Subgroups — D₅₃×#2214 #27538474

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#510×D₅₃	C₂	1	2	yes	maximal
2	#494×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20276797	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276796	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276795	C₂	1	2	yes	maximal, complemented
7	#20276794	C₂	1	2	yes	maximal, complemented
8	#20276793	C₂	1	1	yes	maximal, characteristic, complemented
9	#364×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#362×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2214	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×D₅₃	—	2	1	no	—
14	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
15	#131×D₅₃	—	2	1	no	—
16	C₄⋊Q₈×D₅₃	—	2	1	no	—
17	#16811066	C₂²	1	2	yes	—
18	#16811065	—	2	2	no	—
19	#16811064	C₂²	1	2	yes	—
20	#16811063	C₂²	1	2	yes	—
21	#16811062	—	2	2	no	—
22	Q₃₂×D₅₃	—	2	2	no	—
23	SD₃₂×D₅₃	—	2	2	no	—
24	M₃₂×D₅₃	C₂²	1	2	yes	—
25	#16811003	C₂²	1	1	yes	characteristic, complemented
26	#16811002	—	2	1	no	—
27	#16811001	—	2	1	no	—
28	#16811001	C₂²	1	2	yes	complemented
29	#16810741	C₂²	1	1	yes	characteristic
30	#16810740	C₂²	1	1	yes	characteristic
31	#16810739	C₂²	1	1	yes	characteristic
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	#16810440	C₂²	1	2	yes	complemented
40	#16810439	C₂²	1	2	yes	complemented
41	#16810438	C₂²	1	2	yes	complemented
42	C₅₃×#510	C₂²	1	2	yes	—
43	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#443	C₂²	1	1	yes	characteristic
45	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#362	C₂²	1	2	yes	complemented
47	#41×D₅₃	—	2	1	no	—
48	C₂×Q₈×D₅₃	—	2	2	no	—
49	#13133240	D₄	1	1	yes	characteristic
50	#13133239	C₂³	1	1	yes	characteristic
51	#13133238	—	2	1	no	—
52	#13133237	—	2	1	no	—
53	#13133236	D₄	1	1	yes	characteristic
54	C₂×Dic₂₁₂	—	2	1	no	—
55	C₂×Dic₂₁₂	C₂³	1	2	yes	—
56	C₂×#11719925	—	2	2	no	—
57	C₂×#11719925	—	2	2	no	—
58	Q₁₆×D₅₃	—	2	2	no	—
59	Q₁₆×D₅₃	—	2	2	no	—
60	Q₁₆×D₅₃	—	2	2	no	—
61	Q₁₆×D₅₃	—	2	2	no	—
62	Q₁₆×D₅₃	—	4	1	no	—
63	Q₁₆×D₅₃	—	4	1	no	—
64	Q₁₆×D₅₃	—	4	1	no	—
65	Q₁₆×D₅₃	—	4	1	no	—
66	Q₁₆×D₅₃	D₄	1	1	yes	characteristic
67	SD₁₆×D₅₃	—	2	1	no	—
68	D₈×D₅₃	D₄	1	1	yes	characteristic
69	#13133225	—	2	1	no	—
70	#13133225	—	4	1	no	—
71	#13133224	—	2	1	no	—
72	#13133223	—	2	1	no	—
73	#13133223	—	4	1	no	—
74	M₁₆×D₅₃	—	2	1	no	—
75	M₁₆×D₅₃	—	4	1	no	—
76	C₂×C₈×D₅₃	—	4	1	no	—
77	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
