Subgroups — #27353123

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 526 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	#432×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
4	#20273481	C₂	1	2	yes	maximal, complemented
5	#20273479	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273476	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273476	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273466	C₂	1	2	yes	maximal, complemented
10	#20273464	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719912	C₄	1	1	yes	characteristic
13	C₂²×#11719912	C₄	1	1	yes	characteristic
14	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#16810680	C₂²	1	1	yes	characteristic
16	#16810680	C₂²	1	1	yes	characteristic
17	#16810679	C₂²	1	2	yes	—
18	#16810678	—	2	1	no	—
19	#16810678	—	2	1	no	—
20	#16810677	—	2	1	no	—
21	#16810675	C₂²	1	2	yes	—
22	#16810673	—	2	1	no	—
23	#16810645	C₂²	1	2	yes	—
24	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
25	#16810585	—	2	1	no	—
26	C₂²⋊C₈×D₅₃	—	2	1	no	—
27	#16810575	—	2	1	no	—
28	C₄×#11719912	C₄	1	1	yes	characteristic
29	C₄×#11719912	C₄	1	1	yes	characteristic
30	C₄×#11719912	C₄	1	1	yes	characteristic
31	C₄×#11719912	C₄	1	1	yes	characteristic
32	C₄×#11719912	C₂²	1	1	yes	characteristic
33	C₄×#11719912	C₂²	1	1	yes	characteristic
34	C₄×#11719912	C₂²	1	1	yes	characteristic
35	C₄×#11719912	C₂²	1	1	yes	characteristic
36	C₂×#13133127	C₄	1	1	yes	characteristic
37	C₂×#13133127	C₄	1	1	yes	characteristic
38	#97×D₅₃	—	2	1	no	—
39	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
40	#16810376	C₂²	1	2	yes	complemented
41	#16810375	C₂²	1	2	yes	complemented
42	#16810375	C₂²	1	2	yes	complemented
43	#16810374	C₂²	1	2	yes	complemented
44	#16810373	C₂²	1	2	yes	complemented
45	#16810373	C₂²	1	2	yes	complemented
46	C₅₃×#432	C₂²	1	1	yes	characteristic
47	C₅₃×#432	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₅₃×#335	C₂²	1	2	yes	complemented
50	C₅₃×#334	C₂²	1	2	yes	complemented
51	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
54	C₂×#11719912	—	2	4	no	—
55	C₂×#11719912	—	2	4	no	—
56	C₂×#11719912	—	2	2	no	—
57	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
58	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
59	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
61	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
62	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
63	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
64	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
65	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
66	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×D₅₃	—	2	2	no	—
68	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
70	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
71	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄²×D₅₃	D₄	1	1	yes	characteristic
74	C₄²×D₅₃	D₄	1	1	yes	characteristic
75	C₄²×D₅₃	Q₈	1	1	yes	characteristic
