Subgroups — D₅₃×#1953 #27387656

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430×D₅₃	C₂	1	2	yes	maximal
2	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#20275506	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275505	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275504	C₂	1	2	yes	maximal, complemented
8	#20275503	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275502	C₂	1	2	yes	maximal, complemented
10	#344×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	#117×D₅₃	C₂²	1	1	yes	characteristic
17	#16810672	C₂²	1	2	yes	—
18	#16810671	C₂²	1	2	yes	—
19	#16810671	C₂²	1	2	yes	—
20	#16810670	—	2	4	no	—
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	C₂×#13133153	C₂²	1	1	yes	characteristic
23	C₂×#13133152	C₂²	1	2	yes	—
24	C₂×#13133152	C₂²	1	2	yes	—
25	#108×D₅₃	—	2	4	no	—
26	#108×D₅₃	—	2	1	no	—
27	#108×D₅₃	—	2	1	no	—
28	#108×D₅₃	—	2	1	no	—
29	#108×D₅₃	—	2	1	no	—
30	#108×D₅₃	C₄	1	4	yes	complemented
31	#108×D₅₃	C₄	1	4	yes	complemented
32	#16810581	C₂²	1	1	yes	characteristic
33	#16810580	—	2	1	no	—
34	#16810579	C₂²	1	1	yes	characteristic
35	#16810578	C₂²	1	1	yes	characteristic
36	#16810577	—	2	1	no	—
37	#97×D₅₃	—	2	1	no	—
38	C₄×C₈×D₅₃	—	2	1	no	—
39	#16810395	C₂²	1	2	yes	—
40	#16810394	C₂²	1	2	yes	—
41	#16810393	C₂²	1	2	yes	—
42	C₅₃×#430	C₂²	1	2	yes	—
43	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#405	C₂²	1	1	yes	characteristic
46	C₅₃×#344	C₂²	1	2	yes	—
47	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×#11719917	C₂³	1	2	yes	—
49	C₂×#11719917	C₂³	1	2	yes	—
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719912	—	2	1	no	—
52	C₂×#11719912	—	2	2	no	—
53	C₂×#11719912	—	2	2	no	—
54	M₁₆×D₅₃	—	2	2	no	—
55	M₁₆×D₅₃	—	2	2	no	—
56	M₁₆×D₅₃	—	2	2	no	—
57	M₁₆×D₅₃	—	2	2	no	—
58	M₁₆×D₅₃	—	2	2	no	—
59	M₁₆×D₅₃	—	2	2	no	—
60	M₁₆×D₅₃	—	2	1	no	—
61	M₁₆×D₅₃	—	2	1	no	—
62	M₁₆×D₅₃	C₂×C₄	1	4	yes	—
63	M₁₆×D₅₃	C₂×C₄	1	4	yes	—
64	M₁₆×D₅₃	D₄	1	2	yes	—
65	M₁₆×D₅₃	D₄	1	2	yes	—
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂×C₈×D₅₃	—	2	1	no	—
68	C₂×C₈×D₅₃	—	2	2	no	—
69	C₂×C₈×D₅₃	—	2	2	no	—
70	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
74	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
75	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
76	C₂×C₈×D₅₃	Q₈	1	2	yes	—
77	C₂×C₈×D₅₃	Q₈	1	2	yes	—
78	#13133176	C₂³	1	1	yes	characteristic
79	#13133175	—	2	1	no	—
80	#13133174	—	2	1	no	—
81	#13133173	Q₈	1	2	yes	—
82	C₄⋊C₄×D₅₃	—	2	1	no	—
83	C₂²⋊C₄×D₅₃	Q₈	1	2	yes	—
84	C₄²×D₅₃	—	2	1	no	—
85	#13133153	—	2	1	no	—
86	#13133153	—	2	1	no	—
87	#13133153	—	2	4	no	—
88	#13133153	C₂×C₄	1	4	yes	complemented
89	#13133152	—	2	4	no	—
90	#13133152	—	2	4	no	—
91	#13133152	—	2	2	no	—
92	#13133152	—	2	2	no	—
93	#13133152	C₂×C₄	1	8	yes	complemented
94	#13133128	—	2	1	no	—
95	#13133127	—	2	1	no	—
96	#13133127	—	2	1	no	—
97	C₄×#88322	—	2	1	no	—
98	C₈×Dic₅₃	—	2	1	no	—
99	C₈×Dic₅₃	—	2	1	no	—
100	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
101	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
102	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	#117×C₅₃	C₂³	1	1	yes	characteristic
