Subgroups — D₅₃×#3177 #28253928

Order 13568.

1842 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×D₅₃	C₂	1	2	yes	maximal
2	#20285375	C₂	1	1	yes	maximal, characteristic, complemented
3	#20285374	C₂	1	1	yes	maximal, characteristic, complemented
4	#20285373	C₂	1	2	yes	maximal, complemented
5	#20285372	C₂	1	2	yes	maximal, complemented
6	#20285371	C₂	1	1	yes	maximal, characteristic, complemented
7	#20285370	C₂	1	2	yes	maximal, complemented
8	#20285369	C₂	1	2	yes	maximal, complemented
9	#20285368	C₂	1	2	yes	maximal, complemented
10	#20285367	C₂	1	2	yes	maximal, complemented
11	#499×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#494×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#491×D₅₃	C₂	1	2	yes	maximal, complemented
14	#470×D₅₃	C₂	1	2	yes	maximal
15	#467×D₅₃	C₂	1	2	yes	maximal
16	#441×D₅₃	C₂	1	2	yes	maximal, complemented
17	C₄×Q₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
18	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
19	C₅₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
21	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
22	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	#16811236	C₂²	1	2	yes	—
25	#16811235	C₂²	1	2	yes	—
26	#16811234	C₂²	1	2	yes	—
27	#16811233	C₂²	1	4	yes	—
28	#16811232	—	2	4	no	—
29	#16811231	C₂²	1	4	yes	—
30	#16811230	—	2	4	no	—
31	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
32	#126×D₅₃	—	2	4	no	—
33	#125×D₅₃	—	2	4	no	—
34	#125×D₅₃	C₂²	1	2	yes	—
35	C₂²⋊Q₈×D₅₃	C₂²	1	4	yes	—
36	C₄×Q₈×D₅₃	C₂²	1	4	yes	—
37	#117×D₅₃	C₂²	1	1	yes	characteristic
38	#16811025	C₂²	1	1	yes	characteristic, complemented
39	#16811024	C₂²	1	2	yes	complemented
40	#16811023	C₂²	1	2	yes	complemented
41	#16811022	C₂²	1	2	yes	complemented
42	#16811003	C₂²	1	1	yes	characteristic, complemented
43	#16811002	C₂²	1	2	yes	complemented
44	#16811001	C₂²	1	2	yes	complemented
45	#16811001	C₂²	1	2	yes	complemented
46	#16810993	C₂²	1	2	yes	complemented
47	#16810992	C₂²	1	2	yes	complemented
48	#16810991	C₂²	1	2	yes	complemented
49	#16810990	C₂²	1	4	yes	complemented
50	#16810990	C₂²	1	4	yes	complemented
51	#16810882	C₂²	1	2	yes	—
52	#16810881	C₂²	1	2	yes	—
53	#16810880	C₂²	1	4	yes	—
54	#16810879	C₂²	1	4	yes	—
55	#16810878	C₂²	1	2	yes	—
56	#16810865	C₂²	1	2	yes	—
57	#16810864	C₂²	1	2	yes	—
58	#16810863	C₂²	1	4	yes	—
59	#16810862	C₂²	1	4	yes	—
60	#16810861	C₂²	1	2	yes	—
61	#16810731	C₂²	1	2	yes	complemented
62	#16810730	C₂²	1	2	yes	complemented
63	#16810729	C₂²	1	2	yes	complemented
64	#16810728	C₂²	1	4	yes	complemented
65	#16810727	C₂²	1	4	yes	complemented
66	C₄×Dic₂₁₂	C₂²	1	2	yes	complemented
67	Q₁₆×Dic₅₃	C₂²	1	2	yes	complemented
68	#16810717	C₂²	1	2	yes	complemented
69	C₄×#11719925	C₂²	1	4	yes	complemented
70	#16810715	C₂²	1	4	yes	complemented
71	#107×D₅₃	C₂²	1	2	yes	—
72	#106×D₅₃	C₂²	1	2	yes	—
73	#103×D₅₃	C₂²	1	4	yes	—
74	#103×D₅₃	C₂²	1	4	yes	—
75	#16810581	C₂²	1	1	yes	characteristic
