Subgroups — D₅₃×#1668 #27366416

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#493×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#460×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274076	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274075	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274074	C₂	1	2	yes	maximal, complemented
7	#20274073	C₂	1	2	yes	maximal, complemented
8	#20274072	C₂	1	1	yes	maximal, characteristic, complemented
9	#335×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#331×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1668	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×D₅₃	—	2	2	no	—
13	C₂×D₈×D₅₃	—	2	1	no	—
14	C₂×D₈×D₅₃	—	2	1	no	—
15	#127×D₅₃	C₂²	1	2	yes	—
16	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16811000	C₂²	1	1	yes	characteristic
18	#16810999	C₂²	1	2	yes	—
19	#16810998	—	2	1	no	—
20	#16810998	—	2	1	no	—
21	#16810824	C₂²	1	2	yes	complemented
22	#16810823	C₂²	1	2	yes	complemented
23	#16810822	—	2	2	no	—
24	#16810821	—	2	2	no	—
25	#16810820	C₂²	1	2	yes	complemented
26	Q₈×#88322	C₂²	1	1	yes	characteristic, complemented
27	C₈×Dic₁₀₆	—	2	1	no	—
28	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
29	#16810747	—	2	1	no	—
30	#16810747	C₂²	1	1	yes	characteristic, complemented
31	C₄⋊C₈×D₅₃	—	2	1	no	—
32	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
33	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
34	#102×D₅₃	—	2	2	no	—
35	C₄×C₈×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810376	C₂²	1	1	yes	characteristic, complemented
38	#16810375	C₂²	1	2	yes	complemented
39	#16810367	C₂²	1	2	yes	complemented
40	#16810366	C₂²	1	2	yes	complemented
41	#16810366	C₂²	1	2	yes	complemented
42	C₅₃×#493	C₂²	1	1	yes	characteristic
43	C₅₃×#460	C₂²	1	2	yes	complemented
44	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#331	C₂²	1	2	yes	complemented
47	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
48	C₂×D₄×D₅₃	—	2	2	no	—
49	C₂×D₄×D₅₃	—	2	2	no	—
50	C₂×D₄×D₅₃	—	2	2	no	—
51	C₂×#11719913	—	2	2	no	—
52	C₂×#11719923	—	2	2	no	—
53	C₂×#11719924	—	2	2	no	—
54	SD₁₆×D₅₃	—	4	8	no	—
55	C₂×D₄₂₄	—	2	1	no	—
56	C₂×D₄₂₄	—	2	1	no	—
57	C₂×#11719922	—	2	2	no	—
58	C₂×#11719922	—	2	2	no	—
59	D₈×D₅₃	—	4	4	no	—
60	D₈×D₅₃	—	4	4	no	—
61	C₂×C₈×D₅₃	—	2	1	no	—
62	C₂×C₈×D₅₃	—	2	2	no	—
63	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
65	#13133214	C₂³	1	2	yes	—
66	#13133213	—	2	2	no	—
67	#13133213	—	2	2	no	—
68	#13133213	—	2	2	no	—
69	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
70	C₄×Dic₁₀₆	—	2	1	no	—
71	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
72	#13133182	—	2	1	no	—
73	#13133182	D₄	1	1	yes	characteristic
74	C₄⋊C₄×D₅₃	—	2	1	no	—
75	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
76	C₄²×D₅₃	—	2	1	no	—
77	C₄²×D₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊#88322	—	2	1	no	—
79	C₄⋊#88322	C₂³	1	2	yes	—
80	C₄⋊#88322	C₂³	1	1	yes	characteristic
81	#13133146	—	2	1	no	—
82	#13133146	—	2	2	no	—
83	#13133146	—	2	1	no	—
84	#13133146	—	2	1	no	—