78	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
79	#13133217	—	2	1	no	—
80	#13133216	—	4	2	no	—
81	#13133215	—	2	2	no	—
82	C₄⋊C₄×D₅₃	—	4	2	no	—
83	C₄²×D₅₃	D₄	1	1	yes	characteristic
84	Dic₄₂₄	—	2	2	no	—
85	#13133163	—	2	2	no	—
86	#13133163	—	2	2	no	—
87	#13133162	—	2	2	no	—
88	#13133161	—	2	2	no	—
89	#13133160	—	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×Q₁₆×C₅₃	—	2	1	no	—
103	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
104	#131×C₅₃	—	2	1	no	—
105	C₄⋊Q₈×C₅₃	—	2	1	no	—
106	Q₃₂×C₅₃	—	2	2	no	—
107	SD₃₂×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	#41×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	—	2	2	no	—
114	#11719949	—	2	1	no	—
115	Q₈×D₅₃	—	2	1	no	—
116	Q₈×D₅₃	—	4	2	no	—
117	Q₈×D₅₃	—	4	2	no	—
118	Q₈×D₅₃	—	4	2	no	—
119	Q₈×D₅₃	—	4	2	no	—
120	#11719947	—	2	1	no	—
121	#11719947	—	4	1	no	—
122	D₄×D₅₃	—	2	1	no	—
123	D₄×D₅₃	—	4	1	no	—
124	#11719945	—	2	1	no	—
125	#11719945	—	4	1	no	—
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	4	1	no	—
128	C₂×C₄×D₅₃	—	4	2	no	—
129	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₆	—	2	2	no	—
131	C₂×Dic₁₀₆	—	4	2	no	—
132	Q₁₆×C₅₃	—	2	2	no	—
133	Q₁₆×C₅₃	—	2	2	no	—
134	Q₁₆×C₅₃	—	4	1	no	—
135	Q₁₆×C₅₃	—	4	1	no	—
136	Q₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
137	SD₁₆×C₅₃	—	2	1	no	—
138	D₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	M₁₆×C₅₃	—	2	1	no	—
140	M₁₆×C₅₃	—	4	1	no	—
141	C₂×C₈×C₅₃	—	4	1	no	—
142	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₅₃	—	4	2	no	—
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	#11719925	—	2	1	no	—
147	#11719925	—	4	2	no	—
148	#11719925	—	4	2	no	—
149	#11719925	—	4	2	no	—
150	#11719925	—	4	2	no	—
151	#11719924	—	2	1	no	—
152	#11719923	—	2	1	no	—
153	#11719922	—	2	1	no	—
154	C₄⋊Dic₅₃	—	4	2	no	—
155	#11719919	—	4	4	no	—
156	C₄×Dic₅₃	—	2	1	no	—
157	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
158	#11719917	—	2	1	no	—
159	#11719917	—	4	1	no	—
160	C₂×#88322	—	4	1	no	—
161	C₂×#88322	C₂×D₄	1	1	yes	characteristic
162	C₂×#88322	C₂×D₄	1	1	yes	characteristic
163	Dic₂₁₂	—	2	2	no	—
164	Dic₂₁₂	—	2	2	no	—
165	Dic₂₁₂	—	4	1	no	—
166	Dic₂₁₂	—	4	1	no	—
167	Dic₂₁₂	C₂×D₄	1	1	yes	characteristic
168	D₄₂₄	C₂×D₄	1	1	yes	characteristic
169	#11719913	—	2	1	no	—
170	#11719912	—	4	1	no	—
171	#11719912	—	4	1	no	—
172	#11719912	—	4	1	no	—
173	C₈×D₅₃	—	4	1	no	—
174	C₈×D₅₃	—	4	1	no	—
175	C₈×D₅₃	—	4	1	no	—
176	C₈×D₅₃	D₈	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
