Subgroups — #27391505

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810747	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133195	C₂	1	1	yes	maximal, characteristic
4	#20275649	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275432	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275272	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274894	C₂	1	1	yes	maximal, characteristic
8	#20274891	C₂	1	1	yes	maximal, characteristic
9	#20274694	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274576	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274576	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274485	C₂	1	1	yes	maximal, characteristic
14	#20274485	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×#13133195	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
21	#16810747	—	2	4	no	—
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×#13133146	C₂²	1	1	yes	characteristic
24	C₂×#13133146	C₂²	1	1	yes	characteristic
25	C₂×#13133146	C₂²	1	1	yes	characteristic
26	C₂×#13133146	C₂²	1	1	yes	characteristic
27	C₂×#13133129	C₂²	1	1	yes	characteristic
28	C₂×#13133129	C₂²	1	1	yes	characteristic
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₈×D₅₃	—	2	4	no	—
37	#16810527	C₄	1	1	yes	characteristic
38	#16810518	C₄	1	1	yes	characteristic
39	#16810509	C₄	1	1	yes	characteristic
40	#16810505	C₄	1	1	yes	characteristic
41	#16810497	C₄	1	1	yes	characteristic
42	#16810494	C₄	1	1	yes	characteristic
43	#16810482	C₄	1	1	yes	characteristic
44	C₄×#11719919	C₂²	1	1	yes	characteristic
45	C₄×#11719919	C₂²	1	1	yes	characteristic
46	C₄×#11719921	C₂²	1	1	yes	characteristic
47	C₄×#11719921	C₂²	1	1	yes	characteristic
48	#16810378	C₂²	1	1	yes	characteristic
49	#16810378	C₂²	1	1	yes	characteristic
50	#16810377	C₂²	1	1	yes	characteristic
51	#16810377	C₂²	1	1	yes	characteristic
52	#16810377	C₂²	1	1	yes	characteristic
53	#16810377	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×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₅₃×#336	C₂²	1	1	yes	characteristic
60	C₅₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
65	C₂²×#88322	C₂³	1	1	yes	characteristic
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂²×#88322	C₂³	1	1	yes	characteristic
68	C₂²×#88322	C₂³	1	1	yes	characteristic
69	C₂×C₈×D₅₃	—	2	2	no	—
70	C₂×C₈×D₅₃	—	2	2	no	—
71	C₂×C₈×D₅₃	—	2	2	no	—
72	C₂×C₈×D₅₃	—	2	2	no	—
73	C₂×C₈×D₅₃	—	2	2	no	—
74	C₂×C₈×D₅₃	Q₈	1	4	yes	complemented
75	#13133195	C₈	1	8	yes	complemented
76	C₄×Dic₁₀₆	—	2	2	no	—
77	C₄×Dic₁₀₆	—	2	2	no	—
78	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
80	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
81	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
82	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
83	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
84	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
85	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
86	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
