Subgroups — #27353118

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283345	C₂	1	1	yes	maximal, characteristic
2	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273483	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273482	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273480	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273472	C₂	1	2	yes	maximal, complemented
8	#20273469	C₂	1	2	yes	maximal, complemented
9	#20273466	C₂	1	2	yes	maximal, complemented
10	#20273464	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	—	2	1	no	—
13	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
14	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#16810680	C₂²	1	1	yes	characteristic
16	#16810678	—	2	1	no	—
17	#16810675	C₂²	1	1	yes	characteristic
18	#16810675	C₂²	1	1	yes	characteristic
19	#16810673	—	2	1	no	—
20	#16810647	C₂²	1	1	yes	characteristic
21	#16810646	—	2	1	no	—
22	#16810645	—	2	1	no	—
23	#16810645	C₂²	1	1	yes	characteristic
24	#16810645	C₂²	1	2	yes	—
25	#16810645	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×D₅₃	—	2	1	no	—
27	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
28	#16810585	—	2	1	no	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	#16810575	—	2	1	no	—
31	#16810575	C₄	1	1	yes	characteristic
32	#16810575	C₄	1	1	yes	characteristic
33	M₁₆×Dic₅₃	C₄	1	2	yes	—
34	C₄×#11719912	C₂²	1	2	yes	—
35	#97×D₅₃	C₄	1	1	yes	characteristic
36	#97×D₅₃	C₄	1	1	yes	characteristic
37	C₄×C₈×D₅₃	C₂²	1	2	yes	—
38	#16810376	C₂²	1	2	yes	complemented
39	#16810375	C₂²	1	2	yes	complemented
40	#16810375	C₂²	1	2	yes	complemented
41	#16810374	C₂²	1	2	yes	complemented
42	#16810373	C₂²	1	2	yes	complemented
43	#16810373	C₂²	1	2	yes	complemented
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#423	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#335	C₂²	1	2	yes	complemented
48	C₅₃×#334	C₂²	1	2	yes	complemented
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
52	C₂×#11719917	—	2	1	no	—
53	C₂×#11719917	C₂×C₄	1	2	yes	—
54	C₂×#11719912	—	2	1	no	—
55	C₂×#11719912	—	2	1	no	—
56	C₂×#11719912	—	2	2	no	—
57	C₂×#11719912	C₂×C₄	1	2	yes	—
58	C₂×#11719912	C₂×C₄	1	2	yes	—
59	M₁₆×D₅₃	—	2	8	no	—
60	M₁₆×D₅₃	—	4	4	no	—
61	C₂×C₈×D₅₃	—	2	2	no	—
62	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₅₃	D₄	1	1	yes	characteristic
70	C₄²×D₅₃	D₄	1	1	yes	characteristic
71	C₄²×D₅₃	Q₈	1	1	yes	characteristic
72	C₄²×D₅₃	Q₈	1	1	yes	characteristic
73	C₄²×D₅₃	C₂³	1	1	yes	characteristic
74	C₄²×D₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊#88322	—	2	1	no	—
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	2	yes	—
78	C₄⋊#88322	C₂³	1	1	yes	characteristic
79	#13133146	—	2	2	no	—
80	#13133146	—	2	2	no	—
81	#13133146	—	2	1	no	—
82	#13133146	—	2	1	no	—
83	C₂²⋊#88322	—	2	1	no	—
84	#13133129	—	2	1	no	—
85	#13133129	—	2	1	no	—
86	#13133128	—	2	1	no	—
87	#13133128	C₂×C₄	1	1	yes	characteristic
88	#13133128	C₂×C₄	1	1	yes	characteristic
89	#13133127	C₂×C₄	1	2	yes	—
90	#13133127	C₂×C₄	1	2	yes	—
91	C₄×#88322	C₂³	1	2	yes	—
92	C₈×Dic₅₃	—	2	1	no	—
93	C₈×Dic₅₃	—	2	1	no	—
94	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
95	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
96	C₂×M₁₆×C₅₃	—	2	1	no	—
97	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	—	2	1	no	—
100	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