76	C₄²×D₅₃	Q₈	1	1	yes	characteristic
77	C₄²×D₅₃	C₂³	1	1	yes	characteristic
78	C₄²×D₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊#88322	C₂³	1	2	yes	—
80	C₄⋊#88322	C₂³	1	2	yes	—
81	#13133146	—	2	2	no	—
82	#13133146	—	2	2	no	—
83	C₂²⋊#88322	—	2	1	no	—
84	C₂²⋊#88322	—	2	1	no	—
85	#13133129	—	2	2	no	—
86	#13133129	—	2	2	no	—
87	#13133128	—	2	1	no	—
88	#13133128	—	2	1	no	—
89	#13133127	—	2	2	no	—
90	#13133127	C₂×C₄	1	1	yes	characteristic
91	#13133127	C₂×C₄	1	1	yes	characteristic
92	#13133127	C₂×C₄	1	1	yes	characteristic
93	#13133127	C₂×C₄	1	1	yes	characteristic
94	#13133127	C₂×C₄	1	1	yes	characteristic
95	#13133127	C₂×C₄	1	1	yes	characteristic
96	#13133127	C₂×C₄	1	1	yes	characteristic
97	#13133127	C₂×C₄	1	1	yes	characteristic
98	C₄×#88322	C₂×C₄	1	1	yes	characteristic
99	C₄×#88322	C₂×C₄	1	1	yes	characteristic
100	C₄×#88322	C₂³	1	1	yes	characteristic
101	C₄×#88322	C₂³	1	1	yes	characteristic
102	C₈×Dic₅₃	—	2	2	no	—
103	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
107	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
108	C₂²⋊C₈×C₅₃	—	2	1	no	—
109	C₂²⋊C₈×C₅₃	—	2	1	no	—
110	#97×C₅₃	—	2	1	no	—
111	#97×C₅₃	—	2	1	no	—
112	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
115	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
116	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	—	2	2	no	—
124	C₂×C₄×D₅₃	—	2	2	no	—
125	C₂×C₄×D₅₃	—	4	2	no	—
126	C₂×C₄×D₅₃	—	4	2	no	—
127	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	2	no	—
143	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
150	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
155	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
156	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
157	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
161	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
162	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
168	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
169	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
170	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
171	#11719912	—	2	2	no	—
172	#11719912	—	2	2	no	—
173	#11719912	—	2	2	no	—
174	#11719912	—	2	2	no	—
175	#11719912	—	2	2	no	—
176	#11719912	—	2	2	no	—
177	#11719912	—	4	4	no	—
178	#11719912	M₁₆	1	4	yes	complemented
179	#11719912	M₁₆	1	4	yes	complemented
180	C₈×D₅₃	—	4	4	no	—
181	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×D₅₃	—	2	2	no	—
183	C₂²×D₅₃	—	2	2	no	—
184	C₂²×D₅₃	—	4	1	no	—
185	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
197	C₂×Dic₅₃	—	2	2	no	—
198	C₂×Dic₅₃	—	2	2	no	—
199	C₂×Dic₅₃	—	4	1	no	—
200	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄×D₅₃	—	2	2	no	—
207	C₄×D₅₃	—	2	1	no	—
208	C₄×D₅₃	—	2	1	no	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	2	2	no	—
212	C₄×D₅₃	—	4	1	no	—
213	C₄×D₅₃	—	4	1	no	—
214	C₄×D₅₃	—	4	1	no	—
215	C₄×D₅₃	—	4	1	no	—
216	C₄×D₅₃	—	4	1	no	—