87	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
88	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
94	C₄²×D₅₃	—	2	2	no	—
95	C₄²×D₅₃	—	2	2	no	—
96	C₄⋊#88322	—	2	2	no	—
97	#13133146	—	2	2	no	—
98	#13133146	—	2	2	no	—
99	#13133146	—	2	2	no	—
100	#13133146	—	2	2	no	—
101	#13133129	—	2	2	no	—
102	#13133129	—	2	2	no	—
103	C₄×#88322	—	2	2	no	—
104	C₄×#88322	—	2	2	no	—
105	C₄×#88322	—	2	2	no	—
106	C₈×Dic₅₃	—	2	2	no	—
107	C₈×Dic₅₃	—	2	2	no	—
108	C₈×Dic₅₃	—	2	2	no	—
109	C₈×Dic₅₃	D₄	1	4	yes	—
110	#13133124	C₂×C₄	1	1	yes	characteristic
111	#13133124	C₂×C₄	1	1	yes	characteristic
112	#13133124	C₂×C₄	1	1	yes	characteristic
113	#13133124	C₂×C₄	1	1	yes	characteristic
114	#13133124	C₂×C₄	1	1	yes	characteristic
115	#13133124	C₂×C₄	1	1	yes	characteristic
116	#13133124	C₂×C₄	1	1	yes	characteristic
117	#13133124	C₂×C₄	1	1	yes	characteristic
118	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
120	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
121	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₅₃	—	2	2	no	—
128	C₄×C₈×C₅₃	—	2	2	no	—
129	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
130	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
131	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₄×D₅₃	—	2	2	no	—
147	C₂×C₄×D₅₃	—	2	2	no	—
148	C₂×C₄×D₅₃	—	2	1	no	—
149	C₂×C₄×D₅₃	—	2	1	no	—
150	C₂×C₄×D₅₃	—	2	1	no	—
151	C₂×C₄×D₅₃	—	2	1	no	—
152	C₂×C₄×D₅₃	—	2	2	no	—
153	C₂×C₄×D₅₃	—	2	2	no	—
154	C₂×C₄×D₅₃	—	2	2	no	—
155	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
156	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
157	C₂×C₄×D₅₃	#41	1	2	yes	—
158	C₂×Dic₁₀₆	—	2	2	no	—
159	C₂×Dic₁₀₆	—	2	2	no	—
160	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	complemented
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	1	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	2	1	no	—
171	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
175	C₄²×C₅₃	—	2	1	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₄²×C₅₃	—	2	1	no	—
179	C₄²×C₅₃	—	2	1	no	—
180	C₄²×C₅₃	—	2	1	no	—
181	#11719921	—	2	2	no	—
182	#11719921	—	2	2	no	—
183	#11719921	C₂×C₈	1	4	yes	—
184	#11719921	C₂×C₈	1	4	yes	—
185	C₄⋊Dic₅₃	—	2	2	no	—
186	#11719919	—	2	2	no	—
187	#11719919	—	2	2	no	—
188	#11719919	—	2	2	no	—
189	#11719919	—	2	2	no	—
190	#11719919	C₂×C₈	1	4	yes	complemented
191	#11719919	C₂×C₈	1	4	yes	complemented
192	C₄×Dic₅₃	—	2	1	no	—
193	C₄×Dic₅₃	—	2	1	no	—
194	C₄×Dic₅₃	—	2	1	no	—
195	C₄×Dic₅₃	—	2	1	no	—
196	C₄×Dic₅₃	—	2	2	no	—
197	C₄×Dic₅₃	—	2	2	no	—
198	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
199	C₄×Dic₅₃	#41	1	2	yes	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	2	1	no	—
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	2	1	no	—
207	C₂×#88322	—	2	1	no	—
208	C₂×#88322	C₂×D₄	1	2	yes	—
209	C₂×#88322	C₂×Q₈	1	2	yes	—
210	C₂×#88322	#41	1	2	yes	—
211	C₂×#88322	#41	1	2	yes	—
212	C₈×D₅₃	—	2	8	no	—
213	C₈×D₅₃	—	2	8	no	—
214	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
