Subgroups — D₅₃×#2213 #27538468

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#509×D₅₃	C₂	1	2	yes	maximal, complemented
2	#493×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276792	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276791	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276790	C₂	1	2	yes	maximal, complemented
7	#20276789	C₂	1	2	yes	maximal, complemented
8	#20276788	C₂	1	1	yes	maximal, characteristic, complemented
9	#364×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#361×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2213	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₈×D₅₃	—	2	1	no	—
14	C₂×D₈×D₅₃	C₂²	1	2	yes	—
15	#131×D₅₃	—	2	1	no	—
16	#127×D₅₃	—	2	1	no	—
17	#16811061	C₂²	1	2	yes	complemented
18	#16811060	—	2	2	no	—
19	#16811059	C₂²	1	2	yes	complemented
20	#16811058	C₂²	1	2	yes	complemented
21	#16811057	—	2	2	no	—
22	SD₃₂×D₅₃	—	2	2	no	—
23	D₁₆×D₅₃	—	2	2	no	—
24	M₃₂×D₅₃	C₂²	1	2	yes	—
25	#16811000	C₂²	1	1	yes	characteristic, complemented
26	#16810999	—	2	1	no	—
27	#16810998	—	2	1	no	—
28	#16810998	C₂²	1	2	yes	complemented
29	#16810741	C₂²	1	1	yes	characteristic, complemented
30	#16810740	C₂²	1	1	yes	characteristic, complemented
31	#16810739	C₂²	1	1	yes	characteristic, complemented
32	#16810738	—	2	1	no	—
33	#16810737	—	2	1	no	—
34	#108×D₅₃	C₂²	1	1	yes	characteristic
35	#104×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810444	C₂²	1	1	yes	characteristic, complemented
38	#16810443	C₂²	1	2	yes	complemented
39	#16810437	C₂²	1	2	yes	complemented
40	#16810436	C₂²	1	2	yes	complemented
41	#16810435	C₂²	1	2	yes	complemented
42	C₅₃×#509	C₂²	1	2	yes	complemented
43	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#443	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#361	C₂²	1	2	yes	complemented
47	#41×D₅₃	—	2	1	no	—
48	C₂×D₄×D₅₃	—	2	2	no	—
49	C₂×D₄×D₅₃	—	4	2	no	—
50	#13133240	D₄	1	1	yes	characteristic, complemented
51	#13133239	C₂³	1	1	yes	characteristic
52	#13133238	—	2	1	no	—
53	#13133237	—	2	1	no	—
54	#13133236	D₄	1	1	yes	characteristic, complemented
55	Q₁₆×D₅₃	D₄	1	1	yes	characteristic, complemented
56	SD₁₆×D₅₃	—	2	1	no	—
57	C₂×D₄₂₄	—	2	1	no	—
58	C₂×D₄₂₄	C₂³	1	2	yes	—
59	C₂×#11719922	—	2	2	no	—
60	C₂×#11719922	—	2	2	no	—
61	D₈×D₅₃	—	2	2	no	—
62	D₈×D₅₃	—	2	2	no	—
63	D₈×D₅₃	—	2	2	no	—
64	D₈×D₅₃	—	2	2	no	—
65	D₈×D₅₃	—	4	1	no	—
66	D₈×D₅₃	—	4	1	no	—
67	D₈×D₅₃	—	4	1	no	—
