Subgroups — D₅₃×#3160 #28253691

Order 13568.

1942 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×D₅₃	C₂	1	2	yes	maximal
2	#20285229	C₂	1	1	yes	maximal, characteristic, complemented
3	#20285228	C₂	1	4	yes	maximal, complemented
4	#20285227	C₂	1	4	yes	maximal, complemented
5	#20285226	C₂	1	2	yes	maximal, complemented
6	#20285225	C₂	1	2	yes	maximal, complemented
7	#20285224	C₂	1	2	yes	maximal, complemented
8	#500×D₅₃	C₂	1	2	yes	maximal, complemented
9	#499×D₅₃	C₂	1	4	yes	maximal, complemented
10	#498×D₅₃	C₂	1	2	yes	maximal, complemented
11	#427×D₅₃	C₂	1	4	yes	maximal
12	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
15	#16811180	C₂²	1	2	yes	—
16	#16811179	C₂²	1	4	yes	—
17	#16811178	C₂²	1	4	yes	—
18	#16811177	—	2	4	no	—
19	#16811176	C₂²	1	4	yes	—
20	#16811175	—	2	4	no	—
21	C₄⋊Q₈×D₅₃	C₂²	1	4	yes	—
22	#125×D₅₃	C₂²	1	4	yes	—
23	C₂²⋊Q₈×D₅₃	—	2	4	no	—
24	C₄×Q₈×D₅₃	—	2	4	no	—
25	#117×D₅₃	C₂²	1	4	yes	—
26	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
27	#16811028	C₂²	1	2	yes	complemented
28	#16811027	C₂²	1	8	yes	complemented
29	#16811026	C₂²	1	4	yes	complemented
30	#16811025	C₂²	1	4	yes	complemented
31	#16811024	C₂²	1	8	yes	complemented
32	#16811023	C₂²	1	8	yes	complemented
33	#16811022	C₂²	1	8	yes	complemented
34	#16811021	C₂²	1	2	yes	complemented
35	#16811020	C₂²	1	8	yes	complemented
36	#16811019	C₂²	1	4	yes	complemented
37	#16810661	C₂²	1	4	yes	—
38	#16810660	C₂²	1	8	yes	—
39	#16810659	C₂²	1	8	yes	—
40	#16810658	C₂²	1	8	yes	—
41	#107×D₅₃	C₂²	1	8	yes	—
42	#106×D₅₃	C₂²	1	8	yes	—
43	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
44	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
45	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
46	C₄×C₈×D₅₃	C₂²	1	2	yes	—
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
49	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
50	C₄×C₈×D₅₃	C₂²	1	2	yes	—
51	C₅₃×#533	C₂²	1	2	yes	—
52	C₅₃×#500	C₂²	1	2	yes	complemented
53	C₅₃×#499	C₂²	1	4	yes	complemented
54	C₅₃×#498	C₂²	1	2	yes	complemented
55	C₅₃×#427	C₂²	1	4	yes	—
56	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×Q₈×D₅₃	—	2	4	no	—
58	C₂²×C₄×D₅₃	D₄	1	2	yes	—
59	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
60	C₂²×#88322	C₂³	1	2	yes	—
61	C₂×C₈×D₅₃	Q₈	1	16	yes	complemented
62	C₂×C₈×D₅₃	C₂³	1	2	yes	—
63	C₂×C₈×D₅₃	C₂³	1	2	yes	—
64	C₂×C₈×D₅₃	C₂³	1	2	yes	—
65	C₂×C₈×D₅₃	C₂³	1	2	yes	—
66	#13133217	C₂³	1	4	yes	—
67	#13133216	—	2	4	no	—
68	#13133216	C₂³	1	8	yes	—
69	#13133215	—	2	4	no	—
70	#13133209	C₂³	1	4	yes	—
71	#13133208	C₂³	1	8	yes	—
72	#13133207	—	2	8	no	—
73	#13133197	—	2	4	no	—
74	#13133196	—	2	4	no	—