85	#13133139	—	2	2	no	—
86	#13133138	—	2	2	no	—
87	#13133137	—	2	2	no	—
88	C₄×#88322	—	2	1	no	—
89	C₄×#88322	C₂³	1	1	yes	characteristic
90	C₈×Dic₅₃	—	2	1	no	—
91	C₈×Dic₅₃	—	2	1	no	—
92	C₈×Dic₅₃	D₄	1	1	yes	characteristic
93	C₈×Dic₅₃	D₄	1	1	yes	characteristic
94	C₂×SD₁₆×C₅₃	—	2	2	no	—
95	C₂×D₈×C₅₃	—	2	1	no	—
96	C₂×D₈×C₅₃	—	2	1	no	—
97	#127×C₅₃	C₂³	1	2	yes	—
98	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	—	2	1	no	—
100	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
101	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
102	#102×C₅₃	—	2	2	no	—
103	C₄×C₈×C₅₃	—	2	1	no	—
104	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₂³×D₅₃	—	4	2	no	—
106	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	2	no	—
110	C₂×C₂²⋊D₅₃	—	4	2	no	—
111	C₂×C₂²⋊D₅₃	—	4	2	no	—
112	C₂×C₂²⋊D₅₃	—	4	2	no	—
113	Q₈×D₅₃	—	2	2	no	—
114	Q₈×D₅₃	D₈	1	4	yes	complemented
115	D₄×D₅₃	—	4	4	no	—
116	D₄×D₅₃	—	4	4	no	—
117	D₄×D₅₃	—	4	4	no	—
118	D₄×D₅₃	—	4	4	no	—
119	D₄×D₅₃	—	8	4	no	—
120	C₂×D₂₁₂	—	2	2	no	—
121	C₂×D₂₁₂	—	2	2	no	—
122	C₂×D₂₁₂	—	2	2	no	—
123	C₂×C₄×D₅₃	—	2	1	no	—
124	C₂×C₄×D₅₃	—	2	1	no	—
125	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
128	C₂×Dic₁₀₆	—	2	1	no	—
129	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
130	SD₁₆×C₅₃	—	4	4	no	—
131	D₈×C₅₃	—	4	2	no	—
132	D₈×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
141	#11719924	—	4	4	no	—
142	#11719923	—	4	4	no	—
143	#11719922	—	4	4	no	—
144	#11719922	—	4	4	no	—
145	C₄⋊Dic₅₃	—	2	1	no	—
146	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
147	#11719919	—	2	1	no	—
148	#11719919	—	2	1	no	—
149	#11719919	—	2	1	no	—
150	C₄×Dic₅₃	—	2	1	no	—
151	C₄×Dic₅₃	—	2	1	no	—
152	C₄×Dic₅₃	—	2	1	no	—
153	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄×Dic₅₃	#41	1	1	yes	characteristic
156	C₂×#88322	—	2	1	no	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	C₂×D₄	1	1	yes	characteristic
159	C₂×#88322	C₂×D₄	1	1	yes	characteristic
160	D₄₂₄	—	4	2	no	—
161	D₄₂₄	—	4	2	no	—
162	#11719913	—	4	4	no	—
163	C₈×D₅₃	—	2	2	no	—
164	C₈×D₅₃	—	4	4	no	—
165	C₈×D₅₃	—	4	2	no	—
166	C₈×D₅₃	SD₁₆	1	4	yes	complemented
167	C₂³×C₅₃	—	4	2	no	—
168	C₂²×D₅₃	—	4	2	no	—
169	C₂²×D₅₃	—	8	4	no	—
170	C₂²×D₅₃	—	8	4	no	—
171	C₂²×D₅₃	—	8	4	no	—
172	C₂²×D₅₃	#121	1	1	yes	characteristic
173	Q₈×C₅₃	—	2	1	no	—
174	Q₈×C₅₃	C₂×D₈	1	2	yes	complemented
175	D₄×C₅₃	—	4	2	no	—
176	D₄×C₅₃	—	4	2	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	4	2	no	—
179	D₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	8	4	no	—
186	C₂²⋊D₅₃	—	8	4	no	—
187	C₂²⋊D₅₃	—	8	4	no	—
188	C₂×Dic₅₃	—	2	1	no	—
189	C₂×Dic₅₃	—	2	1	no	—
190	C₂×Dic₅₃	#121	1	1	yes	characteristic
191	C₂×Dic₅₃	#121	1	1	yes	characteristic
192	C₂×Dic₅₃	#121	1	1	yes	characteristic
193	D₂₁₂	—	4	2	no	—
194	D₂₁₂	—	4	2	no	—
195	D₂₁₂	—	4	2	no	—
196	D₂₁₂	—	4	2	no	—
197	D₂₁₂	—	8	2	no	—
198	C₄×D₅₃	—	2	4	no	—
199	C₄×D₅₃	—	2	2	no	—
200	C₄×D₅₃	—	4	2	no	—
201	C₄×D₅₃	C₂×D₈	1	2	yes	—
202	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
203	C₄×D₅₃	#135	1	2	yes	—
204	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