105	#108×C₅₃	—	2	4	no	—
106	#108×C₅₃	—	2	1	no	—
107	#108×C₅₃	—	2	1	no	—
108	#108×C₅₃	C₂×C₄	1	4	yes	complemented
109	#97×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	—	2	1	no	—
111	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₄×D₅₃	—	2	2	no	—
115	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
123	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
124	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
125	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
126	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	1	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
132	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	C₂×Q₈	1	2	yes	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₂²⋊Dic₅₃	C₂×Q₈	1	2	yes	—
144	#11719921	—	2	2	no	—
145	C₄⋊Dic₅₃	—	2	1	no	—
146	#11719919	—	2	2	no	—
147	C₄×Dic₅₃	—	2	1	no	—
148	C₄×Dic₅₃	—	2	2	no	—
149	#11719917	—	2	2	no	—
150	#11719917	—	2	1	no	—
151	#11719917	—	2	2	no	—
152	#11719917	—	2	2	no	—
153	#11719917	C₂²×C₄	1	4	yes	—
154	#11719917	C₂×D₄	1	2	yes	—
155	C₂×#88322	—	2	1	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	C₂²×C₄	1	2	yes	—
159	C₂×#88322	C₂×D₄	1	1	yes	characteristic
160	C₂×#88322	C₂×D₄	1	1	yes	characteristic
161	C₂×#88322	C₂×Q₈	1	2	yes	—
162	#11719912	—	2	4	no	—
163	#11719912	—	2	4	no	—
164	#11719912	—	2	2	no	—
165	#11719912	—	2	2	no	—
166	#11719912	—	4	2	no	—
167	#11719912	—	4	2	no	—
168	C₈×D₅₃	—	2	4	no	—
169	C₈×D₅₃	—	2	4	no	—
170	C₈×D₅₃	—	2	2	no	—
171	C₈×D₅₃	—	2	2	no	—
172	C₈×D₅₃	—	4	2	no	—
173	C₈×D₅₃	—	4	2	no	—
174	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
175	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
176	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
177	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
178	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
179	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
180	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
181	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
182	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
183	C₂²×D₅₃	—	4	2	no	—
184	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
185	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
186	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
188	C₂²×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
189	C₂²×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×Dic₅₃	—	2	2	no	—
198	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
199	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
203	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
204	C₄×D₅₃	—	4	4	no	—
205	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
210	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
211	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
212	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
213	C₄×D₅₃	#121	1	1	yes	characteristic
214	C₄×D₅₃	#121	1	1	yes	characteristic
215	C₄×D₅₃	#121	1	1	yes	characteristic
216	C₄×D₅₃	#121	1	1	yes	characteristic
217	C₄×D₅₃	#125	1	1	yes	characteristic