102	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²⋊C₈×C₅₃	—	2	1	no	—
104	#97×C₅₃	—	2	1	no	—
105	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
106	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	—	2	4	no	—
116	C₂×C₄×D₅₃	—	2	2	no	—
117	C₂×C₄×D₅₃	—	2	2	no	—
118	C₂×C₄×D₅₃	—	4	2	no	—
119	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
131	M₁₆×C₅₃	—	2	4	no	—
132	M₁₆×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
141	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
144	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
154	#11719917	—	2	4	no	—
155	#11719917	—	4	2	no	—
156	C₂×#88322	—	2	1	no	—
157	C₂×#88322	—	2	1	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	C₂²×C₄	1	2	yes	—
160	C₂×#88322	C₂²×C₄	1	2	yes	—
161	#11719912	—	2	4	no	—
162	#11719912	—	4	4	no	—
163	#11719912	—	4	2	no	—
164	#11719912	—	4	2	no	—
165	#11719912	M₁₆	1	8	yes	complemented
166	C₈×D₅₃	—	2	4	no	—
167	C₈×D₅₃	—	2	4	no	—
168	C₈×D₅₃	—	4	4	no	—
169	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×D₅₃	—	2	2	no	—
171	C₂²×D₅₃	—	2	2	no	—
172	C₂²×D₅₃	—	4	1	no	—
173	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
185	C₂×Dic₅₃	—	2	2	no	—
186	C₂×Dic₅₃	—	2	2	no	—
187	C₂×Dic₅₃	—	2	2	no	—
188	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₄×D₅₃	—	2	2	no	—
195	C₄×D₅₃	—	2	2	no	—
196	C₄×D₅₃	—	2	2	no	—
197	C₄×D₅₃	—	2	1	no	—
198	C₄×D₅₃	—	2	1	no	—
199	C₄×D₅₃	—	2	2	no	—
200	C₄×D₅₃	—	2	2	no	—
201	C₄×D₅₃	—	2	2	no	—
202	C₄×D₅₃	—	2	2	no	—
203	C₄×D₅₃	—	4	1	no	—
204	C₄×D₅₃	—	4	1	no	—
205	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
206	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
207	C₄×D₅₃	#137	1	2	yes	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	4	2	no	—
210	C₈×C₅₃	—	4	1	no	—
211	C₈×C₅₃	—	4	1	no	—
212	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
213	#88322	—	2	2	no	—
214	#88322	—	4	2	no	—
215	#88322	—	4	1	no	—
216	#88322	—	4	1	no	—
217	#88322	C₂×M₁₆	1	4	yes	complemented
218	#17796	—	53	1	no	maximal, Sylow
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₂×D₅₃	—	2	1	no	—
224	C₂×D₅₃	—	2	1	no	—
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	4	4	no	—
228	C₂×D₅₃	#423	1	2	yes	—
229	C₂×D₅₃	#428	1	2	yes	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
235	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
237	Dic₅₃	—	2	1	no	—
238	Dic₅₃	—	2	1	no	—
239	Dic₅₃	—	2	1	no	—
240	Dic₅₃	#423	1	2	yes	—
241	Dic₅₃	#428	1	2	yes	—
242	Dic₅₃	#430	1	2	yes	—
243	#2920	—	53	1	no	—
244	C₂×#423	—	53	1	no	—
245	C₂×C₄×M₁₆	—	53	1	no	—
246	#1569	—	53	1	no	—
247	#1569	—	53	1	no	—
248	#1569	—	53	1	no	—
249	#1569	D₅₃	1	1	yes	characteristic, complemented
250	#1567	—	53	2	no	—
251	#1566	—	53	2	no	—
252	#1565	—	53	2	no	—
253	#1565	—	53	2	no	—
254	C₂×C₅₃	—	4	1	no	—
255	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
256	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
257	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
258	D₅₃	—	2	2	no	—
259	D₅₃	—	2	2	no	—
260	C₂²×M₁₆	—	53	2	no	—
261	C₂²×M₁₆	—	106	1	no	—
262	C₂²×C₄²	—	53	1	no	—
263	#432	—	53	1	no	—
264	#432	—	106	1	no	—
265	#432	C₂×D₅₃	1	1	yes	characteristic
266	#431	—	53	1	no	—
267	#431	—	53	1	no	—
268	#431	—	106	1	no	—
269	#423	—	53	1	no	—
270	#423	—	53	1	no	—
271	#423	—	53	2	no	—
272	#423	—	53	1	no	—
273	#423	—	106	1	no	—
274	#423	—	106	1	no	—