76	#16810580	C₂²	1	1	yes	characteristic
77	#16810580	C₂²	1	1	yes	characteristic
78	#16810579	C₂²	1	1	yes	characteristic
79	#16810578	C₂²	1	1	yes	characteristic
80	#16810577	C₂²	1	1	yes	characteristic
81	#16810577	C₂²	1	1	yes	characteristic
82	#97×D₅₃	C₂²	1	1	yes	characteristic
83	#97×D₅₃	C₂²	1	1	yes	characteristic
84	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
85	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
86	C₅₃×#541	C₂²	1	2	yes	—
87	C₅₃×#499	C₂²	1	1	yes	characteristic, complemented
88	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
89	C₅₃×#491	C₂²	1	2	yes	complemented
90	C₅₃×#470	C₂²	1	2	yes	—
91	C₅₃×#467	C₂²	1	2	yes	—
92	C₅₃×#441	C₂²	1	2	yes	complemented
93	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
94	C₅₃×#405	C₂²	1	1	yes	characteristic
95	C₂×Q₈×D₅₃	C₂³	1	4	yes	—
96	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
97	C₂×Dic₂₁₂	C₂³	1	2	yes	—
98	C₂×Dic₂₁₂	C₂³	1	2	yes	—
99	C₂×#11719925	C₂³	1	4	yes	—
100	C₂×#11719925	C₂³	1	4	yes	—
101	Q₁₆×D₅₃	—	2	8	no	—
102	Q₁₆×D₅₃	—	2	8	no	—
103	C₂×#11719917	C₂³	1	1	yes	characteristic
104	C₂×#11719912	C₂³	1	1	yes	characteristic
105	C₂×#11719912	C₂³	1	1	yes	characteristic
106	M₁₆×D₅₃	—	2	4	no	—
107	C₂²×#88322	C₂³	1	1	yes	characteristic
108	C₂×C₈×D₅₃	—	2	4	no	—
109	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
110	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
115	#13133217	C₂³	1	2	yes	—
116	#13133216	—	2	4	no	—
117	#13133215	C₂³	1	4	yes	—
118	#13133212	—	2	4	no	—
119	#13133212	—	2	4	no	—
120	#13133212	—	2	4	no	—
121	#13133211	—	2	4	no	—
122	#13133210	—	2	4	no	—
123	#13133210	—	2	4	no	—
124	#13133210	—	2	4	no	—
125	#13133209	—	2	4	no	—
126	#13133209	C₂³	1	2	yes	—
127	#13133208	—	2	4	no	—
128	#13133208	—	2	4	no	—
129	#13133208	C₂³	1	4	yes	—
130	#13133207	—	2	4	no	—
131	#13133207	—	2	4	no	—
132	#13133207	—	2	4	no	—
133	#13133207	—	2	4	no	—
134	#13133207	—	2	4	no	—
135	#13133197	C₂³	1	4	yes	—
136	#13133196	C₂³	1	4	yes	—
137	#13133195	—	2	4	no	—
138	#13133194	—	2	4	no	—
139	#13133193	C₂³	1	4	yes	—
140	Q₈×Dic₅₃	C₂³	1	4	yes	—
141	C₄×Dic₁₀₆	—	2	4	no	—
142	C₄×Dic₁₀₆	C₂³	1	4	yes	—
143	#13133182	—	2	4	no	—
144	#13133182	C₂³	1	4	yes	—
145	#13133176	C₂³	1	1	yes	characteristic
146	#13133175	D₄	1	2	yes	—
147	#13133174	C₂³	1	1	yes	characteristic
148	#13133174	C₂³	1	1	yes	characteristic
149	#13133173	D₄	1	2	yes	—
150	C₄⋊C₄×D₅₃	—	2	4	no	—
151	C₄⋊C₄×D₅₃	—	2	4	no	—
152	C₄⋊C₄×D₅₃	—	2	4	no	—
153	C₄⋊C₄×D₅₃	—	2	4	no	—
154	C₄⋊C₄×D₅₃	D₄	1	2	yes	—
155	C₄⋊C₄×D₅₃	C₂³	1	4	yes	—
156	C₂²⋊C₄×D₅₃	—	2	4	no	—
157	C₂²⋊C₄×D₅₃	D₄	1	2	yes	—
158	C₄²×D₅₃	—	2	4	no	—
159	C₄²×D₅₃	C₂³	1	1	yes	characteristic
160	C₄²×D₅₃	C₂³	1	1	yes	characteristic
161	#13133151	C₂³	1	2	yes	—
162	#13133150	C₂³	1	4	yes	—
163	#13133149	C₂³	1	2	yes	—
164	#13133148	C₂³	1	4	yes	—
165	#13133142	C₂³	1	4	yes	—
166	#13133142	C₂³	1	4	yes	—
167	#13133141	C₂³	1	4	yes	—
168	#13133141	C₂³	1	4	yes	—