215	C₂²×D₅₃	—	2	1	no	—
216	C₂²×D₅₃	—	2	1	no	—
217	C₂²×D₅₃	—	2	1	no	—
218	C₂²×D₅₃	—	2	1	no	—
219	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
220	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
221	C₂²×D₅₃	#131	1	2	yes	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
237	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
238	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
239	C₂×C₄×C₅₃	#131	1	2	yes	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
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	—	2	1	no	—
252	C₂×Dic₅₃	—	2	1	no	—
253	C₂×Dic₅₃	—	2	1	no	—
254	C₂×Dic₅₃	—	2	1	no	—
255	C₂×Dic₅₃	—	2	1	no	—
256	C₂×Dic₅₃	—	2	1	no	—
257	C₂×Dic₅₃	—	2	1	no	—
258	C₂×Dic₅₃	—	2	1	no	—
259	C₂×Dic₅₃	—	2	1	no	—
260	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
261	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
262	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
263	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
264	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
265	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
266	C₂×Dic₅₃	#131	1	2	yes	—
267	C₄×D₅₃	—	2	8	no	—
268	C₄×D₅₃	—	2	2	no	—
269	C₄×D₅₃	—	2	2	no	—
270	C₄×D₅₃	—	2	2	no	—
271	C₄×D₅₃	—	2	2	no	—
272	C₄×D₅₃	—	2	8	no	—
273	Dic₁₀₆	—	2	8	no	—
274	C₈×C₅₃	—	2	2	no	—
275	C₈×C₅₃	—	2	2	no	—
276	C₈×C₅₃	—	4	1	no	—
277	C₈×C₅₃	—	4	1	no	—
278	#88322	—	2	2	no	—
279	#88322	—	4	1	no	—
280	#88322	—	4	1	no	—
281	#88322	C₂²⋊Q₈	1	4	yes	complemented
282	C₈×C₂²⋊Q₈	—	53	1	no	maximal, Sylow
283	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
284	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
285	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
286	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
287	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
288	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
289	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
290	C₂×D₅₃	—	2	2	no	—
291	C₂×D₅₃	—	2	2	no	—
292	C₂×D₅₃	—	2	2	no	—
293	C₂×D₅₃	—	2	2	no	—
294	C₂×D₅₃	—	2	2	no	—
295	C₂×D₅₃	—	2	2	no	—
296	C₂×D₅₃	C₈×Q₈	1	4	yes	complemented
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	2	2	no	—
300	C₄×C₅₃	—	2	2	no	—
301	C₄×C₅₃	—	4	1	no	—
302	C₄×C₅₃	—	4	1	no	—
303	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
304	C₄×C₅₃	#525	1	1	yes	characteristic
305	C₄×C₅₃	#529	1	1	yes	characteristic
306	C₄×C₅₃	#533	1	1	yes	characteristic
307	Dic₅₃	—	2	2	no	—
308	Dic₅₃	—	2	2	no	—
309	Dic₅₃	—	2	2	no	—
310	Dic₅₃	—	2	2	no	—
311	Dic₅₃	—	4	1	no	—
312	Dic₅₃	—	4	1	no	—
313	Dic₅₃	C₈×D₄	1	4	yes	—
314	C₂×C₈×Q₈	—	53	1	no	—
315	C₂²×C₄×C₈	—	53	1	no	—
316	C₄×C₂²⋊Q₈	—	53	1	no	—
317	#1984	—	53	1	no	—
318	#1939	—	53	1	no	—
319	#1916	—	53	1	no	—
320	#1916	D₅₃	1	1	yes	characteristic, complemented
321	#1850	—	53	1	no	—
322	#1850	—	53	1	no	—
323	#1797	—	53	1	no	—
324	C₈×C₄⋊C₄	—	53	1	no	—
325	C₈×C₄⋊C₄	—	53	1	no	—
326	C₈×C₄⋊C₄	—	53	1	no	—
327	C₈×C₂²⋊C₄	—	53	1	no	—
328	C₈×C₂²⋊C₄	—	53	1	no	—