68	D₈×D₅₃	—	4	1	no	—
69	D₈×D₅₃	D₄	1	1	yes	characteristic, complemented
70	#13133225	—	2	1	no	—
71	#13133225	—	4	1	no	—
72	#13133224	—	2	1	no	—
73	#13133223	—	2	1	no	—
74	#13133223	—	4	1	no	—
75	M₁₆×D₅₃	—	2	1	no	—
76	M₁₆×D₅₃	—	4	1	no	—
77	C₂×C₈×D₅₃	—	4	1	no	—
78	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
79	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
80	#13133214	—	2	1	no	—
81	#13133213	—	2	2	no	—
82	#13133213	—	4	2	no	—
83	C₄²×D₅₃	D₄	1	1	yes	characteristic
84	#13133162	—	2	2	no	—
85	#13133161	—	2	2	no	—
86	#13133160	—	2	2	no	—
87	D₈₄₈	—	2	2	no	—
88	#13133158	—	2	2	no	—
89	#13133158	—	2	2	no	—
90	#13133157	C₂³	1	2	yes	—
91	#13133156	—	2	2	no	—
92	C₁₆×D₅₃	—	2	2	no	—
93	#13133153	C₂³	1	1	yes	characteristic
94	#13133152	—	2	1	no	—
95	#13133145	—	2	1	no	—
96	#13133144	—	2	1	no	—
97	#13133143	—	2	1	no	—
98	C₄×#88322	D₄	1	1	yes	characteristic
99	C₈×Dic₅₃	D₄	1	1	yes	characteristic
100	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
101	#135×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×D₈×C₅₃	—	2	1	no	—
103	C₂×D₈×C₅₃	C₂³	1	2	yes	—
104	#131×C₅₃	—	2	1	no	—
105	#127×C₅₃	—	2	1	no	—
106	SD₃₂×C₅₃	—	2	2	no	—
107	D₁₆×C₅₃	—	2	2	no	—
108	M₃₂×C₅₃	C₂³	1	2	yes	—
109	#108×C₅₃	C₂³	1	1	yes	characteristic
110	#104×C₅₃	—	2	1	no	—
111	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₂³×D₅₃	—	4	2	no	—
113	#41×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	4	2	no	—
116	C₂×C₂²⋊D₅₃	—	4	4	no	—
117	C₂×C₂²⋊D₅₃	—	4	2	no	—
118	#11719949	—	2	1	no	—
119	Q₈×D₅₃	—	2	1	no	—
120	#11719947	—	2	1	no	—
121	#11719947	—	4	1	no	—
122	D₄×D₅₃	—	2	1	no	—
123	D₄×D₅₃	—	4	1	no	—
124	D₄×D₅₃	—	4	2	no	—
125	D₄×D₅₃	—	4	2	no	—
126	D₄×D₅₃	—	4	2	no	—
127	D₄×D₅₃	—	4	2	no	—
128	D₄×D₅₃	—	8	8	no	—
129	#11719945	—	2	1	no	—
130	#11719945	—	4	1	no	—
131	C₂×D₂₁₂	—	2	2	no	—
132	C₂×D₂₁₂	—	4	2	no	—
133	C₂×C₄×D₅₃	—	2	1	no	—
134	C₂×C₄×D₅₃	—	4	1	no	—
135	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
136	Q₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic, complemented
137	SD₁₆×C₅₃	—	2	1	no	—
138	D₈×C₅₃	—	2	2	no	—
139	D₈×C₅₃	—	2	2	no	—
140	D₈×C₅₃	—	4	1	no	—
141	D₈×C₅₃	—	4	1	no	—
142	D₈×C₅₃	C₂×D₄	1	1	yes	characteristic, complemented
143	M₁₆×C₅₃	—	2	1	no	—
144	M₁₆×C₅₃	—	4	1	no	—
145	C₂×C₈×C₅₃	—	4	1	no	—
146	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	#11719925	—	2	1	no	—
150	#11719924	—	2	1	no	—
151	#11719923	—	2	1	no	—