75	#13133195	—	2	8	no	—
76	#13133194	—	2	8	no	—
77	#13133193	—	2	4	no	—
78	Q₈×Dic₅₃	—	2	4	no	—
79	C₄×Dic₁₀₆	—	2	4	no	—
80	C₄×Dic₁₀₆	—	2	8	no	—
81	#13133182	—	2	4	no	—
82	#13133182	—	2	8	no	—
83	#13133176	C₂³	1	4	yes	—
84	#13133175	C₂³	1	8	yes	—
85	#13133174	—	2	4	no	—
86	#13133173	—	2	4	no	—
87	C₄⋊C₄×D₅₃	—	2	8	no	—
88	C₄⋊C₄×D₅₃	—	2	4	no	—
89	C₄⋊C₄×D₅₃	C₂³	1	8	yes	—
90	C₂²⋊C₄×D₅₃	—	2	4	no	—
91	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
92	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
93	C₄²×D₅₃	—	2	4	no	—
94	C₄²×D₅₃	D₄	1	2	yes	—
95	C₄²×D₅₃	D₄	1	2	yes	—
96	C₄²×D₅₃	C₂³	1	1	yes	characteristic
97	C₄²×D₅₃	C₂³	1	1	yes	characteristic
98	C₄²×D₅₃	C₂³	1	1	yes	characteristic
99	C₄²×D₅₃	C₂³	1	1	yes	characteristic
100	#13133151	C₂³	1	8	yes	—
101	#13133150	C₂³	1	16	yes	—
102	#13133149	C₂³	1	8	yes	—
103	#13133148	C₂³	1	16	yes	—
104	C₄×#88322	C₂³	1	1	yes	characteristic
105	C₄×#88322	C₂³	1	2	yes	—
106	C₄×#88322	C₂³	1	1	yes	characteristic
107	C₈×Dic₅₃	C₂³	1	2	yes	—
108	C₈×Dic₅₃	C₂³	1	2	yes	—
109	C₈×Dic₅₃	C₂³	1	2	yes	—
110	C₈×Dic₅₃	C₂³	1	2	yes	—
111	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
112	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
113	#125×C₅₃	C₂³	1	4	yes	—
114	C₂²⋊Q₈×C₅₃	—	2	4	no	—
115	C₄×Q₈×C₅₃	—	2	4	no	—
116	#117×C₅₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	#107×C₅₃	C₂³	1	8	yes	—
119	#106×C₅₃	C₂³	1	8	yes	—
120	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
121	C₄×C₈×C₅₃	C₂³	1	2	yes	—
122	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
123	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
128	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
129	Q₈×D₅₃	—	4	16	no	—
130	C₂×C₄×D₅₃	—	2	8	no	—
131	C₂×C₄×D₅₃	—	2	8	no	—
132	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
133	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
134	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
135	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
138	C₂×C₄×D₅₃	C₂×Q₈	1	8	yes	—
139	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
140	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
141	C₂×Dic₁₀₆	—	2	4	no	—
142	C₂×Dic₁₀₆	—	2	8	no	—
143	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	complemented
144	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
145	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	2	4	no	—
148	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
149	C₂²⋊C₄×C₅₃	—	2	4	no	—
150	C₄²×C₅₃	—	2	4	no	—
151	C₄²×C₅₃	C₂×D₄	1	2	yes	—
152	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂²⋊Dic₅₃	—	2	4	no	—
155	#11719921	—	2	8	no	—
