Subgroups — #27382653

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810743	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810738	C₂	1	1	yes	maximal, characteristic
3	#20283513	C₂	1	1	yes	maximal, characteristic
4	C₄×#13133223	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274622	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274600	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274595	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274561	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274557	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274554	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274550	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274528	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274525	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274401	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133237	C₂²	1	1	yes	characteristic
17	C₂×#13133223	C₂²	1	1	yes	characteristic
18	C₂×#13133223	C₂²	1	1	yes	characteristic
19	C₄×#11719945	C₂²	1	1	yes	characteristic
20	#16810743	C₄	1	8	yes	complemented
21	#16810738	C₄	1	8	yes	complemented
22	#16810733	C₄	1	2	yes	complemented
23	#16810728	C₄	1	2	yes	complemented
24	#16810723	C₄	1	2	yes	complemented
25	#16810721	C₄	1	2	yes	complemented
26	C₂×#13133152	C₂²	1	1	yes	characteristic
27	C₂×#13133152	C₂²	1	1	yes	characteristic
28	#16810659	C₂²	1	1	yes	characteristic
29	C₂×#13133144	C₂²	1	1	yes	characteristic
30	C₂×#13133144	C₂²	1	1	yes	characteristic
31	C₂×#13133143	C₂²	1	1	yes	characteristic
32	C₂×#13133143	C₂²	1	1	yes	characteristic
33	#16810619	C₂²	1	1	yes	characteristic
34	#16810617	C₂²	1	1	yes	characteristic
35	#16810578	C₂²	1	1	yes	characteristic
36	#16810578	C₂²	1	1	yes	characteristic
37	#16810577	—	2	1	no	—
38	#16810577	—	2	1	no	—
39	C₄×#11719917	C₂²	1	1	yes	characteristic
40	C₄×#11719912	—	2	1	no	—
41	C₂×#13133128	C₂²	1	1	yes	characteristic
42	C₂×#13133127	C₂²	1	1	yes	characteristic
43	C₂×#13133127	C₂²	1	1	yes	characteristic
44	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
46	C₄×C₈×D₅₃	—	2	1	no	—
47	#16810469	C₂²	1	1	yes	characteristic
48	#16810387	C₂²	1	1	yes	characteristic
49	#16810387	C₂²	1	1	yes	characteristic
50	#16810384	C₂²	1	1	yes	characteristic
51	#16810384	C₂²	1	1	yes	characteristic
52	#16810382	C₂²	1	1	yes	characteristic
53	#16810382	C₂²	1	1	yes	characteristic
54	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
56	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×#405	—	2	1	no	—
58	C₅₃×#405	—	2	1	no	—
59	C₅₃×#405	C₂²	1	1	yes	characteristic
60	C₅₃×#405	C₂²	1	1	yes	characteristic
61	C₄×M₁₆×C₅₃	—	2	1	no	—
62	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×C₄×C₈×C₅₃	—	2	1	no	—
64	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
65	C₂×#11719945	C₂³	1	1	yes	characteristic
66	#13133237	C₂×C₄	1	4	yes	—
67	#13133237	C₂×C₄	1	4	yes	—
68	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
72	#13133223	—	2	2	no	—
73	#13133223	—	2	2	no	—
74	#13133223	C₂×C₄	1	4	yes	complemented
75	#13133223	C₂×C₄	1	4	yes	complemented
76	C₂×#11719917	C₂³	1	1	yes	characteristic
77	C₂×#11719917	C₂³	1	1	yes	characteristic
78	C₂×#11719912	—	2	1	no	—
79	C₂×#11719912	—	2	1	no	—