275	#423	C₂×D₅₃	1	1	yes	characteristic
276	C₂×C₄⋊C₈	—	53	2	no	—
277	C₂×C₄⋊C₈	—	106	1	no	—
278	#407	—	106	1	no	—
279	C₄×M₁₆	—	53	2	no	—
280	C₄×M₁₆	—	53	1	no	—
281	C₄×M₁₆	—	53	1	no	—
282	C₄×M₁₆	—	53	1	no	—
283	C₄×M₁₆	—	53	2	no	—
284	C₄×M₁₆	—	106	1	no	—
285	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
286	C₂×#97	—	53	1	no	—
287	C₂×#97	—	53	1	no	—
288	C₂×C₄×C₈	—	53	2	no	—
289	#335	—	53	2	no	—
290	#335	—	53	2	no	—
291	#335	—	53	2	no	—
292	#335	C₂×D₅₃	1	2	yes	complemented
293	#334	—	53	2	no	—
294	#334	—	53	2	no	—
295	#334	—	53	2	no	—
296	#334	C₂×D₅₃	1	2	yes	complemented
297	C₅₃	#17796	1	1	yes	characteristic, complemented, Sylow
298	C₂³×C₄	—	53	1	no	—
299	C₂³×C₄	—	53	1	no	—
300	C₂³×C₄	—	53	1	no	—
301	C₂×M₁₆	—	2	1	no	—
302	C₂×M₁₆	—	53	2	no	—
303	C₂×M₁₆	—	53	2	no	—
304	C₂×M₁₆	—	53	2	no	—
305	C₂×M₁₆	—	106	2	no	—
306	C₂×M₁₆	—	106	1	no	—
307	C₂×M₁₆	—	106	1	no	—
308	C₂×M₁₆	—	106	8	no	—
309	C₂×M₁₆	—	106	1	no	—
310	C₂×M₁₆	—	212	4	no	—
311	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
312	C₂²×C₈	—	53	2	no	—
313	C₂²×C₈	—	53	2	no	—
314	C₂²×C₈	—	106	2	no	—
315	C₂×C₄²	—	53	1	no	—
316	C₂×C₄²	—	53	1	no	—
317	C₂×C₄²	—	53	1	no	—
318	C₂×C₄²	—	53	1	no	—
319	C₂×C₄²	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	1	no	—
323	C₂×C₄²	—	53	1	no	—
324	C₂×C₄²	—	53	1	no	—
325	C₂×C₄²	—	53	1	no	—
326	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	53	1	no	—
329	C₄⋊C₈	—	53	2	no	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	106	2	no	—
332	C₄⋊C₈	—	106	2	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	—	106	1	no	—
335	C₄⋊C₈	—	106	1	no	—
336	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
337	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
338	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
339	C₂²⋊C₈	—	2	1	no	—
340	C₂²⋊C₈	—	106	1	no	—
341	C₂²⋊C₈	—	106	1	no	—
342	C₂²⋊C₈	—	106	1	no	—
343	#97	—	2	1	no	—
344	#97	—	53	2	no	—
345	#97	—	53	2	no	—
346	#97	—	53	1	no	—
347	#97	—	53	1	no	—
348	#97	—	106	1	no	—
349	#97	C₄×D₅₃	1	1	yes	characteristic
350	#97	C₄×D₅₃	1	1	yes	characteristic
351	C₄×C₈	—	53	2	no	—
352	C₄×C₈	—	53	2	no	—
353	C₄×C₈	—	53	2	no	—
354	C₄×C₈	—	106	1	no	—
355	C₄×C₈	—	106	1	no	—
356	C₄×C₈	C₂²×D₅₃	1	2	yes	—
357	C₂⁴	—	53	1	no	—
358	C₂²×C₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	—	53	1	no	—
364	C₂²×C₄	—	53	1	no	—
365	C₂²×C₄	—	53	1	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	1	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	106	4	no	—
374	C₂²×C₄	—	106	2	no	—
375	C₂²×C₄	—	106	2	no	—
376	C₂²×C₄	—	212	2	no	—
377	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
378	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
379	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
380	M₁₆	—	2	4	no	—
381	M₁₆	—	4	2	no	—
382	M₁₆	—	53	8	no	—
383	M₁₆	—	106	4	no	—
384	M₁₆	—	106	4	no	—
385	M₁₆	—	212	4	no	—
386	M₁₆	—	212	2	no	—
387	M₁₆	—	212	2	no	—
388	M₁₆	—	212	2	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	53	2	no	—
393	C₂×C₈	—	53	2	no	—
394	C₂×C₈	—	106	4	no	—
395	C₂×C₈	—	106	4	no	—
396	C₂×C₈	—	106	1	no	—
397	C₂×C₈	—	106	1	no	—
398	C₂×C₈	—	106	2	no	—
399	C₂×C₈	—	212	4	no	—
400	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
401	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
402	C₄²	—	53	1	no	—
403	C₄²	—	53	1	no	—
404	C₄²	—	53	1	no	—
405	C₄²	—	53	1	no	—