217	C₄×D₅₃	—	4	1	no	—
218	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
219	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
220	C₈×C₅₃	—	2	1	no	—
221	C₈×C₅₃	—	2	1	no	—
222	C₈×C₅₃	—	4	2	no	—
223	C₈×C₅₃	—	4	2	no	—
224	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
225	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
226	#88322	—	2	1	no	—
227	#88322	—	2	1	no	—
228	#88322	—	4	2	no	—
229	#88322	—	4	2	no	—
230	#88322	C₂×M₁₆	1	2	yes	—
231	#88322	C₂×M₁₆	1	2	yes	—
232	#17797	—	53	1	no	maximal, Sylow
233	C₂²×C₅₃	—	2	1	no	—
234	C₂²×C₅₃	—	2	1	no	—
235	C₂²×C₅₃	#430	1	2	yes	—
236	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
237	C₂×D₅₃	—	2	1	no	—
238	C₂×D₅₃	—	2	1	no	—
239	C₂×D₅₃	—	2	1	no	—
240	C₂×D₅₃	—	2	1	no	—
241	C₂×D₅₃	—	2	1	no	—
242	C₂×D₅₃	—	4	4	no	—
243	C₂×D₅₃	#423	1	2	yes	—
244	C₄×C₅₃	—	2	1	no	—
245	C₄×C₅₃	—	2	1	no	—
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
249	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
250	C₄×C₅₃	#575	1	2	yes	—
251	Dic₅₃	—	2	1	no	—
252	Dic₅₃	—	2	1	no	—
253	Dic₅₃	—	2	1	no	—
254	Dic₅₃	—	2	1	no	—
255	Dic₅₃	—	2	1	no	—
256	Dic₅₃	#423	1	2	yes	—
257	#2920	—	53	1	no	—
258	C₂×#432	—	53	1	no	—
259	C₂×C₄×M₁₆	—	53	1	no	—
260	#1569	—	53	2	no	—
261	#1569	—	53	2	no	—
262	#1568	—	53	1	no	—
263	#1568	—	53	1	no	—
264	#1568	—	53	1	no	—
265	#1568	D₅₃	1	1	yes	characteristic, complemented
266	#1565	—	53	2	no	—
267	#1565	—	53	2	no	—
268	C₂×C₅₃	—	2	2	no	—
269	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
270	C₂×C₅₃	#2911	1	1	yes	characteristic
271	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
272	D₅₃	—	2	2	no	—
273	D₅₃	—	4	1	no	—
274	C₂²×M₁₆	—	53	1	no	—
275	C₂²×M₁₆	—	53	1	no	—
276	C₂²×C₄²	—	53	1	no	—
277	#432	—	53	1	no	—
278	#432	—	53	1	no	—
279	#432	—	53	2	no	—
280	#432	—	106	1	no	—
281	#432	—	106	1	no	—
282	#432	—	106	1	no	—
283	#432	C₂×D₅₃	1	1	yes	characteristic
284	#432	C₂×D₅₃	1	1	yes	characteristic
285	#431	—	53	2	no	—
286	#431	—	106	1	no	—
287	#423	—	53	2	no	—
288	C₂×C₄⋊C₈	—	53	2	no	—
289	#407	—	106	1	no	—
290	C₂×C₂²⋊C₈	—	106	1	no	—
291	C₄×M₁₆	—	53	1	no	—
292	C₄×M₁₆	—	53	1	no	—
293	C₄×M₁₆	—	53	1	no	—
294	C₄×M₁₆	—	53	1	no	—
295	C₄×M₁₆	—	53	1	no	—
296	C₄×M₁₆	—	53	1	no	—
297	C₄×M₁₆	—	53	1	no	—
298	C₄×M₁₆	—	53	1	no	—
299	C₄×M₁₆	—	106	1	no	—
300	C₂×#97	—	53	1	no	—
301	C₂×#97	—	53	1	no	—
302	C₂×#97	—	106	1	no	—
303	C₂×C₄×C₈	—	53	1	no	—
304	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
305	#335	—	53	2	no	—
306	#335	—	53	2	no	—
307	#335	—	53	2	no	—
308	#335	C₂×D₅₃	1	2	yes	complemented
309	#334	—	53	2	no	—
310	#334	—	53	2	no	—
311	#334	—	53	2	no	—
312	#334	C₂×D₅₃	1	2	yes	complemented
313	C₅₃	#17797	1	1	yes	characteristic, complemented, Sylow
314	C₂³×C₄	—	53	1	no	—
315	C₂³×C₄	—	53	1	no	—
316	C₂³×C₄	—	53	1	no	—
317	C₂×M₁₆	—	53	1	no	—
318	C₂×M₁₆	—	53	1	no	—
319	C₂×M₁₆	—	53	1	no	—
320	C₂×M₁₆	—	53	1	no	—
321	C₂×M₁₆	—	53	1	no	—
322	C₂×M₁₆	—	53	1	no	—
323	C₂×M₁₆	—	53	1	no	—