181	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
182	C₈×D₅₃	#41	1	1	yes	characteristic
183	C₈×D₅₃	#41	1	1	yes	characteristic
184	C₁₆×C₅₃	—	2	2	no	—
185	#11719908	—	2	2	no	—
186	C₂²×D₅₃	—	4	1	no	—
187	C₂²×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
188	Q₈×C₅₃	—	2	1	no	—
189	Q₈×C₅₃	—	4	2	no	—
190	Q₈×C₅₃	—	4	2	no	—
191	D₄×C₅₃	—	2	1	no	—
192	D₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂²⋊D₅₃	—	4	1	no	—
198	C₂²⋊D₅₃	—	4	1	no	—
199	C₂²⋊D₅₃	—	4	1	no	—
200	C₂×Dic₅₃	—	2	1	no	—
201	C₂×Dic₅₃	—	4	1	no	—
202	C₂×Dic₅₃	—	4	2	no	—
203	C₂×Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
204	D₂₁₂	—	2	1	no	—
205	D₂₁₂	—	4	1	no	—
206	C₄×D₅₃	—	2	4	no	—
207	C₄×D₅₃	—	4	1	no	—
208	C₄×D₅₃	—	4	1	no	—
209	C₄×D₅₃	—	4	1	no	—
210	C₄×D₅₃	—	8	2	no	—
211	C₄×D₅₃	—	8	2	no	—
212	C₄×D₅₃	#121	1	1	yes	characteristic
213	C₄×D₅₃	#121	1	1	yes	characteristic
214	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
215	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
216	Dic₁₀₆	—	2	1	no	—
217	Dic₁₀₆	—	4	1	no	—
218	Dic₁₀₆	—	4	2	no	—
219	Dic₁₀₆	—	4	2	no	—
220	Dic₁₀₆	—	8	2	no	—
221	Dic₁₀₆	—	8	2	no	—
222	C₈×C₅₃	—	4	1	no	—
223	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
224	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
225	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
227	#88322	—	4	1	no	—
228	#88322	#121	1	1	yes	characteristic
229	#88322	#121	1	1	yes	characteristic
230	#88322	C₂×D₈	1	1	yes	characteristic
231	#88322	C₂×D₈	1	1	yes	characteristic
232	C₂×#2214	—	53	1	no	maximal, Sylow
233	C₂²×C₅₃	—	4	1	no	—
234	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂×D₅₃	—	2	2	no	—
236	C₂×D₅₃	—	4	1	no	—
237	C₂×D₅₃	—	8	2	no	—
238	C₂×D₅₃	#459	1	1	yes	characteristic
239	C₂×D₅₃	#459	1	1	yes	characteristic
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	4	1	no	—
242	C₄×C₅₃	—	8	2	no	—
243	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
244	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
245	Dic₅₃	—	2	2	no	—
246	Dic₅₃	—	4	1	no	—
247	Dic₅₃	—	8	2	no	—
248	Dic₅₃	#459	1	1	yes	characteristic
249	Dic₅₃	#459	1	1	yes	characteristic
250	C₂×#510	—	53	2	no	—
251	C₂×#494	—	53	1	no	—
252	C₂×#443	—	53	1	no	—
253	#2214	—	53	2	no	—
254	#2214	—	53	1	no	—
255	#2214	—	53	1	no	—
256	#2214	—	53	1	no	—
257	#2214	—	53	2	no	—
258	#2214	D₅₃	1	1	yes	characteristic, direct product factor
259	C₂×#364	—	53	1	no	—
260	C₂×#362	—	53	2	no	—
261	C₂×C₅₃	—	2	1	no	—
262	C₂×C₅₃	—	8	1	no	—