152	#11719922	—	2	1	no	—
153	#11719922	—	4	2	no	—
154	#11719922	—	4	2	no	—
155	#11719922	—	4	2	no	—
156	#11719922	—	4	2	no	—
157	C₄×Dic₅₃	—	2	1	no	—
158	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
159	#11719917	—	2	1	no	—
160	#11719917	—	4	1	no	—
161	C₂×#88322	—	4	1	no	—
162	C₂×#88322	C₂×D₄	1	1	yes	characteristic
163	C₂×#88322	C₂×D₄	1	1	yes	characteristic
164	Dic₂₁₂	C₂×D₄	1	1	yes	characteristic, complemented
165	D₄₂₄	—	2	2	no	—
166	D₄₂₄	—	2	2	no	—
167	D₄₂₄	—	4	1	no	—
168	D₄₂₄	—	4	1	no	—
169	D₄₂₄	C₂×D₄	1	1	yes	characteristic, complemented
170	#11719913	—	2	1	no	—
171	#11719912	—	4	1	no	—
172	#11719912	—	4	1	no	—
173	#11719912	—	4	1	no	—
174	C₈×D₅₃	—	4	1	no	—
175	C₈×D₅₃	—	4	1	no	—
176	C₈×D₅₃	—	4	1	no	—
177	C₈×D₅₃	D₈	1	1	yes	characteristic
178	C₈×D₅₃	D₈	1	1	yes	characteristic
179	C₈×D₅₃	D₈	1	1	yes	characteristic
180	C₈×D₅₃	D₈	1	1	yes	characteristic
181	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
182	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
183	C₈×D₅₃	#41	1	1	yes	characteristic
184	C₈×D₅₃	#41	1	1	yes	characteristic
185	C₁₆×C₅₃	—	2	2	no	—
186	#11719908	—	2	2	no	—
187	C₂³×C₅₃	—	4	2	no	—
188	C₂²×D₅₃	—	4	1	no	—
189	C₂²×D₅₃	—	4	2	no	—
190	C₂²×D₅₃	—	8	8	no	—
191	C₂²×D₅₃	—	8	2	no	—
192	C₂²×D₅₃	—	8	2	no	—
193	C₂²×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
194	Q₈×C₅₃	—	2	1	no	—
195	D₄×C₅₃	—	2	1	no	—
196	D₄×C₅₃	—	4	1	no	—
197	D₄×C₅₃	—	4	2	no	—
198	D₄×C₅₃	—	4	2	no	—
199	D₄×C₅₃	—	8	4	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	4	1	no	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
203	C₂²⋊D₅₃	—	4	1	no	—
204	C₂²⋊D₅₃	—	4	1	no	—
205	C₂²⋊D₅₃	—	4	1	no	—
206	C₂²⋊D₅₃	—	8	8	no	—
207	C₂²⋊D₅₃	—	8	2	no	—
208	C₂²⋊D₅₃	—	8	2	no	—
209	C₂×Dic₅₃	—	2	1	no	—
210	C₂×Dic₅₃	—	4	1	no	—
211	C₂×Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
212	D₂₁₂	—	2	1	no	—
213	D₂₁₂	—	4	1	no	—
214	D₂₁₂	—	4	2	no	—
215	D₂₁₂	—	4	2	no	—
216	D₂₁₂	—	8	4	no	—
217	C₄×D₅₃	—	2	4	no	—
218	C₄×D₅₃	—	4	1	no	—
219	C₄×D₅₃	—	4	1	no	—
220	C₄×D₅₃	—	4	1	no	—
221	C₄×D₅₃	#121	1	1	yes	characteristic
222	C₄×D₅₃	#121	1	1	yes	characteristic
223	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
224	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
225	Dic₁₀₆	—	2	1	no	—
226	Dic₁₀₆	—	4	1	no	—
227	C₈×C₅₃	—	4	1	no	—
228	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
229	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