329	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
330	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
331	C₂×C₅₃	#2929	1	1	yes	characteristic
332	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
333	C₂×C₅₃	#2960	1	1	yes	characteristic
334	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
335	C₂×C₅₃	#2966	1	1	yes	characteristic
336	D₅₃	—	2	4	no	—
337	C₂³×C₈	—	53	1	no	—
338	C₂³×C₈	—	53	1	no	—
339	C₂×C₂²⋊Q₈	—	53	1	no	—
340	C₂×C₄×Q₈	—	53	1	no	—
341	C₂²×C₄²	—	53	1	no	—
342	C₈×Q₈	—	106	4	no	—
343	C₂×C₄⋊C₈	—	53	1	no	—
344	C₂×C₄⋊C₈	—	53	1	no	—
345	C₂×C₄⋊C₈	—	53	1	no	—
346	C₂×C₄⋊C₈	—	53	1	no	—
347	C₂×C₄⋊C₈	—	53	1	no	—
348	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
349	C₂×C₂²⋊C₈	—	53	1	no	—
350	C₂×C₂²⋊C₈	—	53	1	no	—
351	C₂×C₄×C₈	—	53	1	no	—
352	C₂×C₄×C₈	—	53	1	no	—
353	C₂×C₄×C₈	—	53	1	no	—
354	C₂×C₄×C₈	—	53	1	no	—
355	C₂×C₄×C₈	—	53	1	no	—
356	C₂×C₄×C₈	—	53	1	no	—
357	C₂×C₄×C₈	—	53	1	no	—
358	C₂×C₄×C₈	—	106	4	no	—
359	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
360	#391	—	53	1	no	—
361	#389	—	53	1	no	—
362	#387	—	53	1	no	—
363	#387	—	53	1	no	—
364	#385	—	53	1	no	—
365	#385	—	53	1	no	—
366	#380	—	53	1	no	—
367	C₄×C₄⋊C₄	—	53	1	no	—
368	C₄×C₄⋊C₄	—	53	1	no	—
369	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
370	C₄×C₂²⋊C₄	—	53	1	no	—
371	C₄×C₂²⋊C₄	—	53	1	no	—
372	#336	—	53	1	no	—
373	#336	—	53	1	no	—
374	#336	—	53	1	no	—
375	#336	—	53	1	no	—
376	#336	—	53	1	no	—
377	#336	—	53	1	no	—
378	#336	C₂×D₅₃	1	1	yes	characteristic
379	#336	C₂×D₅₃	1	1	yes	characteristic
380	#336	C₂×D₅₃	1	1	yes	characteristic
381	#336	C₂×D₅₃	1	1	yes	characteristic
382	C₅₃	C₈×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
383	C₂²×Q₈	—	53	1	no	—
384	C₂³×C₄	—	53	1	no	—
385	C₂³×C₄	—	53	1	no	—
386	C₂³×C₄	—	53	1	no	—
387	C₂²×C₈	—	53	1	no	—
388	C₂²×C₈	—	53	1	no	—
389	C₂²×C₈	—	53	4	no	—
390	C₂²×C₈	—	53	1	no	—
391	C₂²×C₈	—	53	1	no	—
392	C₂²×C₈	—	106	2	no	—
393	C₂²×C₈	—	106	2	no	—
394	C₂²×C₈	—	106	2	no	—
395	C₂²×C₈	—	106	2	no	—
396	C₂²×C₈	—	106	2	no	—
397	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
398	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
399	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
400	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
401	C₂²⋊Q₈	—	53	8	no	—
402	C₄×Q₈	—	106	2	no	—
403	C₄×Q₈	—	106	2	no	—
404	C₂×C₄⋊C₄	—	53	1	no	—
405	C₂×C₄⋊C₄	—	53	1	no	—
406	C₂×C₄⋊C₄	—	53	1	no	—
407	C₂×C₄⋊C₄	—	53	1	no	—
408	C₂×C₄⋊C₄	—	53	1	no	—
409	C₂×C₄⋊C₄	—	53	1	no	—
410	C₂×C₄⋊C₄	—	53	1	no	—
411	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
413	C₂×C₂²⋊C₄	—	53	1	no	—
414	C₂×C₂²⋊C₄	—	53	1	no	—
415	C₂×C₂²⋊C₄	—	53	1	no	—
416	C₂×C₂²⋊C₄	—	53	1	no	—
417	C₂×C₄²	—	53	1	no	—
418	C₂×C₄²	—	53	1	no	—
419	C₂×C₄²	—	53	1	no	—
420	C₂×C₄²	—	53	1	no	—
421	C₂×C₄²	—	53	1	no	—
422	C₂×C₄²	—	106	2	no	—
423	C₂×C₄²	—	106	2	no	—
424	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
425	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
426	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