80	C₂²×#88322	C₂³	1	1	yes	characteristic
81	C₂²×#88322	C₂³	1	1	yes	characteristic
82	C₂×C₈×D₅₃	—	2	1	no	—
83	C₂×C₈×D₅₃	—	2	1	no	—
84	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
85	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
86	C₄×C₂²⋊D₅₃	—	2	2	no	—
87	#13133176	C₂³	1	1	yes	characteristic
88	#13133174	—	2	1	no	—
89	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
91	C₄²×D₅₃	—	2	1	no	—
92	#13133152	C₂×C₄	1	4	yes	complemented
93	#13133152	C₂×C₄	1	4	yes	complemented
94	#13133150	C₂×C₄	1	2	yes	—
95	#13133148	C₂×C₄	1	2	yes	—
96	#13133144	C₂×C₄	1	4	yes	complemented
97	#13133144	C₂×C₄	1	4	yes	complemented
98	#13133143	C₂×C₄	1	4	yes	complemented
99	#13133143	C₂×C₄	1	4	yes	complemented
100	#13133141	C₂×C₄	1	2	yes	—
101	#13133140	C₂×C₄	1	2	yes	—
102	#13133138	C₂×C₄	1	2	yes	—
103	#13133137	C₂×C₄	1	2	yes	—
104	#13133128	—	2	1	no	—
105	#13133128	C₂×C₄	1	2	yes	—
106	#13133128	C₂×C₄	1	2	yes	—
107	#13133127	—	2	1	no	—
108	#13133127	—	2	1	no	—
109	#13133127	—	2	2	no	—
110	#13133127	C₂×C₄	1	4	yes	complemented
111	C₄×#88322	—	2	1	no	—
112	C₈×Dic₅₃	—	2	1	no	—
113	C₈×Dic₅₃	—	2	1	no	—
114	C₈×Dic₅₃	—	2	2	no	—
115	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
116	#13133124	—	2	2	no	—
117	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
118	#131×C₅₃	—	2	2	no	—
119	#131×C₅₃	—	2	2	no	—
120	#131×C₅₃	C₂×C₄	1	4	yes	complemented
121	#131×C₅₃	C₂×C₄	1	4	yes	complemented
122	C₂×M₁₆×C₅₃	—	2	1	no	—
123	C₂×M₁₆×C₅₃	—	2	1	no	—
124	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₈×C₅₃	—	2	1	no	—
127	C₂²×C₈×C₅₃	—	2	1	no	—
128	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
130	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
131	C₄×D₄×C₅₃	—	2	2	no	—
132	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
133	#117×C₅₃	—	2	1	no	—
134	#117×C₅₃	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₅₃	—	2	1	no	—
136	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
137	#97×C₅₃	—	2	1	no	—
138	#97×C₅₃	—	2	1	no	—
139	#97×C₅₃	—	2	1	no	—
140	C₄×C₈×C₅₃	—	2	1	no	—
141	C₄×C₈×C₅₃	—	2	1	no	—
142	C₄×C₈×C₅₃	—	2	1	no	—
143	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
144	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
145	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
146	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
147	#41×C₅₃	—	2	1	no	—
148	#41×C₅₃	—	2	1	no	—
149	#41×C₅₃	C₂²×C₄	1	2	yes	—
150	#41×C₅₃	C₂²×C₄	1	2	yes	—
151	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₅₃	—	2	1	no	—
153	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	#41	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂×C₂²⋊D₅₃	—	2	1	no	—
159	C₂²×Dic₅₃	—	2	1	no	—
160	C₂²×Dic₅₃	—	2	1	no	—
161	#11719945	—	2	1	no	—
162	#11719945	—	2	1	no	—
163	#11719945	C₂²×C₄	1	2	yes	—
164	#11719945	C₂²×C₄	1	2	yes	—
165	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₄×D₅₃	—	2	1	no	—
167	C₂×C₄×D₅₃	—	2	2	no	—
168	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
169	M₁₆×C₅₃	—	2	1	no	—
170	M₁₆×C₅₃	—	2	1	no	—
171	M₁₆×C₅₃	—	2	4	no	—
172	M₁₆×C₅₃	—	2	4	no	—
173	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
174	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	—	2	4	no	—
178	C₂×C₈×C₅₃	—	2	4	no	—
