characteristic
61	C₂²×#88322	C₂³	1	1	yes	characteristic
62	C₂²×#88322	C₂³	1	1	yes	characteristic
63	C₂²×#88322	C₂³	1	1	yes	characteristic
64	C₂×C₈×D₅₃	—	2	2	no	—
65	C₂×C₈×D₅₃	—	2	2	no	—
66	C₂×C₈×D₅₃	—	2	2	no	—
67	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
72	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
73	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
75	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
76	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊#88322	—	2	2	no	—
81	#13133146	—	2	2	no	—
82	#13133129	—	2	2	no	—
83	#13133129	—	2	2	no	—
84	#13133129	—	2	2	no	—
85	#13133127	—	2	2	no	—
86	#13133127	—	2	2	no	—
87	C₄×#88322	—	2	2	no	—
88	C₄×#88322	—	2	2	no	—
89	C₄×#88322	—	2	2	no	—
90	C₈×Dic₅₃	—	2	2	no	—
91	C₈×Dic₅₃	—	2	2	no	—
92	C₈×Dic₅₃	—	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	#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	—	2	2	no	—
117	#97×C₅₃	—	2	2	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	—	2	1	no	—
140	C₂×C₄×D₅₃	—	2	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	—	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	#41	1	2	yes	—
152	C₂×C₈×C₅₃	#41	1	2	yes	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	2	2	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	—	2	2	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	1	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	C₂×#88322	—	2	1	no	—
178	C₂×#88322	—	2	1	no	—
179	C₂×#88322	—	2	1	no	—
180	C₂×#88322	—	2	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	#41	1	2	yes	—
186	C₂×#88322	#41	1	2	yes	—
187	C₂×#88322	#41	1	2	yes	—
188	C₂×#88322	#41	1	2	yes	—
189	C₈×D₅₃	—	4	4	no	—
190	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
191	C₂²×D₅₃	—	2	1	no	—
192	C₂²×D₅₃	—	2	1	no	—
193	C₂²×D₅₃	—	2	1	no	—
194	C₂²×D₅₃	—	2	1	no	—
195	C₂²×D₅₃	—	2	1	no	—
196	C₂²×D₅₃	—	2	1	no	—
197	C₂²×D₅₃	#131	1	2	yes	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
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	#131	1	2	yes	—
215	C₂×C₄×C₅₃	#131	1	2	yes	—
216	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
222	C₂×Dic₅₃	—	2	1	no	—
223	C₂×Dic₅₃	—	2	1	no	—
224	C₂×Dic₅₃	—	2	1	no	—
225	C₂×Dic₅₃	—	2	1	no	—
226	C₂×Dic₅₃	—	2	1	no	—
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	#131	1	2	yes	—
243	C₄×D₅₃	—	4	1	no	—
244	C₄×D₅₃	—	4	1	no	—
245	C₄×D₅₃	—	4	1	no	—
246	C₄×D₅₃	—	4	1	no	—
247	C₈×C₅₃	—	2	2	no	—
248	C₈×C₅₃	—	2	2	no	—
249	C₈×C₅₃	—	4	1	no	—
250	C₈×C₅₃	—	4	1	no	—
251	#88322	—	2	2	no	—
252	#88322	—	4	1	no	—
253	#88322	—	4	1	no	—
254	#88322	#126	1	4	yes	—
255	#21668	—	53	1	no	maximal, Sylow
256	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
257	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
258	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