218	C₄×D₅₃	#125	1	1	yes	characteristic
219	C₄×D₅₃	#125	1	1	yes	characteristic
220	C₄×D₅₃	#125	1	1	yes	characteristic
221	C₈×C₅₃	—	2	4	no	—
222	C₈×C₅₃	—	2	2	no	—
223	C₈×C₅₃	—	4	2	no	—
224	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
225	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
226	#88322	—	2	4	no	—
227	#88322	—	2	2	no	—
228	#88322	—	4	2	no	—
229	#88322	C₂×C₄⋊C₄	1	2	yes	—
230	#88322	C₂×C₄⋊C₄	1	2	yes	—
231	C₂×#1953	—	53	1	no	maximal, Sylow
232	C₂²×C₅₃	—	4	1	no	—
233	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
234	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
236	C₂×D₅₃	—	2	2	no	—
237	C₂×D₅₃	—	2	2	no	—
238	C₂×D₅₃	—	2	2	no	—
239	C₂×D₅₃	—	4	1	no	—
240	C₂×D₅₃	—	4	1	no	—
241	C₂×D₅₃	—	4	1	no	—
242	C₂×D₅₃	#389	1	1	yes	characteristic
243	C₂×D₅₃	#389	1	1	yes	characteristic
244	C₂×D₅₃	#389	1	1	yes	characteristic
245	C₂×D₅₃	#389	1	1	yes	characteristic
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
248	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
250	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
251	Dic₅₃	—	4	2	no	—
252	Dic₅₃	#389	1	1	yes	characteristic
253	Dic₅₃	#389	1	1	yes	characteristic
254	Dic₅₃	#389	1	1	yes	characteristic
255	Dic₅₃	#389	1	1	yes	characteristic
256	C₂×#430	—	53	2	no	—
257	C₂²×#108	—	53	1	no	—
258	C₂²×#108	—	53	1	no	—
259	C₂×#405	—	53	1	no	—
260	#1953	—	53	2	no	—
261	#1953	—	53	1	no	—
262	#1953	—	53	1	no	—
263	#1953	—	53	1	no	—
264	#1953	—	53	2	no	—
265	#1953	D₅₃	1	1	yes	characteristic, direct product factor
266	C₂×#344	—	53	2	no	—
267	C₂×C₅₃	—	2	1	no	—
268	C₂×C₅₃	—	2	1	no	—
269	C₂×C₅₃	—	2	1	no	—
270	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
271	D₅₃	—	2	1	no	—
272	D₅₃	—	2	1	no	—
273	D₅₃	—	2	1	no	—
274	D₅₃	#1953	1	2	yes	direct product factor
275	C₂²×M₁₆	—	53	1	no	—
276	C₂²×M₁₆	—	53	2	no	—
277	C₂²×M₁₆	—	53	2	no	—
278	C₂³×C₈	—	53	1	no	—
279	C₂×#117	—	53	1	no	—
280	#430	—	53	2	no	—
281	#430	—	53	2	no	—
282	#430	—	53	2	no	—
283	#430	—	106	4	no	—
284	#430	C₂×D₅₃	1	2	yes	—
285	C₂×#108	—	53	2	no	—
286	C₂×#108	—	53	1	no	—
287	C₂×#108	—	53	4	no	—
288	C₂×#108	—	53	4	no	—
289	C₂×#108	—	53	2	no	—
290	C₂×#108	—	53	1	no	—
291	C₂×#108	—	106	1	no	—
292	C₂×#108	—	106	1	no	—
293	C₂×#108	—	106	1	no	—
294	C₂×#108	—	106	1	no	—
295	C₂×#108	—	106	4	no	—
296	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
297	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
298	#405	—	53	1	no	—
299	#405	—	53	1	no	—
300	#405	—	53	1	no	—
301	#405	—	106	1	no	—
302	#405	—	106	1	no	—
303	#405	C₂×D₅₃	1	1	yes	characteristic
304	C₂×#97	—	106	1	no	—
305	C₂×C₄×C₈	—	106	1	no	—
306	#344	—	53	2	no	—
307	#344	—	53	2	no	—
308	#344	—	53	2	no	—
309	#344	C₂×D₅₃	1	2	yes	—
310	C₅₃	C₂×#1953	1	1	yes	characteristic, complemented, Sylow
311	C₂³×C₄	—	53	1	no	—
312	C₂×M₁₆	—	53	2	no	—
313	C₂×M₁₆	—	53	1	no	—
314	C₂×M₁₆	—	53	4	no	—
315	C₂×M₁₆	—	53	4	no	—
316	C₂×M₁₆	—	53	2	no	—
317	C₂×M₁₆	—	53	2	no	—
318	C₂×M₁₆	—	53	2	no	—
319	C₂×M₁₆	—	106	1	no	—
320	C₂×M₁₆	—	106	2	no	—
321	C₂×M₁₆	—	106	2	no	—
322	C₂×M₁₆	—	106	2	no	—
323	C₂×M₁₆	—	106	2	no	—
324	C₂×M₁₆	—	106	1	no	—
325	C₂×M₁₆	—	106	1	no	—