169	#13133140	C₂³	1	4	yes	—
170	#13133140	C₂³	1	4	yes	—
171	#13133128	C₂³	1	1	yes	characteristic
172	#13133128	C₂³	1	1	yes	characteristic
173	#13133127	C₂³	1	1	yes	characteristic
174	#13133127	C₂³	1	1	yes	characteristic
175	#13133127	C₂³	1	1	yes	characteristic
176	#13133127	C₂³	1	1	yes	characteristic
177	C₄×#88322	C₂³	1	1	yes	characteristic
178	C₄×#88322	C₂³	1	1	yes	characteristic
179	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
180	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
181	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
182	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
183	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
184	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
185	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
186	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
187	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
188	#126×C₅₃	—	2	4	no	—
189	#125×C₅₃	—	2	4	no	—
190	#125×C₅₃	C₂³	1	2	yes	—
191	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
192	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
193	#117×C₅₃	C₂³	1	1	yes	characteristic
194	#107×C₅₃	C₂³	1	2	yes	—
195	#106×C₅₃	C₂³	1	2	yes	—
196	#103×C₅₃	C₂³	1	4	yes	—
197	#103×C₅₃	C₂³	1	4	yes	—
198	#97×C₅₃	C₂³	1	1	yes	characteristic
199	#97×C₅₃	C₂³	1	1	yes	characteristic
200	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
201	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
202	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
203	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
205	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
206	Q₈×D₅₃	—	2	8	no	—
207	Q₈×D₅₃	—	2	8	no	—
208	C₂×C₄×D₅₃	—	2	2	no	—
209	C₂×C₄×D₅₃	—	2	2	no	—
210	C₂×C₄×D₅₃	—	2	4	no	—
211	C₂×C₄×D₅₃	—	2	4	no	—
212	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
213	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
214	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
215	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
216	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
217	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
218	C₂×Dic₁₀₆	—	2	4	no	—
219	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
220	Q₁₆×C₅₃	—	2	4	no	—
221	Q₁₆×C₅₃	—	2	4	no	—
222	M₁₆×C₅₃	—	2	2	no	—
223	C₂×C₈×C₅₃	—	2	2	no	—
224	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
225	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
226	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
228	C₄⋊C₄×C₅₃	—	2	4	no	—
229	C₄⋊C₄×C₅₃	—	2	4	no	—
230	C₄⋊C₄×C₅₃	—	2	4	no	—
231	C₄⋊C₄×C₅₃	—	2	4	no	—
232	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
233	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
234	C₂²⋊C₄×C₅₃	—	2	4	no	—
235	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
236	C₄²×C₅₃	—	2	4	no	—
237	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
238	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