427	C₄⋊C₈	—	2	2	no	—
428	C₄⋊C₈	—	106	2	no	—
429	C₄⋊C₈	—	106	2	no	—
430	C₄⋊C₈	—	106	2	no	—
431	C₄⋊C₈	—	106	2	no	—
432	C₄⋊C₈	—	106	2	no	—
433	C₂²⋊C₈	—	106	2	no	—
434	C₂²⋊C₈	—	106	2	no	—
435	C₄×C₈	—	2	2	no	—
436	C₄×C₈	—	53	4	no	—
437	C₄×C₈	—	106	2	no	—
438	C₄×C₈	—	106	2	no	—
439	C₄×C₈	—	106	2	no	—
440	C₄×C₈	—	106	2	no	—
441	C₄×C₈	—	106	2	no	—
442	C₄×C₈	—	106	2	no	—
443	#95	—	53	1	no	—
444	#95	—	53	1	no	—
445	#95	—	53	1	no	—
446	#95	—	53	1	no	—
447	#95	—	53	1	no	—
448	#95	—	53	1	no	—
449	#95	—	53	1	no	—
450	#95	—	53	1	no	—
451	#95	C₄×D₅₃	1	1	yes	characteristic
452	#95	C₄×D₅₃	1	1	yes	characteristic
453	C₂⁴	—	53	1	no	—
454	C₂×Q₈	—	53	4	no	—
455	C₂×Q₈	—	106	2	no	—
456	C₂×Q₈	—	106	2	no	—
457	C₂²×C₄	—	53	1	no	—
458	C₂²×C₄	—	53	1	no	—
459	C₂²×C₄	—	53	1	no	—
460	C₂²×C₄	—	53	1	no	—
461	C₂²×C₄	—	53	1	no	—
462	C₂²×C₄	—	53	4	no	—
463	C₂²×C₄	—	53	2	no	—
464	C₂²×C₄	—	53	2	no	—
465	C₂²×C₄	—	53	1	no	—
466	C₂²×C₄	—	53	1	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	1	no	—
473	C₂²×C₄	—	106	1	no	—
474	C₂²×C₄	—	106	1	no	—
475	C₂²×C₄	—	106	1	no	—
476	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
482	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	53	2	no	—
494	C₂×C₈	—	53	2	no	—
495	C₂×C₈	—	53	2	no	—
496	C₂×C₈	—	53	2	no	—
497	C₂×C₈	—	106	8	no	—
498	C₂×C₈	—	106	8	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	1	no	—
502	C₂×C₈	—	106	1	no	—
503	C₂×C₈	—	106	1	no	—
504	C₂×C₈	—	106	1	no	—
505	C₂×C₈	—	106	1	no	—
506	C₂×C₈	—	106	1	no	—
507	C₂×C₈	D₄×D₅₃	1	2	yes	—
508	C₂×C₈	Q₈×D₅₃	1	2	yes	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	53	4	no	—
511	C₄⋊C₄	—	53	4	no	—
512	C₄⋊C₄	—	106	2	no	—
513	C₄⋊C₄	—	106	2	no	—
514	C₄⋊C₄	—	106	2	no	—
515	C₄⋊C₄	—	106	2	no	—
516	C₄⋊C₄	—	106	2	no	—
517	C₄⋊C₄	C₈×D₅₃	1	4	yes	complemented
518	C₂²⋊C₄	—	53	4	no	—
519	C₂²⋊C₄	—	53	4	no	—
520	C₂²⋊C₄	—	106	2	no	—
521	C₂²⋊C₄	—	106	2	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	53	2	no	—
529	C₄²	—	53	2	no	—
530	C₄²	—	106	2	no	—
531	C₄²	—	106	2	no	—
532	C₄²	—	106	1	no	—
533	C₄²	—	106	1	no	—
534	C₄²	—	106	1	no	—
535	C₄²	—	106	1	no	—
536	C₂³	—	53	2	no	—
537	C₂³	—	53	2	no	—
538	C₂³	—	53	2	no	—
539	C₂³	—	106	1	no	—
540	C₂³	—	106	1	no	—
541	C₂³	—	106	1	no	—
542	C₂³	—	106	1	no	—
543	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
544	Q₈	—	106	8	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	53	2	no	—
560	C₂×C₄	—	53	2	no	—
561	C₂×C₄	—	53	2	no	—
562	C₂×C₄	—	53	2	no	—
563	C₂×C₄	—	53	2	no	—
564	C₂×C₄	—	53	2	no	—
565	C₂×C₄	—	53	2	no	—
566	C₂×C₄	—	106	8	no	—
567	C₂×C₄	—	106	8	no	—
568	C₂×C₄	—	106	1	no	—
569	C₂×C₄	—	106	1	no	—
570	C₂×C₄	—	106	1	no	—
571	C₂×C₄	—	106	1	no	—
572	C₂×C₄	—	106	1	no	—
573	C₂×C₄	—	106	1	no	—
574	C₂×C₄	—	106	1	no	—
575	C₂×C₄	—	106	1	no	—
576	C₂×C₄	—	106	2	no	—
577	C₂×C₄	—	106	2	no	—
578	C₂×C₄	—	106	2	no	—
579	C₂×C₄	—	106	2	no	—
580	C₂×C₄	—	106	1	no	—
581	C₂×C₄	—	106	1	no	—
582	C₂×C₄	—	106	1	no	—
583	C₂×C₄	—	106	1	no	—
584	C₂×C₄	—	106	1	no	—
585	C₂×C₄	—	106	1	no	—