259	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
260	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
261	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
262	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
263	C₂×D₅₃	—	2	2	no	—
264	C₂×D₅₃	—	2	2	no	—
265	C₂×D₅₃	—	2	2	no	—
266	C₂×D₅₃	—	4	1	no	—
267	C₂×D₅₃	—	4	1	no	—
268	C₂×D₅₃	—	4	1	no	—
269	C₂×D₅₃	—	4	1	no	—
270	C₄×C₅₃	—	2	2	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	4	1	no	—
273	C₄×C₅₃	—	4	1	no	—
274	C₄×C₅₃	—	4	1	no	—
275	C₄×C₅₃	—	4	1	no	—
276	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
277	C₄×C₅₃	#529	1	1	yes	characteristic
278	C₄×C₅₃	#529	1	1	yes	characteristic
279	C₄×C₅₃	#529	1	1	yes	characteristic
280	Dic₅₃	—	2	2	no	—
281	Dic₅₃	—	4	1	no	—
282	Dic₅₃	—	4	1	no	—
283	Dic₅₃	—	4	1	no	—
284	Dic₅₃	—	4	1	no	—
285	Dic₅₃	—	4	1	no	—
286	Dic₅₃	—	4	1	no	—
287	#2401	—	53	1	no	—
288	#1924	—	53	1	no	—
289	#1924	—	53	1	no	—
290	#1924	—	53	1	no	—
291	#1917	—	53	1	no	—
292	#1917	—	53	1	no	—
293	#1917	D₅₃	1	1	yes	characteristic, complemented
294	#1840	—	53	1	no	—
295	C₈×C₄⋊C₄	—	53	1	no	—
296	C₈×C₄⋊C₄	—	53	1	no	—
297	C₈×C₄⋊C₄	—	53	1	no	—
298	C₈×C₂²⋊C₄	—	53	1	no	—
299	C₈×C₂²⋊C₄	—	53	1	no	—
300	C₈×C₂²⋊C₄	—	53	1	no	—
301	#1726	—	53	1	no	—
302	C₂×C₅₃	C₄×#126	1	1	yes	characteristic
303	C₂×C₅₃	#2966	1	1	yes	characteristic
304	C₂×C₅₃	#2966	1	1	yes	characteristic
305	C₂×C₅₃	#2966	1	1	yes	characteristic
306	C₂×C₅₃	#2966	1	1	yes	characteristic
307	C₂×C₅₃	#2966	1	1	yes	characteristic
308	C₂×C₅₃	#2966	1	1	yes	characteristic
309	D₅₃	—	4	2	no	—
310	C₂³×C₈	—	53	1	no	—
311	C₂×C₄⋊C₈	—	53	1	no	—
312	C₂×C₄⋊C₈	—	53	1	no	—
313	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
314	C₂×C₂²⋊C₈	—	53	1	no	—
315	C₂×C₂²⋊C₈	—	53	1	no	—
316	C₂×C₂²⋊C₈	—	53	1	no	—
317	C₂×#97	—	53	1	no	—
318	C₂×#97	—	53	1	no	—
319	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
320	C₂×C₄×C₈	—	53	1	no	—
321	C₂×C₄×C₈	—	53	1	no	—
322	C₂×C₄×C₈	—	53	1	no	—
323	C₂×C₄×C₈	—	53	1	no	—
324	C₂×C₄×C₈	—	53	1	no	—
325	C₂×C₄×C₈	—	53	1	no	—
326	#401	—	53	1	no	—
327	#400	—	53	1	no	—
328	#400	—	53	1	no	—
329	#400	—	53	1	no	—
330	#397	—	53	1	no	—
331	#397	—	53	1	no	—
332	#397	—	53	1	no	—
333	#396	—	53	1	no	—
334	C₄×C₄⋊C₄	—	53	1	no	—
335	C₄×C₄⋊C₄	—	53	1	no	—
336	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
337	C₄×C₂²⋊C₄	—	53	1	no	—
338	C₄×C₂²⋊C₄	—	53	1	no	—
339	C₄×C₂²⋊C₄	—	53	1	no	—
340	#376	—	53	1	no	—
341	#336	—	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	C₂×D₅₃	1	1	yes	characteristic
350	#336	C₂×D₅₃	1	1	yes	characteristic
351	#336	C₂×D₅₃	1	1	yes	characteristic
352	#336	C₂×D₅₃	1	1	yes	characteristic
353	C₅₃	#21668	1	1	yes	characteristic, complemented, Sylow
354	C₂³×C₄	—	53	1	no	—
355	C₂²×C₈	—	53	1	no	—
356	C₂²×C₈	—	53	1	no	—
