Subgroups — #27382662

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 608 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×#16810742	C₂	1	1	yes	maximal, characteristic
3	#20283483	C₂	1	1	yes	maximal, characteristic
4	#20283137	C₂	1	1	yes	maximal, characteristic, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274616	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274600	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274598	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274562	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274559	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274558	C₂	1	1	yes	maximal, characteristic, complemented
12	#104×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274519	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1765	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133237	C₂²	1	1	yes	characteristic
17	C₂×#13133225	C₂²	1	1	yes	characteristic
18	C₂×#13133224	C₂²	1	1	yes	characteristic
19	C₂×#13133223	C₂²	1	1	yes	characteristic
20	#41×Dic₅₃	C₂²	1	1	yes	characteristic
21	#16811091	C₂²	1	1	yes	characteristic
22	#16810743	C₄	1	8	yes	complemented
23	#16810742	C₄	1	8	yes	complemented
24	#16810715	C₄	1	2	yes	complemented
25	#16810714	C₄	1	2	yes	complemented
26	#16810708	C₄	1	2	yes	complemented
27	#16810703	C₄	1	2	yes	complemented
28	C₂×#13133152	C₂²	1	1	yes	characteristic
29	C₂×#13133152	C₂²	1	1	yes	characteristic
30	#16810659	C₂²	1	1	yes	characteristic
31	C₂×#13133145	C₂²	1	1	yes	characteristic
32	C₂×#13133144	C₂²	1	1	yes	characteristic
33	C₂×#13133143	C₂²	1	1	yes	characteristic
34	#16810620	C₂²	1	1	yes	characteristic
35	#16810581	—	2	1	no	—
36	#16810581	C₂²	1	1	yes	characteristic
37	#16810580	—	2	1	no	—
38	#16810579	C₂²	1	1	yes	characteristic
39	#16810578	—	2	1	no	—
40	#16810578	C₂²	1	1	yes	characteristic
41	#16810577	—	2	1	no	—
42	#16810575	—	2	1	no	—
43	M₁₆×Dic₅₃	—	2	1	no	—
44	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
45	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
46	C₂×#13133128	C₂²	1	1	yes	characteristic
47	C₂×#13133127	C₂²	1	1	yes	characteristic
48	C₂×#13133127	C₂²	1	1	yes	characteristic
49	C₂×C₈×Dic₅₃	—	2	1	no	—
50	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₄×C₈×D₅₃	—	2	1	no	—
53	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
54	#16810388	C₂²	1	1	yes	characteristic
55	#16810384	C₂²	1	1	yes	characteristic
56	#16810383	C₂²	1	1	yes	characteristic
57	#16810382	C₂²	1	1	yes	characteristic
58	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
60	C₅₃×#416	C₂²	1	1	yes	characteristic
61	C₅₃×#405	C₂²	1	1	yes	characteristic
62	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
63	C₅₃×#341	C₂²	1	1	yes	characteristic
64	C₅₃×#339	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	#13133225	—	2	2	no	—
73	#13133225	C₂×C₄	1	4	yes	complemented
74	#13133224	—	2	2	no	—
75	#13133224	C₂×C₄	1	4	yes	complemented
76	#13133223	—	2	2	no	—
77	#13133223	C₂×C₄	1	4	yes	complemented
78	C₂×#11719917	—	2	1	no	—
79	C₂×#11719917	C₂³	1	1	yes	characteristic
80	C₂×#11719917	C₂³	1	1	yes	characteristic
81	C₂×#11719912	—	2	1	no	—
82	C₂×#11719912	—	2	1	no	—
83	C₂²×#88322	—	2	1	no	—
84	C₂²×#88322	C₂³	1	1	yes	characteristic
85	C₂²×#88322	C₂³	1	1	yes	characteristic
86	C₂×C₈×D₅₃	—	2	1	no	—
87	C₂×C₈×D₅₃	—	2	1	no	—
88	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
89	#13133182	C₂×C₄	1	2	yes	—
90	D₄×Dic₅₃	—	2	2	no	—
91	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
92	#13133179	C₂×C₄	1	2	yes	—
93	#13133177	—	2	2	no	—
94	#13133176	—	2	1	no	—
95	#13133176	C₂³	1	1	yes	characteristic
96	#13133174	—	2	1	no	—
97	C₂×C₄×Dic₅₃	—	2	1	no	—
98	C₂×C₄×Dic₅₃	—	2	2	no	—