586	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
587	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
588	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
589	C₂×C₄	#13133225	1	2	yes	—
590	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
591	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
592	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
593	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
594	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
595	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
596	C₈	—	2	2	no	—
597	C₈	—	2	2	no	—
598	C₈	—	4	1	no	—
599	C₈	—	4	1	no	—
600	C₈	—	53	4	no	—
601	C₈	—	106	2	no	—
602	C₈	—	212	1	no	—
603	C₈	—	212	1	no	—
604	C₂²	—	53	4	no	—
605	C₂²	—	106	2	no	—
606	C₂²	—	106	2	no	—
607	C₂²	—	106	2	no	—
608	C₂²	—	106	2	no	—
609	C₂²	—	106	2	no	—
610	C₂²	—	106	2	no	—
611	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
612	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
613	C₂²	#16811091	1	1	yes	characteristic
614	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
615	C₂²	#131×D₅₃	1	1	yes	characteristic
616	C₂²	#131×D₅₃	1	1	yes	characteristic
617	C₂²	C₂×#13133225	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	4	1	no	—
623	C₄	—	4	1	no	—
624	C₄	—	53	4	no	—
625	C₄	—	106	2	no	—
626	C₄	—	106	2	no	—
627	C₄	—	106	2	no	—
628	C₄	—	106	2	no	—
629	C₄	—	212	1	no	—
630	C₄	—	212	1	no	—
631	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
632	C₄	#16811131	1	1	yes	characteristic
633	C₄	#16811156	1	1	yes	characteristic
634	C₄	#16811179	1	1	yes	characteristic
635	C₂	—	106	4	no	—
636	C₂	#20277249	1	1	yes	characteristic
637	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
638	C₂	#20283406	1	1	yes	characteristic
639	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
640	C₂	#20283646	1	1	yes	characteristic
641	C₂	#20283671	1	1	yes	characteristic
642	C₂	#20283688	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#1916	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1916	D₅₃
hypocenter	C₅₃	C₈×C₂²⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1916

Complements

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

N	H ≅ G/N	type
1. C₂×#16810747	C₂	semidirect
4. #20275649	C₂	semidirect
5. #20275432	C₂	semidirect
6. #20275272	C₂	semidirect
10. C₄⋊C₄×#88322	C₂	semidirect
11. #20274576	C₂	semidirect
12. #20274576	C₂	semidirect
15. C₅₃×#1916	C₂	semidirect
74. C₂×C₈×D₅₃	Q₈	semidirect
75. #13133195	C₈	semidirect
160. C₂×Dic₁₀₆	C₂×C₈	semidirect
174. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
190. #11719919	C₂×C₈	semidirect
191. #11719919	C₂×C₈	semidirect
281. #88322	C₂²⋊Q₈	semidirect
296. C₂×D₅₃	C₈×Q₈	semidirect
320. #1916	D₅₃	semidirect
382. C₅₃	C₈×C₂²⋊Q₈	semidirect
517. C₄⋊C₄	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810747	2
C₂×C₄×C₈×D₅₃	2
C₄×#13133195	2
#20275649	2
#20275432	2
#20275272	2
#20274894	2
#20274891	2
#20274694	2
C₄⋊C₄×#88322	2
#20274576	2
#20274485	2
C₅₃×#1916	2
C₈×C₂²⋊Q₈	53