205	Dic₁₀₆	—	2	1	no	—
206	Dic₁₀₆	—	2	4	no	—
207	Dic₁₀₆	C₂×D₈	1	2	yes	complemented
208	C₈×C₅₃	—	2	1	no	—
209	C₈×C₅₃	—	4	2	no	—
210	C₈×C₅₃	—	4	1	no	—
211	C₈×C₅₃	C₂×SD₁₆	1	2	yes	complemented
212	#88322	—	2	1	no	—
213	#88322	—	4	2	no	—
214	#88322	—	4	1	no	—
215	#88322	C₂×SD₁₆	1	2	yes	complemented
216	C₂×#1668	—	53	1	no	maximal, Sylow
217	C₂²×C₅₃	—	8	2	no	—
218	C₂²×C₅₃	—	8	2	no	—
219	C₂²×C₅₃	—	8	2	no	—
220	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
221	C₂×D₅₃	—	8	2	no	—
222	C₂×D₅₃	—	8	2	no	—
223	C₂×D₅₃	—	8	2	no	—
224	C₂×D₅₃	—	16	8	no	—
225	C₂×D₅₃	#460	1	2	yes	—
226	C₂×D₅₃	#466	1	2	yes	—
227	C₂×D₅₃	#472	1	2	yes	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	4	1	no	—
231	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
232	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
233	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
234	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	Dic₅₃	—	2	2	no	—
236	Dic₅₃	—	2	1	no	—
237	Dic₅₃	—	4	1	no	—
238	Dic₅₃	#460	1	2	yes	—
239	Dic₅₃	#466	1	2	yes	—
240	C₂×#493	—	53	1	no	—
241	C₂×#460	—	53	2	no	—
242	C₂×C₈×Q₈	—	53	1	no	—
243	#1668	—	53	1	no	—
244	#1668	—	53	2	no	—
245	#1668	—	53	1	no	—
246	#1668	—	53	1	no	—
247	#1668	—	53	2	no	—
248	#1668	D₅₃	1	1	yes	characteristic, direct product factor
249	C₂×#335	—	53	1	no	—
250	C₂×#331	—	53	2	no	—
251	C₂×C₅₃	—	16	2	no	—
252	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
253	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
254	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
255	D₅₃	—	16	2	no	—
256	D₅₃	#1668	1	4	yes	direct product factor
257	C₂²×SD₁₆	—	106	2	no	—
258	C₂²×D₈	—	106	1	no	—
259	C₂²×D₈	—	106	1	no	—
260	C₂×#127	—	53	2	no	—
261	C₂×C₄×Q₈	—	53	1	no	—
262	#493	—	53	2	no	—
263	#493	—	53	1	no	—
264	#493	—	106	1	no	—
265	#493	—	106	1	no	—
266	#493	C₂×D₅₃	1	1	yes	characteristic
267	#460	—	53	2	no	—
268	#460	—	53	2	no	—
269	#460	—	53	2	no	—
270	#460	—	106	2	no	—
271	#460	—	106	2	no	—
272	#460	C₂×D₅₃	1	2	yes	complemented
273	C₈×Q₈	—	53	1	no	—
274	C₈×Q₈	—	53	1	no	—
275	C₈×Q₈	—	53	1	no	—
276	C₈×Q₈	—	106	1	no	—
277	C₈×Q₈	—	106	1	no	—
278	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
279	C₂×C₄⋊C₈	—	53	1	no	—
280	C₂×C₄⋊C₈	—	53	2	no	—
281	C₂×C₄⋊C₈	—	106	1	no	—
282	C₂×#102	—	106	2	no	—
283	C₂×C₄×C₈	—	53	1	no	—
284	C₂×C₄×C₈	—	106	1	no	—
285	#335	—	53	2	no	—
286	#335	—	53	1	no	—
287	#335	C₂×D₅₃	1	1	yes	characteristic, complemented
288	#331	—	53	2	no	—
289	#331	—	53	2	no	—
290	#331	—	53	2	no	—
291	#331	C₂×D₅₃	1	2	yes	complemented
292	C₅₃	C₂×#1668	1	1	yes	characteristic, complemented, Sylow
293	C₂²×Q₈	—	53	1	no	—
294	C₂²×D₄	—	106	2	no	—
295	C₂²×D₄	—	106	2	no	—
296	C₂²×D₄	—	106	2	no	—
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×SD₁₆	—	106	2	no	—
299	C₂×SD₁₆	—	106	2	no	—
300	C₂×SD₁₆	—	106	2	no	—
301	C₂×SD₁₆	—	212	8	no	—
302	C₂×D₈	—	2	1	no	—
303	C₂×D₈	—	2	1	no	—
304	C₂×D₈	—	106	2	no	—
305	C₂×D₈	—	106	2	no	—
306	C₂×D₈	—	106	1	no	—
307	C₂×D₈	—	106	1	no	—
308	C₂×D₈	—	212	4	no	—
309	C₂×D₈	—	212	4	no	—