324	C₂×M₁₆	—	53	1	no	—
325	C₂×M₁₆	—	106	4	no	—
326	C₂×M₁₆	—	106	4	no	—
327	C₂×M₁₆	—	106	2	no	—
328	C₂²×C₈	—	53	1	no	—
329	C₂²×C₈	—	53	1	no	—
330	C₂²×C₈	—	106	2	no	—
331	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
332	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
333	C₂×C₄²	—	53	1	no	—
334	C₂×C₄²	—	53	1	no	—
335	C₂×C₄²	—	53	1	no	—
336	C₂×C₄²	—	53	1	no	—
337	C₂×C₄²	—	53	1	no	—
338	C₂×C₄²	—	53	1	no	—
339	C₂×C₄²	—	53	1	no	—
340	C₂×C₄²	—	53	1	no	—
341	C₂×C₄²	—	53	1	no	—
342	C₂×C₄²	—	53	1	no	—
343	C₂×C₄²	—	53	1	no	—
344	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
345	C₄⋊C₈	—	53	2	no	—
346	C₄⋊C₈	—	53	2	no	—
347	C₄⋊C₈	—	106	2	no	—
348	C₄⋊C₈	—	106	2	no	—
349	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
350	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	106	2	no	—
354	C₂²⋊C₈	—	106	2	no	—
355	C₂²⋊C₈	—	106	1	no	—
356	C₂²⋊C₈	—	106	1	no	—
357	#97	—	2	1	no	—
358	#97	—	2	1	no	—
359	#97	—	53	1	no	—
360	#97	—	53	1	no	—
361	#97	—	53	1	no	—
362	#97	—	53	1	no	—
363	#97	—	53	1	no	—
364	#97	—	53	1	no	—
365	#97	—	53	1	no	—
366	#97	—	53	1	no	—
367	#97	—	106	2	no	—
368	#97	—	106	1	no	—
369	#97	—	106	1	no	—
370	C₄×C₈	—	53	1	no	—
371	C₄×C₈	—	53	1	no	—
372	C₄×C₈	—	53	1	no	—
373	C₄×C₈	—	53	1	no	—
374	C₄×C₈	—	106	2	no	—
375	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
376	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
377	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
378	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
379	C₂⁴	—	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	—	53	1	no	—
390	C₂²×C₄	—	53	1	no	—
391	C₂²×C₄	—	53	1	no	—
392	C₂²×C₄	—	53	1	no	—
393	C₂²×C₄	—	53	1	no	—
394	C₂²×C₄	—	53	1	no	—
395	C₂²×C₄	—	106	2	no	—
396	C₂²×C₄	—	106	2	no	—
397	C₂²×C₄	—	212	2	no	—
398	C₂²×C₄	—	212	2	no	—
399	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
400	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
401	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
402	M₁₆	—	53	4	no	—
403	M₁₆	—	53	4	no	—
404	M₁₆	—	106	2	no	—
405	M₁₆	—	106	2	no	—
406	M₁₆	—	106	2	no	—
407	M₁₆	—	106	2	no	—
408	M₁₆	—	106	2	no	—
409	M₁₆	—	106	2	no	—
410	M₁₆	—	212	4	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	53	1	no	—
416	C₂×C₈	—	53	1	no	—
417	C₂×C₈	—	53	1	no	—
418	C₂×C₈	—	53	1	no	—
419	C₂×C₈	—	106	2	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	—	106	2	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	212	4	no	—
424	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
425	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
426	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
428	C₄²	—	53	1	no	—
429	C₄²	—	53	1	no	—
430	C₄²	—	53	1	no	—
431	C₄²	—	53	1	no	—
432	C₄²	—	53	1	no	—
433	C₄²	—	53	1	no	—
434	C₄²	—	53	1	no	—
435	C₄²	—	53	1	no	—
436	C₄²	—	53	1	no	—
437	C₄²	—	53	1	no	—
438	C₄²	—	53	1	no	—
439	C₄²	—	53	1	no	—
440	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