239	C₂²⋊Dic₅₃	—	2	4	no	—
240	C₂²⋊Dic₅₃	C₂×D₄	1	2	yes	—
241	#11719925	—	2	8	no	—
242	#11719925	—	2	8	no	—
243	#11719921	—	2	4	no	—
244	#11719921	—	2	4	no	—
245	#11719921	C₂×D₄	1	2	yes	—
246	#11719921	C₂×D₄	1	2	yes	—
247	C₄⋊Dic₅₃	—	2	4	no	—
248	C₄⋊Dic₅₃	—	2	4	no	—
249	C₄⋊Dic₅₃	—	2	4	no	—
250	C₄⋊Dic₅₃	—	2	4	no	—
251	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
252	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
253	#11719919	—	2	4	no	—
254	#11719919	—	2	4	no	—
255	#11719919	—	2	4	no	—
256	#11719919	—	2	4	no	—
257	#11719919	—	2	4	no	—
258	#11719919	—	2	4	no	—
259	#11719919	—	2	4	no	—
260	#11719919	—	2	4	no	—
261	#11719919	—	2	4	no	—
262	#11719919	C₂×D₄	1	2	yes	—
263	#11719919	C₂×D₄	1	2	yes	—
264	C₄×Dic₅₃	—	2	4	no	—
265	C₄×Dic₅₃	—	2	4	no	—
266	C₄×Dic₅₃	—	2	4	no	—
267	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
268	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
269	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
270	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
271	#11719917	—	2	2	no	—
272	C₂×#88322	—	2	2	no	—
273	C₂×#88322	C₂⁴	1	1	yes	characteristic
274	C₂×#88322	C₂⁴	1	1	yes	characteristic
275	C₂×#88322	C₂⁴	1	1	yes	characteristic
276	C₂×#88322	C₂⁴	1	1	yes	characteristic
277	Dic₂₁₂	—	2	4	no	—
278	Dic₂₁₂	—	2	4	no	—
279	#11719912	—	2	2	no	—
280	#11719912	—	2	2	no	—
281	C₈×D₅₃	—	2	2	no	—
282	C₈×D₅₃	—	2	2	no	—
283	C₈×D₅₃	—	2	2	no	—
284	C₈×D₅₃	—	2	2	no	—
285	C₈×D₅₃	#41	1	4	yes	—
286	C₈×D₅₃	#41	1	4	yes	—
287	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
288	C₂²×D₅₃	—	2	2	no	—
289	C₂²×D₅₃	—	2	4	no	—
290	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
291	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
292	Q₈×C₅₃	—	2	4	no	—
293	Q₈×C₅₃	—	2	4	no	—
294	C₂×C₄×C₅₃	—	2	1	no	—
295	C₂×C₄×C₅₃	—	2	1	no	—
296	C₂×C₄×C₅₃	—	2	4	no	—
297	C₂×C₄×C₅₃	—	2	4	no	—
298	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
302	C₂×Dic₅₃	—	2	2	no	—
303	C₂×Dic₅₃	—	2	4	no	—
304	C₂×Dic₅₃	—	2	4	no	—
305	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
307	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
308	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
309	C₄×D₅₃	—	2	2	no	—
310	C₄×D₅₃	—	2	4	no	—
311	C₄×D₅₃	—	2	4	no	—
312	C₄×D₅₃	—	2	1	no	—
313	C₄×D₅₃	—	2	1	no	—
314	C₄×D₅₃	—	2	2	no	—
315	C₄×D₅₃	—	4	8	no	—
316	C₄×D₅₃	—	4	8	no	—
317	C₄×D₅₃	C₂×#41	1	2	yes	—
318	C₄×D₅₃	C₂×#41	1	2	yes	—
319	Dic₁₀₆	—	2	4	no	—
320	Dic₁₀₆	—	2	4	no	—
321	Dic₁₀₆	—	4	8	no	—
322	C₈×C₅₃	—	2	1	no	—
323	C₈×C₅₃	—	2	1	no	—
324	C₈×C₅₃	C₂×#41	1	2	yes	—
325	C₈×C₅₃	C₂×#41	1	2	yes	—
326	#88322	—	2	1	no	—
327	#88322	—	2	1	no	—
328	#88322	C₂×#41	1	2	yes	—
329	#88322	C₂×#41	1	2	yes	—
330	C₂×#3177	—	53	1	no	maximal, Sylow
331	C₂²×C₅₃	—	2	1	no	—
332	C₂²×C₅₃	—	2	2	no	—
333	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
334	C₂×D₅₃	—	2	1	no	—
335	C₂×D₅₃	—	2	2	no	—
336	C₂×D₅₃	—	4	4	no	—