310	C₂²×C₈	—	53	1	no	—
311	C₂²×C₈	—	53	1	no	—
312	C₂²×C₈	—	106	2	no	—
313	C₂²×C₈	—	106	1	no	—
314	#127	—	53	2	no	—
315	#127	—	106	2	no	—
316	#127	—	106	2	no	—
317	#127	—	106	2	no	—
318	#127	C₂²×D₅₃	1	2	yes	—
319	C₄×Q₈	—	53	1	no	—
320	C₄×Q₈	—	53	1	no	—
321	C₄×Q₈	—	53	1	no	—
322	C₄×Q₈	—	106	1	no	—
323	C₄×Q₈	—	106	1	no	—
324	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
325	C₂×C₄⋊C₄	—	53	1	no	—
326	C₂×C₄⋊C₄	—	106	1	no	—
327	C₂×C₄²	—	53	1	no	—
328	C₂×C₄²	—	106	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	53	2	no	—
332	C₄⋊C₈	—	106	1	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	—	106	2	no	—
335	C₄⋊C₈	—	106	1	no	—
336	C₄⋊C₈	—	106	1	no	—
337	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
338	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
339	#102	—	2	2	no	—
340	#102	—	106	2	no	—
341	#102	—	106	2	no	—
342	#102	—	106	2	no	—
343	C₄×C₈	—	2	1	no	—
344	C₄×C₈	—	53	1	no	—
345	C₄×C₈	—	53	1	no	—
346	C₄×C₈	—	53	1	no	—
347	C₄×C₈	—	106	1	no	—
348	C₄×C₈	—	106	1	no	—
349	C₄×C₈	—	106	1	no	—
350	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
351	C₂⁴	—	212	2	no	—
352	C₂×Q₈	—	53	4	no	—
353	C₂×Q₈	—	53	1	no	—
354	C₂×Q₈	—	106	1	no	—
355	C₂×Q₈	—	106	2	no	—
356	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	106	2	no	—
361	C₂×D₄	—	106	2	no	—
362	C₂×D₄	—	106	2	no	—
363	C₂×D₄	—	424	4	no	—
364	C₂×D₄	—	212	4	no	—
365	C₂×D₄	—	212	4	no	—
366	C₂×D₄	—	212	4	no	—
367	C₂×D₄	—	212	4	no	—
368	C₂×D₄	—	212	2	no	—
369	C₂×D₄	—	212	2	no	—
370	C₂×D₄	—	212	2	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	106	1	no	—
375	C₂²×C₄	—	106	1	no	—
376	SD₁₆	—	4	4	no	—
377	SD₁₆	—	212	4	no	—
378	SD₁₆	—	212	4	no	—
379	SD₁₆	—	212	4	no	—
380	D₈	—	4	2	no	—
381	D₈	—	4	2	no	—
382	D₈	—	212	2	no	—
383	D₈	—	212	4	no	—
384	D₈	—	212	4	no	—
385	D₈	—	212	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	53	1	no	—
389	C₂×C₈	—	53	4	no	—
390	C₂×C₈	—	53	1	no	—
391	C₂×C₈	—	106	2	no	—
392	C₂×C₈	—	106	1	no	—
393	C₂×C₈	—	106	2	no	—
394	C₂×C₈	—	212	4	no	—
395	C₂×C₈	—	212	2	no	—
396	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
397	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	53	1	no	—
400	C₄⋊C₄	—	106	1	no	—
401	C₄⋊C₄	—	106	1	no	—
402	C₄⋊C₄	—	106	1	no	—
403	C₄⋊C₄	—	106	1	no	—
404	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
405	C₄²	—	2	1	no	—
406	C₄²	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	—	53	1	no	—
409	C₄²	—	106	1	no	—
410	C₄²	—	106	1	no	—
411	C₄²	—	106	1	no	—
412	C₄²	C₂³×D₅₃	1	1	yes	characteristic
413	C₂³	—	4	2	no	—
414	C₂³	—	53	1	no	—
415	C₂³	—	424	4	no	—
416	C₂³	—	424	4	no	—
417	C₂³	—	424	4	no	—
418	C₂³	—	212	2	no	—
419	Q₈	—	2	1	no	—
420	Q₈	—	53	2	no	—
421	Q₈	—	106	4	no	—
422	Q₈	—	106	1	no	—
423	Q₈	D₈×D₅₃	1	2	yes	complemented
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	8	2	no	—
429	D₄	—	424	2	no	—
430	D₄	—	424	4	no	—
431	D₄	—	424	4	no	—
432	D₄	—	424	4	no	—
433	D₄	—	212	2	no	—
434	D₄	—	212	2	no	—
435	D₄	—	212	2	no	—
436	D₄	—	212	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	53	1	no	—
440	C₂×C₄	—	53	1	no	—