179	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
180	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
181	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₅₃	#41	1	2	yes	—
186	C₂×C₈×C₅₃	#41	1	2	yes	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
189	C₂²⋊C₄×C₅₃	—	2	1	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₄²×C₅₃	—	2	2	no	—
192	C₄²×C₅₃	—	2	1	no	—
193	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
194	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊Dic₅₃	—	2	1	no	—
197	#11719925	C₄²	1	4	yes	—
198	#11719924	C₄²	1	4	yes	—
199	#11719923	C₄²	1	4	yes	—
200	#11719922	C₄²	1	4	yes	—
201	#11719921	—	2	2	no	—
202	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
203	#11719919	—	2	2	no	—
204	C₄×Dic₅₃	—	2	2	no	—
205	C₄×Dic₅₃	—	2	1	no	—
206	C₄×Dic₅₃	—	2	1	no	—
207	C₄×Dic₅₃	—	2	1	no	—
208	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
209	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
210	#11719917	—	2	1	no	—
211	#11719917	—	2	1	no	—
212	#11719917	C₂²×C₄	1	2	yes	—
213	#11719917	C₂²×C₄	1	2	yes	—
214	C₂×#88322	—	2	1	no	—
215	C₂×#88322	—	2	1	no	—
216	C₂×#88322	—	2	1	no	—
217	C₂×#88322	—	2	1	no	—
218	C₂×#88322	C₂²×C₄	1	2	yes	—
219	C₂×#88322	C₂²×C₄	1	2	yes	—
220	C₂×#88322	C₂²×C₄	1	2	yes	—
221	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
222	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
223	#11719912	—	2	4	no	—
224	#11719912	—	2	4	no	—
225	C₈×D₅₃	—	2	4	no	—
226	C₈×D₅₃	—	2	4	no	—
227	C₂³×C₅₃	—	2	1	no	—
228	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
229	C₂²×D₅₃	—	2	1	no	—
230	Q₈×C₅₃	—	2	1	no	—
231	Q₈×C₅₃	C₂×C₄²	1	2	yes	—
232	D₄×C₅₃	—	2	2	no	—
233	D₄×C₅₃	—	2	2	no	—
234	D₄×C₅₃	—	2	1	no	—
235	D₄×C₅₃	C₂×C₄²	1	2	yes	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	4	2	no	—
241	C₂×C₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
250	C₂²⋊D₅₃	—	2	2	no	—
251	C₂²⋊D₅₃	—	2	2	no	—
252	C₂×Dic₅₃	—	2	1	no	—
253	C₂×Dic₅₃	—	2	2	no	—
254	C₂×Dic₅₃	—	2	2	no	—
255	C₂×Dic₅₃	—	2	2	no	—
256	C₂×Dic₅₃	—	4	2	no	—
257	C₂×Dic₅₃	—	4	2	no	—
258	D₂₁₂	—	2	1	no	—
259	D₂₁₂	C₂×C₄²	1	2	yes	—
260	C₄×D₅₃	—	2	2	no	—
261	C₄×D₅₃	—	2	2	no	—
262	C₄×D₅₃	—	4	4	no	—
263	Dic₁₀₆	—	2	1	no	—
264	Dic₁₀₆	C₂×C₄²	1	2	yes	—
265	C₈×C₅₃	—	2	2	no	—
266	C₈×C₅₃	—	2	2	no	—
267	C₈×C₅₃	C₄×D₄	1	2	yes	—
268	C₈×C₅₃	C₄×D₄	1	2	yes	—
269	C₈×C₅₃	C₄×D₄	1	2	yes	—
270	C₈×C₅₃	C₄×D₄	1	2	yes	—
271	#88322	—	2	2	no	—
272	#88322	—	2	2	no	—
273	#88322	—	2	2	no	—
274	#88322	C₂×C₄²	1	2	yes	—
275	#88322	C₂×C₄²	1	2	yes	—
276	#20510	—	53	1	no	maximal, Sylow
277	C₂²×C₅₃	—	2	2	no	—
278	C₂²×C₅₃	—	2	2	no	—
279	C₂²×C₅₃	—	4	2	no	—
280	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
282	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
283	C₂×D₅₃	—	2	2	no	—
284	C₂×D₅₃	—	4	2	no	—
285	C₄×C₅₃	—	2	2	no	—
286	C₄×C₅₃	—	2	2	no	—
287	C₄×C₅₃	—	2	2	no	—
288	C₄×C₅₃	—	4	2	no	—
289	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
290	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
291	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
292	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
293	Dic₅₃	—	2	2	no	—
294	Dic₅₃	—	4	2	no	—
295	Dic₅₃	—	4	2	no	—
296	Dic₅₃	—	4	2	no	—
297	C₂×#444	—	53	1	no	—
298	C₂×#443	—	53	1	no	—