357	C₂²×C₈	—	53	1	no	—
358	C₂²×C₈	—	53	1	no	—
359	C₂²×C₈	—	106	2	no	—
360	C₂²×C₈	—	106	2	no	—
361	C₂²×C₈	—	106	2	no	—
362	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
364	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
365	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	—	53	1	no	—
367	C₂×C₄⋊C₄	—	53	1	no	—
368	C₂×C₄⋊C₄	—	53	1	no	—
369	C₂×C₄⋊C₄	—	53	1	no	—
370	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	53	1	no	—
373	C₂×C₂²⋊C₄	—	53	1	no	—
374	C₂×C₂²⋊C₄	—	53	1	no	—
375	C₂×C₂²⋊C₄	—	53	1	no	—
376	C₂×C₂²⋊C₄	—	53	1	no	—
377	C₂×C₂²⋊C₄	—	53	1	no	—
378	C₂×C₄²	—	53	1	no	—
379	C₂×C₄²	—	53	1	no	—
380	C₂×C₄²	—	53	1	no	—
381	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
382	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
383	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
384	C₄⋊C₈	—	2	2	no	—
385	C₄⋊C₈	—	106	2	no	—
386	C₄⋊C₈	—	106	2	no	—
387	C₂²⋊C₈	—	106	2	no	—
388	C₂²⋊C₈	—	106	2	no	—
389	C₂²⋊C₈	—	106	2	no	—
390	#97	—	2	2	no	—
391	#97	—	106	2	no	—
392	#97	—	106	2	no	—
393	C₄×C₈	—	106	2	no	—
394	C₄×C₈	—	106	2	no	—
395	C₄×C₈	—	106	2	no	—
396	C₄×C₈	—	106	2	no	—
397	C₄×C₈	—	106	2	no	—
398	C₄×C₈	—	106	2	no	—
399	#95	—	53	1	no	—
400	#95	—	53	1	no	—
401	#95	—	53	1	no	—
402	#95	—	53	1	no	—
403	#95	—	53	1	no	—
404	#95	—	53	1	no	—
405	#95	—	53	1	no	—
406	#95	—	53	1	no	—
407	#95	—	53	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	C₄×D₅₃	1	1	yes	characteristic
414	#95	C₄×D₅₃	1	1	yes	characteristic
415	C₂⁴	—	53	1	no	—
416	C₂²×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	C₂²×C₄	—	106	1	no	—
424	C₂²×C₄	—	106	1	no	—
425	C₂²×C₄	—	106	1	no	—
426	C₂²×C₄	—	106	1	no	—
427	C₂²×C₄	—	106	1	no	—
428	C₂²×C₄	—	106	1	no	—
429	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
433	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	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	—	53	2	no	—
447	C₂×C₈	—	53	2	no	—
448	C₂×C₈	—	53	2	no	—
449	C₂×C₈	—	53	2	no	—
450	C₂×C₈	—	106	1	no	—
451	C₂×C₈	—	106	1	no	—
452	C₂×C₈	—	106	1	no	—
453	C₂×C₈	—	106	1	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	—	212	4	no	—
459	C₂×C₈	#11719947	1	2	yes	—
460	C₂×C₈	#11719949	1	2	yes	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	106	2	no	—
464	C₄⋊C₄	—	106	2	no	—
465	C₄⋊C₄	—	106	2	no	—
466	C₄⋊C₄	—	106	2	no	—
467	C₂²⋊C₄	—	106	2	no	—
468	C₂²⋊C₄	—	106	2	no	—
469	C₂²⋊C₄	—	106	2	no	—
470	C₂²⋊C₄	—	106	2	no	—
471	C₂²⋊C₄	—	106	2	no	—
472	C₂²⋊C₄	—	106	2	no	—
473	C₄²	—	2	1	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	106	1	no	—
480	C₄²	—	106	1	no	—
481	C₄²	—	106	1	no	—
482	C₄²	—	106	1	no	—
483	C₄²	—	106	1	no	—
484	C₄²	—	106	1	no	—
485	C₂³	—	53	2	no	—
486	C₂³	—	106	1	no	—
487	C₂³	—	106	1	no	—
488	C₂³	—	106	1	no	—
489	C₂³	—	106	1	no	—
490	C₂³	—	106	1	no	—
491	C₂³	—	106	1	no	—