441	C₂×C₄	—	53	2	no	—
442	C₂×C₄	—	53	2	no	—
443	C₂×C₄	—	53	2	no	—
444	C₂×C₄	—	53	2	no	—
445	C₂×C₄	—	53	1	no	—
446	C₂×C₄	—	106	2	no	—
447	C₂×C₄	—	106	4	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	106	1	no	—
450	C₂×C₄	—	212	2	no	—
451	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
453	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
454	C₈	—	2	1	no	—
455	C₈	—	4	2	no	—
456	C₈	—	4	1	no	—
457	C₈	—	53	2	no	—
458	C₈	—	106	1	no	—
459	C₈	—	212	2	no	—
460	C₈	—	212	1	no	—
461	C₈	SD₁₆×D₅₃	1	2	yes	complemented
462	C₂²	—	8	2	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	53	2	no	—
466	C₂²	—	53	2	no	—
467	C₂²	—	53	2	no	—
468	C₂²	—	848	8	no	—
469	C₂²	—	424	2	no	—
470	C₂²	—	424	2	no	—
471	C₂²	—	424	2	no	—
472	C₂²	#121×D₅₃	1	1	yes	characteristic
473	C₄	—	2	1	no	—
474	C₄	—	2	2	no	—
475	C₄	—	4	1	no	—
476	C₄	—	53	2	no	—
477	C₄	—	53	2	no	—
478	C₄	—	106	1	no	—
479	C₄	—	106	2	no	—
480	C₄	—	212	1	no	—
481	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
482	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
483	C₄	#135×D₅₃	1	1	yes	characteristic
484	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
485	C₂	—	16	2	no	—
486	C₂	—	53	4	no	—
487	C₂	—	848	2	no	—
488	C₂	#460×D₅₃	1	1	yes	characteristic
489	C₂	#466×D₅₃	1	1	yes	characteristic
490	C₂	#472×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1668	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#493×D₅₃	C₂
hypercenter	#1668	D₅₃
hypocenter	C₅₃	C₂×#1668
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1668

Complements

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

N	H ≅ G/N	type
2. #460×D₅₃	C₂	semidirect
3. C₈×Q₈×D₅₃	C₂	semidirect
4. #20274076	C₂	semidirect
5. #20274075	C₂	semidirect
6. #20274074	C₂	semidirect
7. #20274073	C₂	semidirect
8. #20274072	C₂	semidirect
9. #335×D₅₃	C₂	semidirect
10. #331×D₅₃	C₂	semidirect
11. C₅₃×#1668	C₂	semidirect
21. #16810824	C₂²	semidirect
22. #16810823	C₂²	semidirect
25. #16810820	C₂²	semidirect
26. Q₈×#88322	C₂²	semidirect
28. C₈×Dic₁₀₆	C₂²	semidirect
30. #16810747	C₂²	semidirect
37. #16810376	C₂²	semidirect
38. #16810375	C₂²	semidirect
39. #16810367	C₂²	semidirect
40. #16810366	C₂²	semidirect
41. #16810366	C₂²	semidirect
43. C₅₃×#460	C₂²	semidirect
44. C₈×Q₈×C₅₃	C₂²	semidirect
45. C₅₃×#335	C₂²	semidirect
46. C₅₃×#331	C₂²	semidirect
114. Q₈×D₅₃	D₈	semidirect
166. C₈×D₅₃	SD₁₆	semidirect
174. Q₈×C₅₃	C₂×D₈	semidirect
207. Dic₁₀₆	C₂×D₈	semidirect
211. C₈×C₅₃	C₂×SD₁₆	semidirect
215. #88322	C₂×SD₁₆	semidirect
248. #1668	D₅₃	direct (N × H)
256. D₅₃	#1668	direct (N × H)
272. #460	C₂×D₅₃	semidirect
278. C₈×Q₈	C₂×D₅₃	semidirect
287. #335	C₂×D₅₃	semidirect
291. #331	C₂×D₅₃	semidirect
292. C₅₃	C₂×#1668	semidirect
423. Q₈	D₈×D₅₃	semidirect
461. C₈	SD₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#493×D₅₃	2
#460×D₅₃	2
C₈×Q₈×D₅₃	2
#20274076	2
#20274075	2
#20274074	2
#20274073	2
#20274072	2
#335×D₅₃	2
#331×D₅₃	2
C₅₃×#1668	2
C₂×#1668	53

Subgroups — D53×#1668 #27366416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1668 #27366416