263	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
264	D₅₃	—	2	1	no	—
265	D₅₃	—	8	1	no	—
266	D₅₃	#2214	1	2	yes	direct product factor
267	C₂×#135	—	53	1	no	—
268	C₂²×Q₁₆	—	53	2	no	—
269	C₂²×Q₁₆	—	106	1	no	—
270	C₂×#131	—	106	1	no	—
271	C₂×C₄⋊Q₈	—	106	1	no	—
272	#510	—	53	2	no	—
273	#510	—	53	2	no	—
274	#510	—	53	2	no	—
275	#510	—	106	2	no	—
276	#510	—	106	2	no	—
277	#510	C₂×D₅₃	1	2	yes	—
278	C₂×Q₃₂	—	106	2	no	—
279	C₂×SD₃₂	—	106	2	no	—
280	C₂×M₃₂	—	53	2	no	—
281	#494	—	53	2	no	—
282	#494	—	53	1	no	—
283	#494	—	106	1	no	—
284	#494	—	106	1	no	—
285	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
286	#443	—	53	1	no	—
287	#443	—	53	1	no	—
288	#443	—	53	1	no	—
289	#443	—	106	1	no	—
290	#443	—	106	1	no	—
291	#443	C₂×D₅₃	1	1	yes	characteristic
292	C₂×#108	—	53	1	no	—
293	C₂×#104	—	106	1	no	—
294	C₂×C₄×C₈	—	53	1	no	—
295	#364	—	53	2	no	—
296	#364	—	53	1	no	—
297	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
298	#362	—	53	2	no	—
299	#362	—	53	2	no	—
300	#362	—	53	2	no	—
301	#362	C₂×D₅₃	1	2	yes	complemented
302	C₅₃	C₂×#2214	1	1	yes	characteristic, complemented, Sylow
303	C₂×#41	—	106	1	no	—
304	C₂²×Q₈	—	106	2	no	—
305	#135	—	53	1	no	—
306	#135	—	53	1	no	—
307	#135	—	53	1	no	—
308	#135	—	106	1	no	—
309	#135	—	106	1	no	—
310	#135	C₂²×D₅₃	1	1	yes	characteristic
311	C₂×Q₁₆	—	2	1	no	—
312	C₂×Q₁₆	—	53	1	no	—
313	C₂×Q₁₆	—	53	2	no	—
314	C₂×Q₁₆	—	106	2	no	—
315	C₂×Q₁₆	—	106	2	no	—
316	C₂×Q₁₆	—	106	2	no	—
317	C₂×Q₁₆	—	106	2	no	—
318	C₂×Q₁₆	—	106	2	no	—
319	C₂×Q₁₆	—	106	2	no	—
320	C₂×Q₁₆	—	106	1	no	—
321	C₂×Q₁₆	—	212	1	no	—
322	C₂×Q₁₆	—	212	1	no	—
323	C₂×Q₁₆	—	212	1	no	—
324	C₂×Q₁₆	—	212	1	no	—
325	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
326	C₂×SD₁₆	—	106	1	no	—
327	C₂×D₈	—	53	1	no	—
328	#131	—	2	1	no	—
329	#131	—	106	1	no	—
330	#131	—	106	1	no	—
331	#131	—	106	1	no	—
332	#131	—	212	1	no	—
333	#131	—	212	1	no	—
334	C₂×M₁₆	—	106	1	no	—
335	C₂×M₁₆	—	212	1	no	—
336	C₂²×C₈	—	53	1	no	—
337	C₂²×C₈	—	53	1	no	—
338	C₂²×C₈	—	212	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	106	2	no	—
341	C₄⋊Q₈	—	106	1	no	—
342	C₄⋊Q₈	—	212	2	no	—
343	C₂×C₄⋊C₄	—	212	2	no	—
344	C₂×C₄²	—	53	1	no	—
345	Q₃₂	—	2	2	no	—
346	Q₃₂	—	106	2	no	—
347	Q₃₂	—	106	2	no	—
348	Q₃₂	—	106	2	no	—
349	SD₃₂	—	2	2	no	—
350	SD₃₂	—	106	2	no	—
351	SD₃₂	—	106	2	no	—
352	SD₃₂	—	106	2	no	—
353	M₃₂	—	53	2	no	—
354	M₃₂	—	106	2	no	—
355	M₃₂	C₂²×D₅₃	1	2	yes	—
356	C₂×C₁₆	—	106	2	no	—
357	#108	—	53	1	no	—