492	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
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	—	53	2	no	—
510	C₂×C₄	—	106	1	no	—
511	C₂×C₄	—	106	1	no	—
512	C₂×C₄	—	106	1	no	—
513	C₂×C₄	—	106	1	no	—
514	C₂×C₄	—	106	1	no	—
515	C₂×C₄	—	106	1	no	—
516	C₂×C₄	—	106	1	no	—
517	C₂×C₄	—	106	1	no	—
518	C₂×C₄	—	106	1	no	—
519	C₂×C₄	—	106	1	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	—	212	1	no	—
531	C₂×C₄	—	212	1	no	—
532	C₂×C₄	—	212	1	no	—
533	C₂×C₄	—	212	1	no	—
534	C₂×C₄	#13133225	1	2	yes	—
535	C₂×C₄	#13133225	1	2	yes	—
536	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
537	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
538	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
539	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
540	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
541	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
542	C₈	—	2	2	no	—
543	C₈	—	2	2	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	53	4	no	—
547	C₈	—	106	2	no	—
548	C₈	—	212	1	no	—
549	C₈	—	212	1	no	—
550	C₂²	—	106	2	no	—
551	C₂²	—	106	2	no	—
552	C₂²	—	106	2	no	—
553	C₂²	—	212	1	no	—
554	C₂²	—	212	1	no	—
555	C₂²	—	212	1	no	—
556	C₂²	—	212	1	no	—
557	C₂²	C₄×#11719947	1	1	yes	characteristic
558	C₂²	#16811091	1	1	yes	characteristic
559	C₂²	C₄×#11719949	1	1	yes	characteristic
560	C₂²	#131×D₅₃	1	1	yes	characteristic
561	C₂²	#131×D₅₃	1	1	yes	characteristic
562	C₂²	C₂×#13133225	1	1	yes	characteristic
563	C₂²	C₂×#13133225	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	106	2	no	—
571	C₄	—	212	1	no	—
572	C₄	—	212	1	no	—
573	C₄	—	212	1	no	—
574	C₄	—	212	1	no	—
575	C₄	—	212	1	no	—
576	C₄	—	212	1	no	—
577	C₄	#126×D₅₃	1	1	yes	characteristic
578	C₄	#16811150	1	1	yes	characteristic
579	C₄	#16811158	1	1	yes	characteristic
580	C₄	#16811159	1	1	yes	characteristic
581	C₂	—	212	2	no	—
582	C₂	#20277265	1	1	yes	characteristic
583	C₂	#20283684	1	1	yes	characteristic
584	C₂	#20283685	1	1	yes	characteristic
585	C₂	#20283688	1	1	yes	characteristic
586	C₂	#20283689	1	1	yes	characteristic
587	C₂	#20283689	1	1	yes	characteristic
588	C₂	#20283693	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	#21668
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
5. #20275282	C₂	semidirect
6. #20275280	C₂	semidirect
7. #20274859	C₂	semidirect
8. #20274580	C₂	semidirect
9. C₄⋊C₄×#88322	C₂	semidirect
10. #20274576	C₂	semidirect
14. #20274403	C₂	semidirect
15. C₅₃×#1917	C₂	semidirect
293. #1917	D₅₃	semidirect
353. C₅₃	#21668	semidirect

Maximal subgroups

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

subgroup	index
#20278016	2
#20275329	2
#20275328	2
#20275327	2
#20275282	2
#20275280	2
#20274859	2
#20274580	2
C₄⋊C₄×#88322	2
#20274576	2
#20274485	2
#20274403	2
C₅₃×#1917	2
#21668	53