326	C₂×M₁₆	—	106	2	no	—
327	C₂×M₁₆	—	106	2	no	—
328	C₂×M₁₆	—	106	2	no	—
329	C₂×M₁₆	—	106	2	no	—
330	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
331	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
332	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
333	C₂²×C₈	—	53	1	no	—
334	C₂²×C₈	—	53	2	no	—
335	C₂²×C₈	—	53	2	no	—
336	C₂²×C₈	—	53	1	no	—
337	C₂²×C₈	—	53	1	no	—
338	C₂²×C₈	—	53	1	no	—
339	C₂²×C₈	—	53	1	no	—
340	C₂²×C₈	—	53	2	no	—
341	C₂²×C₈	—	53	2	no	—
342	C₂²×C₈	—	106	1	no	—
343	C₂²×C₈	—	106	2	no	—
344	C₂²×C₈	—	106	2	no	—
345	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
346	#117	—	53	1	no	—
347	#117	—	53	2	no	—
348	#117	—	106	1	no	—
349	#117	—	106	1	no	—
350	#117	C₂²×D₅₃	1	1	yes	characteristic
351	C₂×C₄⋊C₄	—	106	1	no	—
352	C₂×C₂²⋊C₄	—	53	2	no	—
353	C₂×C₄²	—	106	1	no	—
354	#108	—	2	4	no	—
355	#108	—	2	1	no	—
356	#108	—	2	1	no	—
357	#108	—	53	8	no	—
358	#108	—	53	4	no	—
359	#108	—	106	4	no	—
360	#108	—	106	2	no	—
361	#108	—	106	2	no	—
362	#108	—	106	4	no	—
363	#108	—	106	4	no	—
364	#108	—	106	1	no	—
365	#108	—	106	1	no	—
366	#108	C₄×D₅₃	1	4	yes	complemented
367	#97	—	2	1	no	—
368	#97	—	106	1	no	—
369	#97	—	106	1	no	—
370	#97	—	106	1	no	—
371	C₄×C₈	—	2	1	no	—
372	C₄×C₈	—	106	1	no	—
373	C₄×C₈	—	106	1	no	—
374	C₄×C₈	—	106	1	no	—
375	C₂⁴	—	53	1	no	—
376	C₂²×C₄	—	53	1	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	53	1	no	—
379	C₂²×C₄	—	53	1	no	—
380	C₂²×C₄	—	53	1	no	—
381	C₂²×C₄	—	53	1	no	—
382	C₂²×C₄	—	53	1	no	—
383	C₂²×C₄	—	53	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	53	1	no	—
386	C₂²×C₄	—	53	1	no	—
387	C₂²×C₄	—	53	1	no	—
388	C₂²×C₄	—	53	1	no	—
389	C₂²×C₄	—	106	2	no	—
390	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
391	M₁₆	—	2	2	no	—
392	M₁₆	—	2	1	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	2	no	—
395	M₁₆	—	53	4	no	—
396	M₁₆	—	53	2	no	—
397	M₁₆	—	106	4	no	—
398	M₁₆	—	106	4	no	—
399	M₁₆	—	106	2	no	—
400	M₁₆	—	106	2	no	—
401	M₁₆	—	106	2	no	—
402	M₁₆	—	106	1	no	—
403	M₁₆	—	106	2	no	—
404	M₁₆	—	106	2	no	—
405	M₁₆	—	212	2	no	—
406	M₁₆	—	212	2	no	—
407	M₁₆	C₂×C₄×D₅₃	1	4	yes	—
408	M₁₆	D₄×D₅₃	1	2	yes	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	53	2	no	—
413	C₂×C₈	—	53	1	no	—
414	C₂×C₈	—	53	1	no	—
415	C₂×C₈	—	53	2	no	—
416	C₂×C₈	—	53	2	no	—
417	C₂×C₈	—	53	2	no	—
418	C₂×C₈	—	53	2	no	—
419	C₂×C₈	—	53	2	no	—
420	C₂×C₈	—	53	2	no	—
421	C₂×C₈	—	53	2	no	—
422	C₂×C₈	—	53	2	no	—
423	C₂×C₈	—	53	2	no	—
424	C₂×C₈	—	106	4	no	—
425	C₂×C₈	—	106	4	no	—
426	C₂×C₈	—	106	2	no	—
427	C₂×C₈	—	106	2	no	—
428	C₂×C₈	—	106	1	no	—
429	C₂×C₈	—	106	2	no	—
430	C₂×C₈	—	106	2	no	—
431	C₂×C₈	—	212	2	no	—
432	C₂×C₈	—	212	2	no	—
433	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
434	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
435	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
436	C₂×C₈	Q₈×D₅₃	1	2	yes	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	106	2	no	—
439	C₄⋊C₄	—	106	1	no	—
440	C₂²⋊C₄	—	53	2	no	—
441	C₂²⋊C₄	—	106	2	no	—
442	C₂²⋊C₄	Q₈×D₅₃	1	2	yes	—
443	C₄²	—	2	1	no	—
444	C₄²	—	106	2	no	—
445	C₄²	—	106	1	no	—