441	C₄²	D₄×D₅₃	1	1	yes	characteristic
442	C₄²	Q₈×D₅₃	1	1	yes	characteristic
443	C₄²	C₂³×D₅₃	1	1	yes	characteristic
444	C₂³	—	53	1	no	—
445	C₂³	—	53	1	no	—
446	C₂³	—	106	2	no	—
447	C₂³	—	106	2	no	—
448	C₂³	—	212	1	no	—
449	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	53	1	no	—
455	C₂×C₄	—	53	2	no	—
456	C₂×C₄	—	53	2	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	—	106	2	no	—
463	C₂×C₄	—	106	2	no	—
464	C₂×C₄	—	106	2	no	—
465	C₂×C₄	—	106	2	no	—
466	C₂×C₄	—	106	1	no	—
467	C₂×C₄	—	106	1	no	—
468	C₂×C₄	—	106	2	no	—
469	C₂×C₄	—	106	2	no	—
470	C₂×C₄	—	212	1	no	—
471	C₂×C₄	—	212	1	no	—
472	C₂×C₄	—	212	1	no	—
473	C₂×C₄	—	212	1	no	—
474	C₂×C₄	—	212	1	no	—
475	C₂×C₄	—	212	1	no	—
476	C₂×C₄	—	212	1	no	—
477	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
478	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
479	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
480	C₂×C₄	C₂²×C₄×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	C₈	—	2	1	no	—
484	C₈	—	2	1	no	—
485	C₈	—	4	2	no	—
486	C₈	—	4	2	no	—
487	C₈	—	53	2	no	—
488	C₈	—	53	2	no	—
489	C₈	—	106	1	no	—
490	C₈	—	106	1	no	—
491	C₈	—	212	2	no	—
492	C₈	—	212	2	no	—
493	C₈	M₁₆×D₅₃	1	2	yes	—
494	C₈	M₁₆×D₅₃	1	2	yes	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	53	2	no	—
498	C₂²	—	106	1	no	—
499	C₂²	—	106	1	no	—
500	C₂²	—	106	1	no	—
501	C₂²	—	106	1	no	—
502	C₂²	—	106	1	no	—
503	C₂²	—	212	4	no	—
504	C₂²	#16810670	1	2	yes	—
505	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	53	2	no	—
511	C₄	—	106	1	no	—
512	C₄	—	106	1	no	—
513	C₄	—	106	1	no	—
514	C₄	—	106	1	no	—
515	C₄	—	106	1	no	—
516	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
517	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
518	C₄	#16811420	1	2	yes	—
519	C₂	—	2	2	no	—
520	C₂	—	106	2	no	—
521	C₂	—	212	1	no	—
522	C₂	#423×D₅₃	1	1	yes	characteristic
523	C₂	#20283303	1	1	yes	characteristic
524	C₂	C₂×#16810670	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×#1568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17797
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
4. #20273481	C₂	semidirect
5. #20273479	C₂	semidirect
6. #20273478	C₂	semidirect
7. #20273476	C₂	semidirect
8. #20273476	C₂	semidirect
9. #20273466	C₂	semidirect
10. #20273464	C₂	semidirect
11. C₅₃×#1568	C₂	semidirect
40. #16810376	C₂²	semidirect
41. #16810375	C₂²	semidirect
42. #16810375	C₂²	semidirect
43. #16810374	C₂²	semidirect
44. #16810373	C₂²	semidirect
45. #16810373	C₂²	semidirect
49. C₅₃×#335	C₂²	semidirect
50. C₅₃×#334	C₂²	semidirect
178. #11719912	M₁₆	semidirect
179. #11719912	M₁₆	semidirect
265. #1568	D₅₃	semidirect
308. #335	C₂×D₅₃	semidirect
312. #334	C₂×D₅₃	semidirect
313. C₅₃	#17797	semidirect

Maximal subgroups

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

subgroup	index
#20283345	2
#432×D₅₃	2
C₂×C₄×#11719912	2
#20273481	2
#20273479	2
#20273478	2
#20273476	2
#20273466	2
#20273464	2
C₅₃×#1568	2
#17797	53