337	C₂×D₅₃	#532	1	2	yes	—
338	C₂×D₅₃	#578	1	4	yes	—
339	C₄×C₅₃	—	2	1	no	—
340	C₄×C₅₃	—	2	2	no	—
341	C₄×C₅₃	—	2	2	no	—
342	C₄×C₅₃	—	4	4	no	—
343	C₄×C₅₃	—	4	4	no	—
344	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
345	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
346	Dic₅₃	—	2	1	no	—
347	Dic₅₃	—	2	2	no	—
348	Dic₅₃	—	2	2	no	—
349	Dic₅₃	—	4	4	no	—
350	Dic₅₃	—	4	4	no	—
351	Dic₅₃	#532	1	2	yes	—
352	C₂×#541	—	53	2	no	—
353	#3177	—	53	1	no	—
354	#3177	—	53	1	no	—
355	#3177	—	53	2	no	—
356	#3177	—	53	2	no	—
357	#3177	—	53	2	no	—
358	#3177	—	53	2	no	—
359	#3177	—	53	2	no	—
360	#3177	—	53	1	no	—
361	#3177	—	53	2	no	—
362	#3177	D₅₃	1	1	yes	characteristic, direct product factor
363	C₂×#499	—	53	1	no	—
364	C₂×#494	—	53	1	no	—
365	C₂×#491	—	53	2	no	—
366	C₂×#470	—	53	2	no	—
367	C₂×#467	—	53	2	no	—
368	C₂×#441	—	53	2	no	—
369	C₂×C₄×Q₁₆	—	53	2	no	—
370	C₂×#405	—	53	1	no	—
371	C₂×C₅₃	—	4	1	no	—
372	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
373	C₂×C₅₃	C₂×#578	1	2	yes	—
374	D₅₃	—	4	1	no	—
375	D₅₃	#3177	1	4	yes	direct product factor
376	C₂²×Q₁₆	—	53	2	no	—
377	C₂²×Q₁₆	—	53	2	no	—
378	C₂²×M₁₆	—	53	1	no	—
379	C₂³×C₈	—	53	1	no	—
380	#541	—	53	2	no	—
381	#541	—	53	2	no	—
382	#541	—	53	4	no	—
383	#541	—	53	4	no	—
384	#541	—	53	2	no	—
385	#541	—	106	4	no	—
386	#541	—	106	4	no	—
387	#541	C₂×D₅₃	1	2	yes	—
388	C₂×C₄⋊Q₈	—	53	2	no	—
389	C₂×#126	—	106	4	no	—
390	C₂×#125	—	53	2	no	—
391	C₂×#125	—	106	4	no	—
392	C₂×C₂²⋊Q₈	—	53	4	no	—
393	C₂×C₄×Q₈	—	53	4	no	—
394	C₂×#117	—	53	1	no	—
395	#499	—	53	2	no	—
396	#499	—	53	2	no	—
397	#499	—	53	2	no	—
398	#499	—	53	1	no	—
399	#499	C₂×D₅₃	1	1	yes	characteristic, complemented
400	#494	—	53	2	no	—
401	#494	—	53	2	no	—
402	#494	—	53	2	no	—
403	#494	—	53	1	no	—
404	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
405	#491	—	53	2	no	—
406	#491	—	53	2	no	—
407	#491	—	53	4	no	—
408	#491	—	53	4	no	—
409	#491	—	53	2	no	—
410	#491	C₂×D₅₃	1	2	yes	complemented
411	#470	—	53	2	no	—
412	#470	—	53	2	no	—
413	#470	—	53	4	no	—
414	#470	—	53	2	no	—
415	#470	—	53	4	no	—
416	#470	C₂×D₅₃	1	2	yes	—
417	#467	—	53	2	no	—
418	#467	—	53	2	no	—
419	#467	—	53	4	no	—
420	#467	—	53	2	no	—
421	#467	—	53	4	no	—
422	#467	C₂×D₅₃	1	2	yes	—
423	#441	—	53	2	no	—
424	#441	—	53	4	no	—
425	#441	—	53	2	no	—
426	#441	—	53	4	no	—
427	#441	—	53	2	no	—
428	#441	C₂×D₅₃	1	2	yes	complemented
429	C₄×Q₁₆	—	53	2	no	—
430	C₄×Q₁₆	—	53	4	no	—
431	C₄×Q₁₆	—	53	2	no	—
432	C₄×Q₁₆	—	53	4	no	—
433	C₄×Q₁₆	—	53	2	no	—
434	C₄×Q₁₆	C₂×D₅₃	1	2	yes	complemented
435	C₂×#107	—	53	2	no	—
436	C₂×#106	—	53	2	no	—
437	C₂×#103	—	53	4	no	—
438	C₂×#103	—	53	4	no	—
439	#405	—	53	1	no	—
440	#405	—	53	1	no	—
441	#405	—	53	1	no	—
442	#405	—	53	1	no	—
443	#405	—	53	1	no	—
444	#405	—	53	1	no	—
445	#405	—	53	1	no	—
446	#405	C₂×D₅₃	1	1	yes	characteristic
447	C₂×#97	—	53	1	no	—
448	C₂×#97	—	53	1	no	—
449	C₂×C₄×C₈	—	53	1	no	—
450	C₂×C₄×C₈	—	53	1	no	—