230	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
231	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
232	#88322	—	4	1	no	—
233	#88322	#121	1	1	yes	characteristic
234	#88322	#121	1	1	yes	characteristic
235	#88322	C₂×D₈	1	1	yes	characteristic
236	#88322	C₂×D₈	1	1	yes	characteristic
237	C₂×#2213	—	53	1	no	maximal, Sylow
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	8	4	no	—
240	C₂²×C₅₃	—	8	2	no	—
241	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×D₅₃	—	2	2	no	—
243	C₂×D₅₃	—	4	1	no	—
244	C₂×D₅₃	—	8	2	no	—
245	C₂×D₅₃	—	8	4	no	—
246	C₂×D₅₃	—	8	2	no	—
247	C₂×D₅₃	—	16	4	no	—
248	C₂×D₅₃	—	16	4	no	—
249	C₂×D₅₃	#459	1	1	yes	characteristic
250	C₂×D₅₃	#459	1	1	yes	characteristic
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
254	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
255	Dic₅₃	—	2	2	no	—
256	Dic₅₃	—	4	1	no	—
257	Dic₅₃	#459	1	1	yes	characteristic
258	Dic₅₃	#459	1	1	yes	characteristic
259	C₂×#509	—	53	2	no	—
260	C₂×#493	—	53	1	no	—
261	C₂×#443	—	53	1	no	—
262	#2213	—	53	2	no	—
263	#2213	—	53	1	no	—
264	#2213	—	53	1	no	—
265	#2213	—	53	1	no	—
266	#2213	—	53	2	no	—
267	#2213	D₅₃	1	1	yes	characteristic, direct product factor
268	C₂×#364	—	53	1	no	—
269	C₂×#361	—	53	2	no	—
270	C₂×C₅₃	—	2	1	no	—
271	C₂×C₅₃	—	8	1	no	—
272	C₂×C₅₃	—	16	2	no	—
273	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
274	D₅₃	—	2	1	no	—
275	D₅₃	—	8	1	no	—
276	D₅₃	—	16	2	no	—
277	D₅₃	#2213	1	2	yes	direct product factor
278	C₂×#135	—	53	1	no	—
279	C₂²×D₈	—	53	2	no	—
280	C₂²×D₈	—	106	1	no	—
281	C₂×#131	—	106	1	no	—
282	C₂×#127	—	106	1	no	—
283	#509	—	53	2	no	—
284	#509	—	53	2	no	—
285	#509	—	53	2	no	—
286	#509	—	106	2	no	—
287	#509	—	106	2	no	—
288	#509	C₂×D₅₃	1	2	yes	complemented
289	C₂×SD₃₂	—	106	2	no	—
290	C₂×D₁₆	—	106	2	no	—
291	C₂×M₃₂	—	53	2	no	—
292	#493	—	53	2	no	—
293	#493	—	53	1	no	—
294	#493	—	106	1	no	—
295	#493	—	106	1	no	—
296	#493	C₂×D₅₃	1	1	yes	characteristic, complemented
297	#443	—	53	1	no	—
298	#443	—	53	1	no	—
299	#443	—	53	1	no	—
300	#443	—	106	1	no	—
301	#443	—	106	1	no	—
302	#443	C₂×D₅₃	1	1	yes	characteristic, complemented
303	C₂×#108	—	53	1	no	—
304	C₂×#104	—	106	1	no	—
305	C₂×C₄×C₈	—	53	1	no	—
306	#364	—	53	2	no	—
307	#364	—	53	1	no	—
308	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
309	#361	—	53	2	no	—
310	#361	—	53	2	no	—
311	#361	—	53	2	no	—
312	#361	C₂×D₅₃	1	2	yes	complemented