358	#108	—	106	1	no	—
359	#108	C₂²×D₅₃	1	1	yes	characteristic
360	#104	—	2	1	no	—
361	#104	—	106	1	no	—
362	#104	—	106	1	no	—
363	#104	—	106	1	no	—
364	C₄×C₈	—	53	1	no	—
365	C₄×C₈	—	53	1	no	—
366	C₄×C₈	—	53	1	no	—
367	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
368	#41	—	2	1	no	—
369	#41	—	106	1	no	—
370	#41	—	106	1	no	—
371	#41	—	106	1	no	—
372	#41	—	212	1	no	—
373	#41	—	212	1	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	106	1	no	—
376	C₂×Q₈	—	106	2	no	—
377	C₂×Q₈	—	212	2	no	—
378	C₂×Q₈	—	212	2	no	—
379	C₂×Q₈	—	212	2	no	—
380	C₂×Q₈	—	212	2	no	—
381	C₂×Q₈	—	212	2	no	—
382	C₂×D₄	—	106	1	no	—
383	C₂×D₄	—	212	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	106	1	no	—
386	C₂²×C₄	—	212	1	no	—
387	C₂²×C₄	—	212	2	no	—
388	Q₁₆	—	2	2	no	—
389	Q₁₆	—	2	2	no	—
390	Q₁₆	—	4	1	no	—
391	Q₁₆	—	4	1	no	—
392	Q₁₆	—	53	1	no	—
393	Q₁₆	—	106	1	no	—
394	Q₁₆	—	106	2	no	—
395	Q₁₆	—	106	2	no	—
396	Q₁₆	—	212	2	no	—
397	Q₁₆	—	212	2	no	—
398	Q₁₆	—	212	2	no	—
399	Q₁₆	—	212	2	no	—
400	Q₁₆	—	212	1	no	—
401	Q₁₆	—	212	1	no	—
402	Q₁₆	D₄×D₅₃	1	1	yes	characteristic
403	SD₁₆	—	2	1	no	—
404	SD₁₆	—	106	1	no	—
405	SD₁₆	—	106	1	no	—
406	SD₁₆	—	106	1	no	—
407	D₈	—	53	1	no	—
408	D₈	—	106	1	no	—
409	D₈	D₄×D₅₃	1	1	yes	characteristic
410	M₁₆	—	2	1	no	—
411	M₁₆	—	4	1	no	—
412	M₁₆	—	106	1	no	—
413	M₁₆	—	212	1	no	—
414	M₁₆	—	212	1	no	—
415	M₁₆	—	212	1	no	—
416	M₁₆	—	212	1	no	—
417	C₂×C₈	—	4	1	no	—
418	C₂×C₈	—	53	1	no	—
419	C₂×C₈	—	53	1	no	—
420	C₂×C₈	—	53	1	no	—
421	C₂×C₈	—	53	1	no	—
422	C₂×C₈	—	53	1	no	—
423	C₂×C₈	—	53	1	no	—
424	C₂×C₈	—	53	1	no	—
425	C₂×C₈	—	53	1	no	—
426	C₂×C₈	—	53	1	no	—
427	C₂×C₈	—	53	1	no	—
428	C₂×C₈	—	212	1	no	—
429	C₂×C₈	—	212	1	no	—
430	C₂×C₈	—	212	1	no	—
431	C₂×C₈	—	212	1	no	—
432	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
433	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	—	212	2	no	—
436	C₄⋊C₄	—	212	4	no	—
437	C₄²	—	53	1	no	—
438	C₄²	—	106	1	no	—
439	C₄²	D₄×D₅₃	1	1	yes	characteristic
440	C₁₆	—	2	2	no	—
441	C₁₆	—	106	2	no	—
442	C₂³	—	53	1	no	—
443	C₂³	—	212	1	no	—
444	Q₈	—	2	1	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	106	1	no	—
448	Q₈	—	424	2	no	—
449	Q₈	—	424	2	no	—
450	Q₈	—	212	1	no	—
451	Q₈	—	212	2	no	—
452	Q₈	—	212	2	no	—
453	D₄	—	2	1	no	—
454	D₄	—	4	1	no	—
455	D₄	—	106	1	no	—
456	D₄	—	212	1	no	—
457	D₄	—	212	1	no	—
458	D₄	—	212	1	no	—
459	D₄	—	212	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	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	—	106	4	no	—