451	C₅₃	C₂×#3177	1	1	yes	characteristic, complemented, Sylow
452	C₂²×Q₈	—	53	4	no	—
453	C₂³×C₄	—	53	1	no	—
454	C₂×Q₁₆	—	53	4	no	—
455	C₂×Q₁₆	—	53	4	no	—
456	C₂×Q₁₆	—	53	2	no	—
457	C₂×Q₁₆	—	53	2	no	—
458	C₂×Q₁₆	—	106	8	no	—
459	C₂×Q₁₆	—	106	8	no	—
460	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
461	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
462	C₂×M₁₆	—	53	1	no	—
463	C₂×M₁₆	—	53	1	no	—
464	C₂×M₁₆	—	53	1	no	—
465	C₂×M₁₆	—	106	4	no	—
466	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
467	C₂²×C₈	—	53	1	no	—
468	C₂²×C₈	—	53	1	no	—
469	C₂²×C₈	—	53	1	no	—
470	C₂²×C₈	—	53	1	no	—
471	C₂²×C₈	—	53	1	no	—
472	C₂²×C₈	—	53	1	no	—
473	C₂²×C₈	—	53	1	no	—
474	C₂²×C₈	—	106	4	no	—
475	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
476	C₄⋊Q₈	—	53	4	no	—
477	C₄⋊Q₈	—	53	2	no	—
478	C₄⋊Q₈	—	106	4	no	—
479	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
480	#126	—	2	4	no	—
481	#126	—	106	4	no	—
482	#126	—	106	4	no	—
483	#126	—	106	4	no	—
484	#126	—	106	4	no	—
485	#126	—	106	4	no	—
486	#126	—	106	4	no	—
487	#126	—	106	4	no	—
488	#125	—	2	4	no	—
489	#125	—	53	4	no	—
490	#125	—	53	2	no	—
491	#125	—	106	4	no	—
492	#125	—	106	4	no	—
493	#125	—	106	4	no	—
494	#125	—	106	4	no	—
495	#125	—	106	4	no	—
496	#125	—	106	4	no	—
497	#125	—	106	4	no	—
498	#125	—	106	4	no	—
499	#125	C₂²×D₅₃	1	2	yes	—
500	C₂²⋊Q₈	—	53	4	no	—
501	C₂²⋊Q₈	—	53	4	no	—
502	C₂²⋊Q₈	—	53	4	no	—
503	C₂²⋊Q₈	—	106	4	no	—
504	C₂²⋊Q₈	—	106	4	no	—
505	C₂²⋊Q₈	C₂²×D₅₃	1	4	yes	—
506	C₄×Q₈	—	53	4	no	—
507	C₄×Q₈	—	53	4	no	—
508	C₄×Q₈	—	53	4	no	—
509	C₄×Q₈	—	106	4	no	—
510	C₄×Q₈	—	106	4	no	—
511	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
512	#117	—	53	1	no	—
513	#117	—	53	2	no	—
514	#117	—	53	2	no	—
515	#117	—	53	1	no	—
516	#117	—	53	1	no	—
517	#117	C₂²×D₅₃	1	1	yes	characteristic
518	C₂×C₄⋊C₄	—	53	2	no	—
519	C₂×C₄⋊C₄	—	53	4	no	—
520	C₂×C₄⋊C₄	—	106	4	no	—
521	C₂×C₄⋊C₄	—	106	4	no	—
522	C₂×C₄⋊C₄	—	106	4	no	—
523	C₂×C₄⋊C₄	—	106	4	no	—
524	C₂×C₂²⋊C₄	—	53	2	no	—
525	C₂×C₂²⋊C₄	—	106	4	no	—
526	C₂×C₄²	—	53	1	no	—
527	C₂×C₄²	—	53	1	no	—
528	C₂×C₄²	—	106	4	no	—
529	#107	—	53	2	no	—
530	#107	—	53	4	no	—
531	#107	C₂²×D₅₃	1	2	yes	—
532	#106	—	53	2	no	—
533	#106	—	53	4	no	—
534	#106	C₂²×D₅₃	1	2	yes	—
535	#103	—	53	4	no	—
536	#103	—	53	4	no	—
537	#103	—	53	4	no	—
538	#103	—	53	4	no	—
539	#103	—	53	4	no	—
540	#103	—	53	4	no	—
541	#103	C₂²×D₅₃	1	4	yes	—
542	#103	C₂²×D₅₃	1	4	yes	—
543	#97	—	53	1	no	—
544	#97	—	53	1	no	—
545	#97	—	53	1	no	—
546	#97	—	53	1	no	—
547	#97	—	53	1	no	—
548	#97	—	53	1	no	—
549	#97	C₂²×D₅₃	1	1	yes	characteristic
550	#97	C₂²×D₅₃	1	1	yes	characteristic
551	C₄×C₈	—	53	1	no	—
552	C₄×C₈	—	53	1	no	—
553	C₄×C₈	—	53	1	no	—
554	C₄×C₈	—	53	1	no	—
555	C₄×C₈	—	53	1	no	—
556	C₄×C₈	—	53	1	no	—
557	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
558	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
559	C₂⁴	—	53	1	no	—
560	C₂×Q₈	—	53	4	no	—
561	C₂×Q₈	—	106	4	no	—