156	C₄⋊Dic₅₃	—	2	4	no	—
157	C₄⋊Dic₅₃	—	2	8	no	—
158	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
159	#11719919	—	2	8	no	—
160	#11719919	—	2	8	no	—
161	#11719919	—	2	8	no	—
162	#11719919	—	2	8	no	—
163	C₄×Dic₅₃	—	2	4	no	—
164	C₄×Dic₅₃	—	2	8	no	—
165	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
166	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
167	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
168	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
169	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
170	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
171	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
172	C₂×#88322	C₂×Q₈	1	8	yes	complemented
173	C₂×#88322	C₂⁴	1	2	yes	—
174	C₂×#88322	C₂⁴	1	2	yes	—
175	C₈×D₅₃	C₂×Q₈	1	8	yes	—
176	C₈×D₅₃	C₂×Q₈	1	8	yes	—
177	C₈×D₅₃	C₂×Q₈	1	8	yes	complemented
178	C₈×D₅₃	C₂×Q₈	1	8	yes	complemented
179	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂²×D₅₃	—	2	4	no	—
181	C₂²×D₅₃	C₄⋊Q₈	1	4	yes	—
182	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₅₃	—	4	8	no	—
185	C₂×C₄×C₅₃	—	2	4	no	—
186	C₂×C₄×C₅₃	—	2	8	no	—
187	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
191	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
192	C₂×Dic₅₃	—	2	4	no	—
193	C₂×Dic₅₃	—	2	8	no	—
194	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
195	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
196	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
199	C₄×D₅₃	—	2	4	no	—
200	C₄×D₅₃	—	2	4	no	—
201	C₄×D₅₃	—	2	4	no	—
202	C₄×D₅₃	—	4	16	no	—
203	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
204	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
205	C₄×D₅₃	#135	1	8	yes	—
206	C₄×D₅₃	C₂²×Q₈	1	4	yes	—
207	C₄×D₅₃	C₂²×Q₈	1	4	yes	—
208	Dic₁₀₆	—	4	8	no	—
209	Dic₁₀₆	—	4	16	no	—
210	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
211	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
212	#88322	C₂²×Q₈	1	4	yes	—
213	#88322	C₂²×Q₈	1	4	yes	—
214	C₂×#3160	—	53	1	no	maximal, Sylow
215	C₂²×C₅₃	—	2	2	no	—
216	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
217	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
218	C₂×D₅₃	—	2	2	no	—
219	C₂×D₅₃	—	2	8	no	—
220	C₂×D₅₃	C₂×C₄⋊Q₈	1	2	yes	—
221	C₂×D₅₃	C₂×C₄⋊Q₈	1	2	yes	—
222	C₂×D₅₃	C₂×#135	1	4	yes	—
223	C₄×C₅₃	—	2	2	no	—
224	C₄×C₅₃	—	4	8	no	—
225	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
226	C₄×C₅₃	C₂×#135	1	4	yes	—
227	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
228	Dic₅₃	—	2	2	no	—
229	Dic₅₃	—	4	8	no	—
230	Dic₅₃	C₂×C₄⋊Q₈	1	2	yes	—
231	Dic₅₃	C₂×C₄⋊Q₈	1	2	yes	—
232	Dic₅₃	C₂×#135	1	4	yes	—