446	C₂³	—	53	1	no	—
447	C₂³	—	53	1	no	—
448	C₂³	—	53	1	no	—
449	C₂³	—	53	1	no	—
450	C₂³	—	53	1	no	—
451	C₂³	—	53	1	no	—
452	C₂³	—	212	2	no	—
453	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	53	1	no	—
456	C₂×C₄	—	53	1	no	—
457	C₂×C₄	—	53	1	no	—
458	C₂×C₄	—	53	1	no	—
459	C₂×C₄	—	53	1	no	—
460	C₂×C₄	—	53	1	no	—
461	C₂×C₄	—	53	1	no	—
462	C₂×C₄	—	53	1	no	—
463	C₂×C₄	—	53	1	no	—
464	C₂×C₄	—	53	1	no	—
465	C₂×C₄	—	53	1	no	—
466	C₂×C₄	—	53	1	no	—
467	C₂×C₄	—	53	1	no	—
468	C₂×C₄	—	53	1	no	—
469	C₂×C₄	—	53	1	no	—
470	C₂×C₄	—	53	1	no	—
471	C₂×C₄	—	53	1	no	—
472	C₂×C₄	—	53	1	no	—
473	C₂×C₄	—	53	1	no	—
474	C₂×C₄	—	53	1	no	—
475	C₂×C₄	—	53	1	no	—
476	C₂×C₄	—	53	1	no	—
477	C₂×C₄	—	106	2	no	—
478	C₂×C₄	—	212	4	no	—
479	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
480	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
482	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
483	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
484	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
485	C₈	—	2	4	no	—
486	C₈	—	2	2	no	—
487	C₈	—	4	2	no	—
488	C₈	—	53	2	no	—
489	C₈	—	53	2	no	—
490	C₈	—	106	4	no	—
491	C₈	—	106	2	no	—
492	C₈	—	212	2	no	—
493	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
494	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
495	C₂²	—	4	1	no	—
496	C₂²	—	53	1	no	—
497	C₂²	—	53	1	no	—
498	C₂²	—	53	1	no	—
499	C₂²	—	53	1	no	—
500	C₂²	—	106	2	no	—
501	C₂²	—	106	2	no	—
502	C₂²	—	106	2	no	—
503	C₂²	—	212	1	no	—
504	C₂²	—	212	1	no	—
505	C₂²	—	212	1	no	—
506	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
507	C₂²	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
508	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
509	C₄	—	4	2	no	—
510	C₄	—	53	1	no	—
511	C₄	—	53	1	no	—
512	C₄	—	53	1	no	—
513	C₄	—	53	1	no	—
514	C₄	—	212	2	no	—
515	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
516	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
517	C₄	#121×D₅₃	1	1	yes	characteristic
518	C₄	#125×D₅₃	1	1	yes	characteristic
519	C₂	—	2	1	no	—
520	C₂	—	2	1	no	—
521	C₂	—	2	1	no	—
522	C₂	—	53	2	no	—
523	C₂	—	106	1	no	—
524	C₂	—	106	1	no	—
525	C₂	—	106	1	no	—
526	C₂	#389×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1953	C₂
socle	C₂×C₅₃	C₂×#389
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1953	D₅₃
hypocenter	C₅₃	C₂×#1953
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
5. #20275506	C₂	semidirect
6. #20275505	C₂	semidirect
7. #20275504	C₂	semidirect
8. #20275503	C₂	semidirect
9. #20275502	C₂	semidirect
11. C₅₃×#1953	C₂	semidirect
30. #108×D₅₃	C₄	semidirect
31. #108×D₅₃	C₄	semidirect
88. #13133153	C₂×C₄	semidirect
93. #13133152	C₂×C₄	semidirect
108. #108×C₅₃	C₂×C₄	semidirect
265. #1953	D₅₃	direct (N × H)
274. D₅₃	#1953	direct (N × H)
310. C₅₃	C₂×#1953	semidirect
366. #108	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#430×D₅₃	2
C₂×#108×D₅₃	2
#405×D₅₃	2
#20275506	2
#20275505	2
#20275504	2
#20275503	2
#20275502	2
#344×D₅₃	2
C₅₃×#1953	2
C₂×#1953	53

Subgroups — D53×#1953 #27387656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1953 #27387656