562	C₂×Q₈	—	106	8	no	—
563	C₂×Q₈	—	106	8	no	—
564	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
565	C₂²×C₄	—	53	1	no	—
566	C₂²×C₄	—	53	1	no	—
567	C₂²×C₄	—	53	2	no	—
568	C₂²×C₄	—	53	2	no	—
569	C₂²×C₄	—	53	1	no	—
570	C₂²×C₄	—	53	1	no	—
571	C₂²×C₄	—	53	1	no	—
572	C₂²×C₄	—	106	2	no	—
573	C₂²×C₄	—	106	2	no	—
574	C₂²×C₄	—	106	4	no	—
575	C₂²×C₄	—	106	4	no	—
576	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
577	Q₁₆	—	2	4	no	—
578	Q₁₆	—	2	4	no	—
579	Q₁₆	—	106	8	no	—
580	Q₁₆	—	106	8	no	—
581	Q₁₆	—	106	4	no	—
582	Q₁₆	—	106	4	no	—
583	M₁₆	—	2	2	no	—
584	M₁₆	—	106	2	no	—
585	M₁₆	—	106	2	no	—
586	M₁₆	—	106	2	no	—
587	C₂×C₈	—	2	2	no	—
588	C₂×C₈	—	53	1	no	—
589	C₂×C₈	—	53	1	no	—
590	C₂×C₈	—	53	4	no	—
591	C₂×C₈	—	53	4	no	—
592	C₂×C₈	—	53	1	no	—
593	C₂×C₈	—	53	1	no	—
594	C₂×C₈	—	106	2	no	—
595	C₂×C₈	—	106	2	no	—
596	C₂×C₈	—	106	2	no	—
597	C₂×C₈	—	106	2	no	—
598	C₂×C₈	—	106	2	no	—
599	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
600	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
601	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
602	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	53	2	no	—
608	C₄⋊C₄	—	53	2	no	—
609	C₄⋊C₄	—	53	2	no	—
610	C₄⋊C₄	—	53	4	no	—
611	C₄⋊C₄	—	106	4	no	—
612	C₄⋊C₄	—	106	4	no	—
613	C₄⋊C₄	—	106	4	no	—
614	C₄⋊C₄	—	106	4	no	—
615	C₄⋊C₄	—	106	4	no	—
616	C₄⋊C₄	—	106	4	no	—
617	C₄⋊C₄	—	106	4	no	—
618	C₄⋊C₄	—	106	4	no	—
619	C₄⋊C₄	—	106	4	no	—
620	C₄⋊C₄	—	106	4	no	—
621	C₄⋊C₄	—	106	4	no	—
622	C₄⋊C₄	—	106	4	no	—
623	C₄⋊C₄	—	106	4	no	—
624	C₄⋊C₄	D₄×D₅₃	1	2	yes	—
625	C₄⋊C₄	C₂³×D₅₃	1	4	yes	—
626	C₂²⋊C₄	—	2	4	no	—
627	C₂²⋊C₄	—	53	2	no	—
628	C₂²⋊C₄	—	53	2	no	—
629	C₂²⋊C₄	—	53	2	no	—
630	C₂²⋊C₄	—	106	4	no	—
631	C₂²⋊C₄	—	106	4	no	—
632	C₂²⋊C₄	—	106	4	no	—
633	C₂²⋊C₄	D₄×D₅₃	1	2	yes	—
634	C₄²	—	2	4	no	—
635	C₄²	—	53	2	no	—
636	C₄²	—	53	2	no	—
637	C₄²	—	53	1	no	—
638	C₄²	—	53	1	no	—
639	C₄²	—	106	4	no	—
640	C₄²	—	106	4	no	—
641	C₄²	—	106	4	no	—
642	C₄²	C₂³×D₅₃	1	1	yes	characteristic
643	C₄²	C₂³×D₅₃	1	1	yes	characteristic
644	C₂³	—	53	1	no	—
645	C₂³	—	53	1	no	—
646	C₂³	—	106	2	no	—
647	C₂³	—	106	4	no	—
648	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
649	Q₈	—	2	4	no	—
650	Q₈	—	2	4	no	—
651	Q₈	—	106	4	no	—
652	Q₈	—	106	4	no	—
653	Q₈	—	212	8	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	53	1	no	—
659	C₂×C₄	—	53	2	no	—
660	C₂×C₄	—	53	2	no	—
661	C₂×C₄	—	53	2	no	—
662	C₂×C₄	—	53	2	no	—
663	C₂×C₄	—	53	1	no	—
664	C₂×C₄	—	106	2	no	—
665	C₂×C₄	—	106	4	no	—
666	C₂×C₄	—	106	4	no	—
667	C₂×C₄	—	106	2	no	—
668	C₂×C₄	—	106	2	no	—
669	C₂×C₄	—	106	1	no	—
670	C₂×C₄	—	106	1	no	—
671	C₂×C₄	—	106	4	no	—
672	C₂×C₄	—	106	4	no	—
673	C₂×C₄	—	212	8	no	—
674	C₂×C₄	—	212	8	no	—
675	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
676	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
677	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
678	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