233	C₂×#533	—	53	2	no	—
234	#3160	—	53	2	no	—
235	#3160	—	53	4	no	—
236	#3160	—	53	2	no	—
237	#3160	—	53	4	no	—
238	#3160	—	53	1	no	—
239	#3160	—	53	2	no	—
240	#3160	D₅₃	1	1	yes	characteristic, direct product factor
241	C₂×#500	—	53	2	no	—
242	C₂×#499	—	53	4	no	—
243	C₂×#498	—	53	2	no	—
244	C₂×#427	—	53	4	no	—
245	C₂²×C₄×C₈	—	53	1	no	—
246	C₂×C₅₃	—	2	2	no	—
247	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₅₃	C₂²×#135	1	2	yes	—
249	D₅₃	—	2	2	no	—
250	D₅₃	#3160	1	4	yes	direct product factor
251	C₂³×C₈	—	53	2	no	—
252	#533	—	53	4	no	—
253	#533	—	53	4	no	—
254	#533	—	53	4	no	—
255	#533	—	53	2	no	—
256	#533	—	106	4	no	—
257	#533	—	106	4	no	—
258	#533	C₂×D₅₃	1	2	yes	—
259	C₂×C₄⋊Q₈	—	53	4	no	—
260	C₂×#125	—	53	4	no	—
261	C₂×C₂²⋊Q₈	—	106	4	no	—
262	C₂×C₄×Q₈	—	106	4	no	—
263	C₂×#117	—	53	4	no	—
264	C₂²×C₄²	—	53	1	no	—
265	#500	—	53	4	no	—
266	#500	—	53	8	no	—
267	#500	—	53	2	no	—
268	#500	C₂×D₅₃	1	2	yes	complemented
269	#499	—	53	8	no	—
270	#499	—	53	8	no	—
271	#499	—	53	8	no	—
272	#499	—	53	4	no	—
273	#499	C₂×D₅₃	1	4	yes	complemented
274	#498	—	53	4	no	—
275	#498	—	53	8	no	—
276	#498	—	53	2	no	—
277	#498	C₂×D₅₃	1	2	yes	complemented
278	#427	—	53	4	no	—
279	#427	—	53	8	no	—
280	#427	—	53	8	no	—
281	#427	—	53	8	no	—
282	#427	C₂×D₅₃	1	4	yes	—
283	C₂×#107	—	53	8	no	—
284	C₂×#106	—	53	8	no	—
285	C₂×C₄×C₈	—	53	2	no	—
286	C₂×C₄×C₈	—	53	1	no	—
287	C₂×C₄×C₈	—	53	1	no	—
288	C₂×C₄×C₈	—	53	2	no	—
289	C₂×C₄×C₈	—	53	1	no	—
290	C₂×C₄×C₈	—	53	2	no	—
291	C₂×C₄×C₈	—	53	1	no	—
292	C₂×C₄×C₈	—	53	1	no	—
293	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
294	C₅₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
295	C₂²×Q₈	—	106	4	no	—
296	C₂³×C₄	—	53	1	no	—
297	C₂³×C₄	—	53	2	no	—
298	C₂²×C₈	—	53	2	no	—
299	C₂²×C₈	—	53	16	no	—
300	C₂²×C₈	—	53	2	no	—
301	C₂²×C₈	—	53	2	no	—
302	C₂²×C₈	—	53	2	no	—
303	C₂²×C₈	—	53	2	no	—
304	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
305	C₄⋊Q₈	—	53	8	no	—
306	C₄⋊Q₈	—	53	4	no	—
307	C₄⋊Q₈	—	106	4	no	—
308	C₄⋊Q₈	—	106	4	no	—
309	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
310	#125	—	53	8	no	—
311	#125	—	53	4	no	—
312	#125	—	106	8	no	—
313	#125	C₂²×D₅₃	1	4	yes	—
314	C₂²⋊Q₈	—	2	4	no	—
315	C₂²⋊Q₈	—	106	8	no	—
316	C₂²⋊Q₈	—	106	8	no	—
317	C₂²⋊Q₈	—	106	4	no	—
318	C₂²⋊Q₈	—	106	4	no	—
319	C₂²⋊Q₈	—	106	4	no	—
320	C₄×Q₈	—	2	4	no	—
321	C₄×Q₈	—	106	8	no	—
322	C₄×Q₈	—	106	8	no	—
323	C₄×Q₈	—	106	4	no	—
324	C₄×Q₈	—	106	4	no	—
325	C₄×Q₈	—	106	4	no	—
326	#117	—	53	4	no	—
327	#117	—	53	8	no	—
328	#117	—	106	4	no	—
329	#117	—	106	4	no	—
330	#117	C₂²×D₅₃	1	4	yes	—
331	C₂×C₄⋊C₄	—	53	8	no	—
332	C₂×C₄⋊C₄	—	106	8	no	—