299	#2943	—	53	1	no	—
300	C₄×#131	—	53	1	no	—
301	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
302	#1778	—	53	1	no	—
303	#1773	—	53	1	no	—
304	#1772	—	53	1	no	—
305	#1765	—	53	1	no	—
306	#1765	—	53	1	no	—
307	#1764	—	53	1	no	—
308	#1764	—	53	1	no	—
309	#1759	—	53	1	no	—
310	#1759	—	53	1	no	—
311	#1726	—	53	1	no	—
312	C₂×C₅₃	—	2	1	no	—
313	C₂×C₅₃	—	2	1	no	—
314	C₂×C₅₃	—	4	2	no	—
315	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
316	C₂×C₅₃	#2955	1	2	yes	—
317	D₅₃	—	4	2	no	—
318	C₂×#135	—	53	1	no	—
319	C₂×#131	—	53	1	no	—
320	C₂×#131	—	53	1	no	—
321	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
322	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
323	C₄×#41	—	53	1	no	—
324	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
325	#444	—	53	8	no	—
326	#443	—	53	8	no	—
327	#442	—	53	2	no	—
328	#441	—	53	2	no	—
329	#440	—	53	2	no	—
330	#440	—	53	2	no	—
331	C₂×#108	—	53	1	no	—
332	C₂×#108	—	53	1	no	—
333	#427	—	53	1	no	—
334	C₂×#104	—	53	1	no	—
335	C₂×#104	—	53	1	no	—
336	C₂×#104	—	53	1	no	—
337	C₂×#104	—	53	1	no	—
338	#416	—	53	1	no	—
339	#416	—	53	1	no	—
340	#405	—	2	1	no	—
341	#405	—	2	1	no	—
342	#405	—	53	1	no	—
343	#405	—	53	1	no	—
344	#405	—	106	1	no	—
345	#405	—	106	1	no	—
346	#405	C₂×D₅₃	1	1	yes	characteristic
347	#405	C₂×D₅₃	1	1	yes	characteristic
348	C₄×M₁₆	—	2	1	no	—
349	C₄×M₁₆	—	53	1	no	—
350	C₄×M₁₆	—	106	1	no	—
351	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
352	C₂×#97	—	53	1	no	—
353	C₂×#97	—	53	1	no	—
354	C₂×#97	—	53	1	no	—
355	C₂×C₄×C₈	—	2	1	no	—
356	C₂×C₄×C₈	—	53	1	no	—
357	C₂×C₄×C₈	—	53	1	no	—
358	C₂×C₄×C₈	—	106	1	no	—
359	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
360	#376	—	53	1	no	—
361	#341	—	53	1	no	—
362	#341	—	53	1	no	—
363	#339	—	53	1	no	—
364	#339	—	53	1	no	—
365	#339	—	53	1	no	—
366	#339	—	53	1	no	—
367	C₅₃	#20510	1	1	yes	characteristic, complemented, Sylow
368	C₂×#41	—	53	1	no	—
369	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
370	#135	—	53	4	no	—
371	#135	—	53	4	no	—
372	C₂×Q₁₆	—	53	1	no	—
373	C₂×SD₁₆	—	53	1	no	—
374	C₂×SD₁₆	—	53	1	no	—
375	C₂×D₈	—	53	1	no	—
376	#131	—	2	2	no	—
377	#131	—	2	2	no	—
378	#131	—	53	4	no	—
379	#131	—	53	4	no	—
380	#131	—	106	2	no	—
381	#131	—	106	2	no	—
382	#131	C₄×D₅₃	1	4	yes	complemented
383	#131	C₄×D₅₃	1	4	yes	complemented
384	C₂×M₁₆	—	2	1	no	—
385	C₂×M₁₆	—	2	1	no	—
386	C₂×M₁₆	—	53	1	no	—
387	C₂×M₁₆	—	53	1	no	—
388	C₂×M₁₆	—	106	1	no	—
389	C₂×M₁₆	—	106	1	no	—
390	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
391	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
392	C₂²×C₈	—	2	1	no	—
393	C₂²×C₈	—	2	1	no	—
394	C₂²×C₈	—	53	1	no	—
395	C₂²×C₈	—	53	1	no	—
396	C₂²×C₈	—	106	1	no	—
397	C₂²×C₈	—	106	1	no	—
398	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
399	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
400	C₄×Q₈	—	53	2	no	—
401	C₄×Q₈	C₄×D₅₃	1	2	yes	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	53	2	no	—
404	C₄×D₄	—	106	2	no	—
405	C₄×D₄	C₄×D₅₃	1	2	yes	—
406	#117	—	2	1	no	—
407	#117	—	53	1	no	—
408	#117	—	106	1	no	—
409	#117	C₂²×D₅₃	1	1	yes	characteristic
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	53	1	no	—
412	C₂×C₄²	—	53	1	no	—
413	C₂×C₄²	—	106	1	no	—