679	C₈	—	2	1	no	—
680	C₈	—	2	1	no	—
681	C₈	—	53	2	no	—
682	C₈	—	53	2	no	—
683	C₈	—	106	1	no	—
684	C₈	—	106	1	no	—
685	C₈	#41×D₅₃	1	2	yes	—
686	C₈	#41×D₅₃	1	2	yes	—
687	C₂²	—	2	2	no	—
688	C₂²	—	2	1	no	—
689	C₂²	—	53	4	no	—
690	C₂²	—	53	2	no	—
691	C₂²	—	106	2	no	—
692	C₂²	—	106	1	no	—
693	C₂²	—	212	4	no	—
694	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
695	C₄	—	2	2	no	—
696	C₄	—	2	2	no	—
697	C₄	—	2	1	no	—
698	C₄	—	4	4	no	—
699	C₄	—	4	4	no	—
700	C₄	—	53	2	no	—
701	C₄	—	106	2	no	—
702	C₄	—	106	2	no	—
703	C₄	—	106	1	no	—
704	C₄	—	212	4	no	—
705	C₄	—	212	4	no	—
706	C₄	C₂×#41×D₅₃	1	1	yes	characteristic
707	C₄	C₂×#41×D₅₃	1	1	yes	characteristic
708	C₂	—	4	1	no	—
709	C₂	—	53	4	no	—
710	C₂	—	212	1	no	—
711	C₂	#532×D₅₃	1	1	yes	characteristic
712	C₂	#578×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₅₃×#3177	C₂
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂³×D₅₃	C₂×D₄
hypercenter	#3177	D₅₃
hypocenter	C₅₃	C₂×#3177
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #20285375	C₂	semidirect
3. #20285374	C₂	semidirect
4. #20285373	C₂	semidirect
5. #20285372	C₂	semidirect
6. #20285371	C₂	semidirect
7. #20285370	C₂	semidirect
8. #20285369	C₂	semidirect
9. #20285368	C₂	semidirect
10. #20285367	C₂	semidirect
11. #499×D₅₃	C₂	semidirect
12. #494×D₅₃	C₂	semidirect
13. #491×D₅₃	C₂	semidirect
16. #441×D₅₃	C₂	semidirect
17. C₄×Q₁₆×D₅₃	C₂	semidirect
19. C₅₃×#3177	C₂	semidirect
38. #16811025	C₂²	semidirect
39. #16811024	C₂²	semidirect
40. #16811023	C₂²	semidirect
41. #16811022	C₂²	semidirect
42. #16811003	C₂²	semidirect
43. #16811002	C₂²	semidirect
44. #16811001	C₂²	semidirect
45. #16811001	C₂²	semidirect
46. #16810993	C₂²	semidirect
47. #16810992	C₂²	semidirect
48. #16810991	C₂²	semidirect
49. #16810990	C₂²	semidirect
50. #16810990	C₂²	semidirect
61. #16810731	C₂²	semidirect
62. #16810730	C₂²	semidirect
63. #16810729	C₂²	semidirect
64. #16810728	C₂²	semidirect
65. #16810727	C₂²	semidirect
66. C₄×Dic₂₁₂	C₂²	semidirect
67. Q₁₆×Dic₅₃	C₂²	semidirect
68. #16810717	C₂²	semidirect
69. C₄×#11719925	C₂²	semidirect
70. #16810715	C₂²	semidirect
87. C₅₃×#499	C₂²	semidirect
88. C₅₃×#494	C₂²	semidirect
89. C₅₃×#491	C₂²	semidirect
92. C₅₃×#441	C₂²	semidirect
93. C₄×Q₁₆×C₅₃	C₂²	semidirect
362. #3177	D₅₃	direct (N × H)
375. D₅₃	#3177	direct (N × H)
399. #499	C₂×D₅₃	semidirect
404. #494	C₂×D₅₃	semidirect
410. #491	C₂×D₅₃	semidirect
428. #441	C₂×D₅₃	semidirect
434. C₄×Q₁₆	C₂×D₅₃	semidirect
451. C₅₃	C₂×#3177	semidirect

Maximal subgroups

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

subgroup	index
#541×D₅₃	2
#20285375	2
#20285374	2
#20285373	2
#20285372	2
#20285371	2
#20285370	2
#20285369	2
#20285368	2
#20285367	2
#499×D₅₃	2
#494×D₅₃	2
#491×D₅₃	2
#470×D₅₃	2
#467×D₅₃	2
#441×D₅₃	2
C₄×Q₁₆×D₅₃	2
#405×D₅₃	2
C₅₃×#3177	2
C₂×#3177	53

Subgroups — D53×#3177 #28253928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#3177 #28253928