406	C₄²	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	—	53	1	no	—
409	C₄²	—	53	1	no	—
410	C₄²	—	53	1	no	—
411	C₄²	—	53	1	no	—
412	C₄²	—	53	1	no	—
413	C₄²	—	53	1	no	—
414	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
415	C₄²	D₄×D₅₃	1	1	yes	characteristic
416	C₄²	Q₈×D₅₃	1	1	yes	characteristic
417	C₄²	C₂³×D₅₃	1	1	yes	characteristic
418	C₂³	—	53	1	no	—
419	C₂³	—	53	1	no	—
420	C₂³	—	106	2	no	—
421	C₂³	—	106	2	no	—
422	C₂³	—	212	1	no	—
423	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	53	1	no	—
429	C₂×C₄	—	53	1	no	—
430	C₂×C₄	—	53	1	no	—
431	C₂×C₄	—	53	4	no	—
432	C₂×C₄	—	53	2	no	—
433	C₂×C₄	—	53	2	no	—
434	C₂×C₄	—	53	1	no	—
435	C₂×C₄	—	53	1	no	—
436	C₂×C₄	—	53	1	no	—
437	C₂×C₄	—	106	2	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	106	2	no	—
440	C₂×C₄	—	106	2	no	—
441	C₂×C₄	—	106	2	no	—
442	C₂×C₄	—	106	2	no	—
443	C₂×C₄	—	106	2	no	—
444	C₂×C₄	—	106	2	no	—
445	C₂×C₄	—	106	2	no	—
446	C₂×C₄	—	106	2	no	—
447	C₂×C₄	—	106	1	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	212	1	no	—
450	C₂×C₄	—	212	1	no	—
451	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
454	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
455	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
456	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
457	C₈	—	2	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	1	no	—
460	C₈	—	4	1	no	—
461	C₈	—	53	4	no	—
462	C₈	—	106	2	no	—
463	C₈	—	212	2	no	—
464	C₈	—	212	1	no	—
465	C₈	—	212	1	no	—
466	C₈	M₁₆×D₅₃	1	4	yes	complemented
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	53	2	no	—
471	C₂²	—	53	2	no	—
472	C₂²	—	106	1	no	—
473	C₂²	—	106	1	no	—
474	C₂²	—	106	1	no	—
475	C₂²	—	106	1	no	—
476	C₂²	—	212	4	no	—
477	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	53	2	no	—
483	C₄	—	53	2	no	—
484	C₄	—	53	2	no	—
485	C₄	—	106	1	no	—
486	C₄	—	106	1	no	—
487	C₄	—	106	1	no	—
488	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
489	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
490	C₄	#137×D₅₃	1	1	yes	characteristic
491	C₂	—	4	1	no	—
492	C₂	—	106	2	no	—
493	C₂	—	106	2	no	—
494	C₂	#423×D₅₃	1	1	yes	characteristic
495	C₂	#428×D₅₃	1	1	yes	characteristic
496	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1569	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1569	D₅₃
hypocenter	C₅₃	#17796
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1569

Complements

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

N	H ≅ G/N	type
4. #20273483	C₂	semidirect
5. #20273482	C₂	semidirect
6. #20273480	C₂	semidirect
7. #20273472	C₂	semidirect
8. #20273469	C₂	semidirect
9. #20273466	C₂	semidirect
10. #20273464	C₂	semidirect
11. C₅₃×#1569	C₂	semidirect
38. #16810376	C₂²	semidirect
39. #16810375	C₂²	semidirect
40. #16810375	C₂²	semidirect
41. #16810374	C₂²	semidirect
42. #16810373	C₂²	semidirect
43. #16810373	C₂²	semidirect
47. C₅₃×#335	C₂²	semidirect
48. C₅₃×#334	C₂²	semidirect
165. #11719912	M₁₆	semidirect
212. C₈×C₅₃	C₂×M₁₆	semidirect
217. #88322	C₂×M₁₆	semidirect
249. #1569	D₅₃	semidirect
292. #335	C₂×D₅₃	semidirect
296. #334	C₂×D₅₃	semidirect
297. C₅₃	#17796	semidirect
466. C₈	M₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283345	2
#423×D₅₃	2
C₄×M₁₆×D₅₃	2
#20273483	2
#20273482	2
#20273480	2
#20273472	2
#20273469	2
#20273466	2
#20273464	2
C₅₃×#1569	2
#17796	53