414	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
415	#108	—	53	4	no	—
416	#108	—	53	4	no	—
417	#107	—	53	2	no	—
418	#106	—	53	2	no	—
419	#104	—	53	4	no	—
420	#104	—	53	4	no	—
421	#104	—	53	4	no	—
422	#104	—	53	4	no	—
423	#103	—	53	2	no	—
424	#103	—	53	2	no	—
425	#102	—	53	2	no	—
426	#102	—	53	2	no	—
427	#97	—	2	1	no	—
428	#97	—	2	1	no	—
429	#97	—	2	1	no	—
430	#97	—	53	4	no	—
431	#97	—	53	2	no	—
432	#97	—	53	2	no	—
433	#97	—	106	2	no	—
434	#97	—	106	1	no	—
435	#97	—	106	1	no	—
436	#97	—	106	1	no	—
437	C₄×C₈	—	2	1	no	—
438	C₄×C₈	—	2	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	—	53	4	no	—
441	C₄×C₈	—	106	2	no	—
442	C₄×C₈	—	106	1	no	—
443	C₄×C₈	—	106	1	no	—
444	C₄×C₈	—	106	1	no	—
445	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
446	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
447	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
448	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
449	#95	—	106	2	no	—
450	#41	—	2	1	no	—
451	#41	—	2	1	no	—
452	#41	—	53	2	no	—
453	#41	—	53	2	no	—
454	#41	—	106	1	no	—
455	#41	—	106	1	no	—
456	#41	C₂×C₄×D₅₃	1	2	yes	—
457	#41	C₂×C₄×D₅₃	1	2	yes	—
458	C₂×Q₈	—	53	1	no	—
459	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	53	1	no	—
462	C₂×D₄	—	106	1	no	—
463	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	106	1	no	—
467	C₂²×C₄	—	106	1	no	—
468	C₂²×C₄	—	106	1	no	—
469	C₂²×C₄	—	106	2	no	—
470	C₂²×C₄	#11719945	1	2	yes	—
471	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
472	Q₁₆	—	53	4	no	—
473	SD₁₆	—	53	4	no	—
474	SD₁₆	—	53	4	no	—
475	D₈	—	53	4	no	—
476	M₁₆	—	2	1	no	—
477	M₁₆	—	2	1	no	—
478	M₁₆	—	2	4	no	—
479	M₁₆	—	2	4	no	—
480	M₁₆	—	53	2	no	—
481	M₁₆	—	53	2	no	—
482	M₁₆	—	106	1	no	—
483	M₁₆	—	106	1	no	—
484	M₁₆	—	106	4	no	—
485	M₁₆	—	106	4	no	—
486	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
487	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	4	no	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	53	2	no	—
493	C₂×C₈	—	53	1	no	—
494	C₂×C₈	—	53	1	no	—
495	C₂×C₈	—	53	2	no	—
496	C₂×C₈	—	53	2	no	—
497	C₂×C₈	—	106	1	no	—
498	C₂×C₈	—	106	1	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	4	no	—
502	C₂×C₈	—	106	4	no	—
503	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
504	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
505	C₂×C₈	#11719945	1	2	yes	—
506	C₂×C₈	#11719945	1	2	yes	—
507	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
508	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
509	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
510	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	106	2	no	—
514	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	106	1	no	—
518	C₂²⋊C₄	—	106	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	53	2	no	—
522	C₄²	—	53	2	no	—
523	C₄²	—	106	2	no	—
524	C₄²	—	106	1	no	—
525	C₄²	—	106	1	no	—
526	C₄²	—	106	1	no	—
527	C₄²	C₂×C₄×D₅₃	1	2	yes	—
528	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
529	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
530	C₂³	—	2	1	no	—
531	C₂³	—	106	1	no	—
532	C₂³	C₂×#11719945	1	1	yes	characteristic
533	Q₈	—	2	1	no	—
534	Q₈	—	53	2	no	—
535	Q₈	—	106	1	no	—
536	Q₈	C₄²×D₅₃	1	2	yes	—
537	D₄	—	2	2	no	—