313	C₅₃	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
314	C₂×#41	—	106	1	no	—
315	C₂²×D₄	—	106	2	no	—
316	C₂²×D₄	—	212	2	no	—
317	#135	—	53	1	no	—
318	#135	—	53	1	no	—
319	#135	—	53	1	no	—
320	#135	—	106	1	no	—
321	#135	—	106	1	no	—
322	#135	C₂²×D₅₃	1	1	yes	characteristic
323	C₂×Q₁₆	—	53	1	no	—
324	C₂×SD₁₆	—	106	1	no	—
325	C₂×D₈	—	2	1	no	—
326	C₂×D₈	—	53	1	no	—
327	C₂×D₈	—	53	2	no	—
328	C₂×D₈	—	106	2	no	—
329	C₂×D₈	—	106	2	no	—
330	C₂×D₈	—	106	2	no	—
331	C₂×D₈	—	106	2	no	—
332	C₂×D₈	—	106	2	no	—
333	C₂×D₈	—	106	2	no	—
334	C₂×D₈	—	106	1	no	—
335	C₂×D₈	—	212	1	no	—
336	C₂×D₈	—	212	1	no	—
337	C₂×D₈	—	212	1	no	—
338	C₂×D₈	—	212	1	no	—
339	C₂×D₈	C₂²×D₅₃	1	2	yes	—
340	#131	—	2	1	no	—
341	#131	—	106	1	no	—
342	#131	—	106	1	no	—
343	#131	—	106	1	no	—
344	#131	—	212	1	no	—
345	#131	—	212	1	no	—
346	C₂×M₁₆	—	106	1	no	—
347	C₂×M₁₆	—	212	1	no	—
348	C₂²×C₈	—	53	1	no	—
349	C₂²×C₈	—	53	1	no	—
350	C₂²×C₈	—	212	1	no	—
351	#127	—	2	1	no	—
352	#127	—	106	2	no	—
353	#127	—	106	1	no	—
354	#127	—	212	2	no	—
355	C₂×C₄²	—	53	1	no	—
356	SD₃₂	—	2	2	no	—
357	SD₃₂	—	106	2	no	—
358	SD₃₂	—	106	2	no	—
359	SD₃₂	—	106	2	no	—
360	D₁₆	—	2	2	no	—
361	D₁₆	—	106	2	no	—
362	D₁₆	—	106	2	no	—
363	D₁₆	—	106	2	no	—
364	M₃₂	—	53	2	no	—
365	M₃₂	—	106	2	no	—
366	M₃₂	C₂²×D₅₃	1	2	yes	—
367	C₂×C₁₆	—	106	2	no	—
368	#108	—	53	1	no	—
369	#108	—	106	1	no	—
370	#108	C₂²×D₅₃	1	1	yes	characteristic
371	#104	—	2	1	no	—
372	#104	—	106	1	no	—
373	#104	—	106	1	no	—
374	#104	—	106	1	no	—
375	C₄×C₈	—	53	1	no	—
376	C₄×C₈	—	53	1	no	—
377	C₄×C₈	—	53	1	no	—
378	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
379	C₂⁴	—	212	2	no	—
380	#41	—	2	1	no	—
381	#41	—	106	1	no	—
382	#41	—	106	1	no	—
383	#41	—	106	1	no	—
384	#41	—	212	1	no	—
385	#41	—	212	1	no	—
386	C₂×Q₈	—	106	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	4	2	no	—
389	C₂×D₄	—	106	1	no	—
390	C₂×D₄	—	106	2	no	—
391	C₂×D₄	—	424	8	no	—
392	C₂×D₄	—	212	2	no	—
393	C₂×D₄	—	212	2	no	—
394	C₂×D₄	—	212	2	no	—
395	C₂×D₄	—	212	2	no	—
396	C₂×D₄	—	212	2	no	—
397	C₂×D₄	—	212	2	no	—
398	C₂×D₄	—	212	4	no	—
399	C₂×D₄	—	212	1	no	—
400	C₂²×C₄	—	53	1	no	—
401	C₂²×C₄	—	106	1	no	—
402	C₂²×C₄	—	212	1	no	—
403	Q₁₆	—	53	1	no	—
404	Q₁₆	—	106	1	no	—
405	Q₁₆	D₄×D₅₃	1	1	yes	characteristic, complemented
406	SD₁₆	—	2	1	no	—
407	SD₁₆	—	106	1	no	—
408	SD₁₆	—	106	1	no	—
409	SD₁₆	—	106	1	no	—
410	D₈	—	2	2	no	—
411	D₈	—	2	2	no	—
412	D₈	—	4	1	no	—
413	D₈	—	4	1	no	—