469	C₂×C₄	—	106	1	no	—
470	C₂×C₄	—	424	2	no	—
471	C₂×C₄	—	424	2	no	—
472	C₂×C₄	—	212	1	no	—
473	C₂×C₄	—	212	1	no	—
474	C₂×C₄	—	212	1	no	—
475	C₂×C₄	—	212	2	no	—
476	C₂×C₄	—	212	1	no	—
477	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
478	C₈	—	4	1	no	—
479	C₈	—	53	1	no	—
480	C₈	—	53	1	no	—
481	C₈	—	53	1	no	—
482	C₈	—	53	1	no	—
483	C₈	—	212	1	no	—
484	C₈	D₈×D₅₃	1	1	yes	characteristic
485	C₈	D₈×D₅₃	1	1	yes	characteristic
486	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
487	C₈	#41×D₅₃	1	1	yes	characteristic
488	C₂²	—	4	1	no	—
489	C₂²	—	53	1	no	—
490	C₂²	—	53	1	no	—
491	C₂²	—	106	2	no	—
492	C₂²	—	424	2	no	—
493	C₂²	—	212	1	no	—
494	C₂²	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
495	C₄	—	2	2	no	—
496	C₄	—	4	1	no	—
497	C₄	—	8	2	no	—
498	C₄	—	53	1	no	—
499	C₄	—	53	1	no	—
500	C₄	—	106	2	no	—
501	C₄	—	424	2	no	—
502	C₄	—	212	1	no	—
503	C₄	#121×D₅₃	1	1	yes	characteristic
504	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
505	C₂	—	2	1	no	—
506	C₂	—	8	1	no	—
507	C₂	—	53	2	no	—
508	C₂	—	106	1	no	—
509	C₂	—	424	1	no	—
510	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×#2214	C₂
socle	C₂×C₅₃	C₂×#459
prime radical	#135×D₅₃	C₂²
hypercenter	#2214	D₅₃
hypocenter	C₅₃	C₂×#2214
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2214

Complements

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

N	H ≅ G/N	type
2. #494×D₅₃	C₂	semidirect
4. #20276797	C₂	semidirect
5. #20276796	C₂	semidirect
6. #20276795	C₂	semidirect
7. #20276794	C₂	semidirect
8. #20276793	C₂	semidirect
9. #364×D₅₃	C₂	semidirect
10. #362×D₅₃	C₂	semidirect
11. C₅₃×#2214	C₂	semidirect
25. #16811003	C₂²	semidirect
28. #16811001	C₂²	semidirect
37. #16810444	C₂²	semidirect
38. #16810443	C₂²	semidirect
39. #16810440	C₂²	semidirect
40. #16810439	C₂²	semidirect
41. #16810438	C₂²	semidirect
43. C₅₃×#494	C₂²	semidirect
45. C₅₃×#364	C₂²	semidirect
46. C₅₃×#362	C₂²	semidirect
258. #2214	D₅₃	direct (N × H)
266. D₅₃	#2214	direct (N × H)
285. #494	C₂×D₅₃	semidirect
297. #364	C₂×D₅₃	semidirect
301. #362	C₂×D₅₃	semidirect
302. C₅₃	C₂×#2214	semidirect

Maximal subgroups

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

subgroup	index
#510×D₅₃	2
#494×D₅₃	2
#443×D₅₃	2
#20276797	2
#20276796	2
#20276795	2
#20276794	2
#20276793	2
#364×D₅₃	2
#362×D₅₃	2
C₅₃×#2214	2
C₂×#2214	53

Subgroups — D53×#2214 #27538474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2214 #27538474