333	C₂×C₄⋊C₄	—	106	4	no	—
334	C₂×C₂²⋊C₄	—	106	4	no	—
335	C₂×C₄²	—	53	2	no	—
336	C₂×C₄²	—	53	1	no	—
337	C₂×C₄²	—	53	2	no	—
338	C₂×C₄²	—	53	2	no	—
339	C₂×C₄²	—	53	1	no	—
340	C₂×C₄²	—	53	1	no	—
341	C₂×C₄²	—	53	1	no	—
342	C₂×C₄²	—	53	1	no	—
343	C₂×C₄²	—	106	4	no	—
344	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
345	#107	—	53	8	no	—
346	#107	—	53	16	no	—
347	#107	C₂²×D₅₃	1	8	yes	—
348	#106	—	53	8	no	—
349	#106	—	53	16	no	—
350	#106	C₂²×D₅₃	1	8	yes	—
351	C₄×C₈	—	53	2	no	—
352	C₄×C₈	—	53	2	no	—
353	C₄×C₈	—	53	2	no	—
354	C₄×C₈	—	53	2	no	—
355	C₄×C₈	—	53	1	no	—
356	C₄×C₈	—	53	2	no	—
357	C₄×C₈	—	53	1	no	—
358	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
359	C₄×C₈	C₂²×D₅₃	1	2	yes	—
360	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
361	C₂⁴	—	53	1	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	106	8	no	—
364	C₂×Q₈	—	106	4	no	—
365	C₂×Q₈	—	212	16	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	2	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	8	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	2	no	—
374	C₂²×C₄	—	53	2	no	—
375	C₂²×C₄	—	53	2	no	—
376	C₂²×C₄	—	53	2	no	—
377	C₂²×C₄	—	106	8	no	—
378	C₂²×C₄	—	106	8	no	—
379	C₂²×C₄	D₄×D₅₃	1	2	yes	—
380	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
381	C₂×C₈	—	53	8	no	—
382	C₂×C₈	—	53	2	no	—
383	C₂×C₈	—	53	2	no	—
384	C₂×C₈	—	53	8	no	—
385	C₂×C₈	—	53	8	no	—
386	C₂×C₈	—	53	8	no	—
387	C₂×C₈	—	53	8	no	—
388	C₂×C₈	Q₈×D₅₃	1	8	yes	complemented
389	C₂×C₈	C₂³×D₅₃	1	2	yes	—
390	C₂×C₈	C₂³×D₅₃	1	2	yes	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	53	8	no	—
394	C₄⋊C₄	—	106	8	no	—
395	C₄⋊C₄	—	106	8	no	—
396	C₄⋊C₄	—	106	8	no	—
397	C₄⋊C₄	—	106	8	no	—
398	C₄⋊C₄	—	106	8	no	—
399	C₄⋊C₄	—	106	4	no	—
400	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
401	C₂²⋊C₄	—	2	4	no	—
402	C₂²⋊C₄	—	106	8	no	—
403	C₂²⋊C₄	—	106	4	no	—
404	C₄²	—	2	4	no	—
405	C₄²	—	53	2	no	—
406	C₄²	—	53	2	no	—
407	C₄²	—	53	2	no	—
408	C₄²	—	53	2	no	—
409	C₄²	—	53	2	no	—
410	C₄²	—	53	1	no	—
411	C₄²	—	53	1	no	—
412	C₄²	—	106	8	no	—
413	C₄²	—	106	4	no	—
414	C₄²	D₄×D₅₃	1	2	yes	—
415	C₄²	C₂³×D₅₃	1	1	yes	characteristic
416	C₄²	C₂³×D₅₃	1	1	yes	characteristic
417	C₂³	—	53	1	no	—
418	C₂³	—	53	4	no	—
419	C₂³	—	53	1	no	—
420	C₂³	—	106	4	no	—
421	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
422	Q₈	—	4	8	no	—
423	Q₈	—	212	16	no	—
424	Q₈	—	212	8	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	53	4	no	—
428	C₂×C₄	—	53	1	no	—
429	C₂×C₄	—	53	1	no	—
430	C₂×C₄	—	53	2	no	—
431	C₂×C₄	—	53	8	no	—
432	C₂×C₄	—	53	4	no	—
433	C₂×C₄	—	53	4	no	—
434	C₂×C₄	—	53	4	no	—
435	C₂×C₄	—	53	4	no	—
436	C₂×C₄	—	53	2	no	—