538	D₄	—	2	2	no	—
539	D₄	—	2	1	no	—
540	D₄	—	53	2	no	—
541	D₄	—	106	2	no	—
542	D₄	—	106	2	no	—
543	D₄	—	106	1	no	—
544	D₄	C₄²×D₅₃	1	2	yes	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	106	1	no	—
552	C₂×C₄	—	106	2	no	—
553	C₂×C₄	—	106	2	no	—
554	C₂×C₄	—	106	2	no	—
555	C₂×C₄	—	106	2	no	—
556	C₂×C₄	—	106	2	no	—
557	C₂×C₄	—	212	4	no	—
558	C₂×C₄	—	212	2	no	—
559	C₂×C₄	—	212	2	no	—
560	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
561	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
562	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
563	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
564	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
565	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
566	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
567	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	53	2	no	—
571	C₈	—	53	2	no	—
572	C₈	—	106	2	no	—
573	C₈	—	106	2	no	—
574	C₈	—	106	2	no	—
575	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
576	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
577	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
578	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	106	2	no	—
583	C₂²	—	212	2	no	—
584	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
585	C₂²	C₄×#11719945	1	1	yes	characteristic
586	C₂²	C₄×#11719945	1	1	yes	characteristic
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	4	2	no	—
591	C₄	—	106	2	no	—
592	C₄	—	212	2	no	—
593	C₄	—	212	2	no	—
594	C₄	—	212	2	no	—
595	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
596	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
597	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
598	C₄	C₄×#11719945	1	1	yes	characteristic
599	C₂	—	2	1	no	—
600	C₂	—	2	1	no	—
601	C₂	—	4	2	no	—
602	C₂	—	212	2	no	—
603	C₂	C₄²×C₂²⋊D₅₃	1	1	yes	characteristic
604	C₂	#20283616	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#20510
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
4. C₄×#13133223	C₂	semidirect
5. #20274622	C₂	semidirect
6. #20274600	C₂	semidirect
7. #20274595	C₂	semidirect
8. #20274561	C₂	semidirect
9. #20274557	C₂	semidirect
10. #20274554	C₂	semidirect
11. #20274550	C₂	semidirect
12. #20274528	C₂	semidirect
13. #20274525	C₂	semidirect
14. #20274401	C₂	semidirect
15. C₄×#131×C₅₃	C₂	semidirect
20. #16810743	C₄	semidirect
21. #16810738	C₄	semidirect
22. #16810733	C₄	semidirect
23. #16810728	C₄	semidirect
24. #16810723	C₄	semidirect
25. #16810721	C₄	semidirect
74. #13133223	C₂×C₄	semidirect
75. #13133223	C₂×C₄	semidirect
92. #13133152	C₂×C₄	semidirect
93. #13133152	C₂×C₄	semidirect
96. #13133144	C₂×C₄	semidirect
97. #13133144	C₂×C₄	semidirect
98. #13133143	C₂×C₄	semidirect
99. #13133143	C₂×C₄	semidirect
110. #13133127	C₂×C₄	semidirect
115. C₈×Dic₅₃	C₂×C₄	semidirect
120. #131×C₅₃	C₂×C₄	semidirect
121. #131×C₅₃	C₂×C₄	semidirect
301. C₄×#131	D₅₃	semidirect
367. C₅₃	#20510	semidirect
382. #131	C₄×D₅₃	semidirect
383. #131	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810743	2
C₂×#16810738	2
#20283513	2
C₄×#13133223	2
#20274622	2
#20274600	2
#20274595	2
#20274561	2
#20274557	2
#20274554	2
#20274550	2
#20274528	2
#20274525	2
#20274401	2
C₄×#131×C₅₃	2
#20510	53