414	D₈	—	53	1	no	—
415	D₈	—	106	1	no	—
416	D₈	—	106	2	no	—
417	D₈	—	106	2	no	—
418	D₈	—	212	2	no	—
419	D₈	—	212	2	no	—
420	D₈	—	212	2	no	—
421	D₈	—	212	2	no	—
422	D₈	—	212	1	no	—
423	D₈	—	212	1	no	—
424	D₈	D₄×D₅₃	1	1	yes	characteristic, complemented
425	M₁₆	—	2	1	no	—
426	M₁₆	—	4	1	no	—
427	M₁₆	—	106	1	no	—
428	M₁₆	—	212	1	no	—
429	M₁₆	—	212	1	no	—
430	M₁₆	—	212	1	no	—
431	M₁₆	—	212	1	no	—
432	C₂×C₈	—	4	1	no	—
433	C₂×C₈	—	53	1	no	—
434	C₂×C₈	—	53	1	no	—
435	C₂×C₈	—	53	1	no	—
436	C₂×C₈	—	53	1	no	—
437	C₂×C₈	—	53	1	no	—
438	C₂×C₈	—	53	1	no	—
439	C₂×C₈	—	53	1	no	—
440	C₂×C₈	—	53	1	no	—
441	C₂×C₈	—	53	1	no	—
442	C₂×C₈	—	53	1	no	—
443	C₂×C₈	—	212	1	no	—
444	C₂×C₈	—	212	1	no	—
445	C₂×C₈	—	212	1	no	—
446	C₂×C₈	—	212	1	no	—
447	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
448	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
449	C₄²	—	53	1	no	—
450	C₄²	—	106	1	no	—
451	C₄²	D₄×D₅₃	1	1	yes	characteristic
452	C₁₆	—	2	2	no	—
453	C₁₆	—	106	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	53	1	no	—
456	C₂³	—	424	2	no	—
457	C₂³	—	424	2	no	—
458	C₂³	—	424	8	no	—
459	C₂³	—	212	1	no	—
460	C₂³	—	212	2	no	—
461	Q₈	—	2	1	no	—
462	Q₈	—	106	1	no	—
463	Q₈	—	212	1	no	—
464	D₄	—	2	1	no	—
465	D₄	—	4	1	no	—
466	D₄	—	4	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	8	4	no	—
469	D₄	—	106	1	no	—
470	D₄	—	424	4	no	—
471	D₄	—	424	2	no	—
472	D₄	—	424	2	no	—
473	D₄	—	424	8	no	—
474	D₄	—	212	1	no	—
475	D₄	—	212	1	no	—
476	D₄	—	212	1	no	—
477	D₄	—	212	1	no	—
478	D₄	—	212	2	no	—
479	D₄	—	212	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	53	1	no	—
483	C₂×C₄	—	53	1	no	—
484	C₂×C₄	—	53	1	no	—
485	C₂×C₄	—	53	1	no	—
486	C₂×C₄	—	53	1	no	—
487	C₂×C₄	—	106	4	no	—
488	C₂×C₄	—	106	1	no	—
489	C₂×C₄	—	212	1	no	—
490	C₂×C₄	—	212	1	no	—
491	C₂×C₄	—	212	1	no	—
492	C₂×C₄	—	212	1	no	—
493	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
494	C₈	—	4	1	no	—
495	C₈	—	53	1	no	—
496	C₈	—	53	1	no	—
497	C₈	—	53	1	no	—
498	C₈	—	53	1	no	—
499	C₈	—	212	1	no	—
500	C₈	D₈×D₅₃	1	1	yes	characteristic
501	C₈	D₈×D₅₃	1	1	yes	characteristic
502	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
503	C₈	#41×D₅₃	1	1	yes	characteristic
504	C₂²	—	4	1	no	—
505	C₂²	—	8	4	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	53	1	no	—
508	C₂²	—	53	1	no	—
509	C₂²	—	848	4	no	—
510	C₂²	—	848	4	no	—
511	C₂²	—	106	2	no	—
512	C₂²	—	424	4	no	—
513	C₂²	—	424	2	no	—
514	C₂²	—	424	2	no	—