437	C₂×C₄	—	106	4	no	—
438	C₂×C₄	—	106	4	no	—
439	C₂×C₄	—	106	4	no	—
440	C₂×C₄	—	106	4	no	—
441	C₂×C₄	—	106	8	no	—
442	C₂×C₄	—	212	16	no	—
443	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
444	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
445	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
446	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
447	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
448	C₈	—	53	4	no	—
449	C₈	—	53	4	no	—
450	C₈	C₂×Q₈×D₅₃	1	4	yes	—
451	C₈	C₂×Q₈×D₅₃	1	4	yes	—
452	C₂²	—	2	2	no	—
453	C₂²	—	53	4	no	—
454	C₂²	—	53	2	no	—
455	C₂²	—	53	2	no	—
456	C₂²	—	106	8	no	—
457	C₂²	—	106	2	no	—
458	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
459	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
460	C₄	—	2	2	no	—
461	C₄	—	4	8	no	—
462	C₄	—	53	4	no	—
463	C₄	—	53	2	no	—
464	C₄	—	53	2	no	—
465	C₄	—	106	2	no	—
466	C₄	—	212	8	no	—
467	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
468	C₄	#135×D₅₃	1	4	yes	—
469	C₄	C₂²×Q₈×D₅₃	1	2	yes	—
470	C₂	—	2	2	no	—
471	C₂	—	53	4	no	—
472	C₂	—	106	2	no	—
473	C₂	C₂×C₄⋊Q₈×D₅₃	1	1	yes	characteristic
474	C₂	C₂×#135×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂³×D₅₃	C₂×D₄
hypercenter	#3160	D₅₃
hypocenter	C₅₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #20285229	C₂	semidirect
3. #20285228	C₂	semidirect
4. #20285227	C₂	semidirect
5. #20285226	C₂	semidirect
6. #20285225	C₂	semidirect
7. #20285224	C₂	semidirect
8. #500×D₅₃	C₂	semidirect
9. #499×D₅₃	C₂	semidirect
10. #498×D₅₃	C₂	semidirect
13. C₅₃×#3160	C₂	semidirect
27. #16811028	C₂²	semidirect
28. #16811027	C₂²	semidirect
29. #16811026	C₂²	semidirect
30. #16811025	C₂²	semidirect
31. #16811024	C₂²	semidirect
32. #16811023	C₂²	semidirect
33. #16811022	C₂²	semidirect
34. #16811021	C₂²	semidirect
35. #16811020	C₂²	semidirect
36. #16811019	C₂²	semidirect
52. C₅₃×#500	C₂²	semidirect
53. C₅₃×#499	C₂²	semidirect
54. C₅₃×#498	C₂²	semidirect
61. C₂×C₈×D₅₃	Q₈	semidirect
143. C₂×C₈×C₅₃	C₂×Q₈	semidirect
172. C₂×#88322	C₂×Q₈	semidirect
177. C₈×D₅₃	C₂×Q₈	semidirect
178. C₈×D₅₃	C₂×Q₈	semidirect
240. #3160	D₅₃	direct (N × H)
250. D₅₃	#3160	direct (N × H)
268. #500	C₂×D₅₃	semidirect
273. #499	C₂×D₅₃	semidirect
277. #498	C₂×D₅₃	semidirect
294. C₅₃	C₂×#3160	semidirect
388. C₂×C₈	Q₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#533×D₅₃	2
#20285229	2
#20285228	2
#20285227	2
#20285226	2
#20285225	2
#20285224	2
#500×D₅₃	2
#499×D₅₃	2
#498×D₅₃	2
#427×D₅₃	2
C₂×C₄×C₈×D₅₃	2
C₅₃×#3160	2
C₂×#3160	53

Subgroups — D53×#3160 #28253691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#3160 #28253691