515	C₂²	—	212	1	no	—
516	C₂²	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
517	C₄	—	2	2	no	—
518	C₄	—	4	1	no	—
519	C₄	—	53	1	no	—
520	C₄	—	53	1	no	—
521	C₄	—	106	2	no	—
522	C₄	—	212	1	no	—
523	C₄	#121×D₅₃	1	1	yes	characteristic
524	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
525	C₂	—	2	1	no	—
526	C₂	—	8	1	no	—
527	C₂	—	16	2	no	—
528	C₂	—	53	2	no	—
529	C₂	—	848	2	no	—
530	C₂	—	106	1	no	—
531	C₂	—	424	1	no	—
532	C₂	#459×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#459×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2213	C₂
socle	C₂×C₅₃	C₂×#459
hypercenter	#2213	D₅₃
hypocenter	C₅₃	C₂×#2213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2213

Complements

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

N	H ≅ G/N	type
1. #509×D₅₃	C₂	semidirect
2. #493×D₅₃	C₂	semidirect
3. #443×D₅₃	C₂	semidirect
4. #20276792	C₂	semidirect
5. #20276791	C₂	semidirect
6. #20276790	C₂	semidirect
7. #20276789	C₂	semidirect
8. #20276788	C₂	semidirect
9. #364×D₅₃	C₂	semidirect
10. #361×D₅₃	C₂	semidirect
11. C₅₃×#2213	C₂	semidirect
17. #16811061	C₂²	semidirect
19. #16811059	C₂²	semidirect
20. #16811058	C₂²	semidirect
25. #16811000	C₂²	semidirect
28. #16810998	C₂²	semidirect
29. #16810741	C₂²	semidirect
30. #16810740	C₂²	semidirect
31. #16810739	C₂²	semidirect
37. #16810444	C₂²	semidirect
38. #16810443	C₂²	semidirect
39. #16810437	C₂²	semidirect
40. #16810436	C₂²	semidirect
41. #16810435	C₂²	semidirect
42. C₅₃×#509	C₂²	semidirect
43. C₅₃×#493	C₂²	semidirect
44. C₅₃×#443	C₂²	semidirect
45. C₅₃×#364	C₂²	semidirect
46. C₅₃×#361	C₂²	semidirect
50. #13133240	D₄	semidirect
54. #13133236	D₄	semidirect
55. Q₁₆×D₅₃	D₄	semidirect
69. D₈×D₅₃	D₄	semidirect
136. Q₁₆×C₅₃	C₂×D₄	semidirect
142. D₈×C₅₃	C₂×D₄	semidirect
164. Dic₂₁₂	C₂×D₄	semidirect
169. D₄₂₄	C₂×D₄	semidirect
267. #2213	D₅₃	direct (N × H)
277. D₅₃	#2213	direct (N × H)
288. #509	C₂×D₅₃	semidirect
296. #493	C₂×D₅₃	semidirect
302. #443	C₂×D₅₃	semidirect
308. #364	C₂×D₅₃	semidirect
312. #361	C₂×D₅₃	semidirect
313. C₅₃	C₂×#2213	semidirect
405. Q₁₆	D₄×D₅₃	semidirect
424. D₈	D₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#509×D₅₃	2
#493×D₅₃	2
#443×D₅₃	2
#20276792	2
#20276791	2
#20276790	2
#20276789	2
#20276788	2
#364×D₅₃	2
#361×D₅₃	2
C₅₃×#2213	2
C₂×#2213	53

Subgroups — D53×#2213 #27538468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2213 #27538468