99	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
101	C₄²×D₅₃	—	2	1	no	—
102	#13133152	C₂×C₄	1	4	yes	complemented
103	#13133152	C₂×C₄	1	4	yes	complemented
104	#13133150	C₂×C₄	1	2	yes	—
105	#13133148	C₂×C₄	1	2	yes	—
106	#13133145	C₂×C₄	1	4	yes	complemented
107	#13133144	C₂×C₄	1	4	yes	complemented
108	#13133143	C₂×C₄	1	4	yes	complemented
109	#13133142	C₂×C₄	1	2	yes	—
110	#13133139	C₂×C₄	1	2	yes	—
111	#13133128	—	2	1	no	—
112	#13133128	C₂×C₄	1	4	yes	complemented
113	#13133127	—	2	1	no	—
114	#13133127	—	2	1	no	—
115	#13133127	—	2	1	no	—
116	#13133127	—	2	1	no	—
117	#13133127	—	2	2	no	—
118	#13133127	C₂×C₄	1	4	yes	complemented
119	C₄×#88322	—	2	1	no	—
120	C₈×Dic₅₃	—	2	1	no	—
121	C₈×Dic₅₃	—	2	1	no	—
122	C₈×Dic₅₃	—	2	1	no	—
123	C₈×Dic₅₃	—	2	1	no	—
124	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
125	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
126	C₈×Dic₅₃	D₄	1	1	yes	characteristic
127	C₈×Dic₅₃	D₄	1	1	yes	characteristic
128	C₈×Dic₅₃	D₄	1	1	yes	characteristic
129	C₈×Dic₅₃	D₄	1	1	yes	characteristic
130	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
131	#131×C₅₃	—	2	2	no	—
132	#131×C₅₃	C₂×C₄	1	4	yes	complemented
133	C₂×M₁₆×C₅₃	—	2	1	no	—
134	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
135	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
136	C₂²×C₈×C₅₃	—	2	1	no	—
137	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
138	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	#117×C₅₃	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
141	#104×C₅₃	C₂×C₄	1	4	yes	complemented
142	#103×C₅₃	C₂×C₄	1	2	yes	—
143	#102×C₅₃	C₂×C₄	1	2	yes	—
144	#97×C₅₃	—	2	1	no	—
145	#97×C₅₃	—	2	2	no	—
146	C₄×C₈×C₅₃	—	2	1	no	—
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	1	no	—
156	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂×C₂²⋊D₅₃	—	2	1	no	—
158	C₂²×Dic₅₃	—	2	1	no	—
159	C₂²×Dic₅₃	—	2	2	no	—
160	C₂²×Dic₅₃	#41	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
173	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	—	2	1	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	#41	1	2	yes	—
184	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
185	C₂²⋊C₄×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	2	1	no	—
187	C₄²×C₅₃	—	2	1	no	—
188	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
189	C₂²⋊Dic₅₃	—	2	1	no	—
190	C₂²⋊Dic₅₃	—	2	2	no	—
191	#11719925	C₄²	1	4	yes	complemented
192	#11719924	C₄²	1	4	yes	complemented
193	#11719923	C₄²	1	4	yes	complemented
194	#11719922	C₄²	1	4	yes	complemented
195	#11719921	—	2	2	no	—
196	C₄⋊Dic₅₃	—	2	2	no	—
197	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
198	#11719919	—	2	2	no	—
199	C₄×Dic₅₃	—	2	1	no	—
200	C₄×Dic₅₃	—	2	1	no	—
201	C₄×Dic₅₃	—	2	2	no	—
202	C₄×Dic₅₃	—	2	2	no	—
203	C₄×Dic₅₃	—	4	4	no	—
204	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
205	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
206	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
207	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
208	#11719917	—	2	1	no	—
209	#11719917	—	2	1	no	—
210	#11719917	—	2	4	no	—
211	#11719917	C₂²×C₄	1	2	yes	—
212	#11719917	C₂²×C₄	1	2	yes	—
213	C₂×#88322	—	2	1	no	—
214	C₂×#88322	—	2	1	no	—
215	C₂×#88322	—	2	4	no	—
216	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
217	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
222	C₂×#88322	C₂×D₄	1	1	yes	characteristic
223	C₂×#88322	#41	1	2	yes	—
224	#11719912	—	2	4	no	—
225	#11719912	—	2	4	no	—
226	C₈×D₅₃	—	2	4	no	—
227	C₈×D₅₃	—	2	4	no	—
228	C₂³×C₅₃	—	2	1	no	—
229	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
230	C₂²×D₅₃	—	2	1	no	—
231	Q₈×C₅₃	—	2	1	no	—
232	Q₈×C₅₃	C₂×C₄²	1	2	yes	—
233	D₄×C₅₃	—	2	2	no	—
234	D₄×C₅₃	—	2	2	no	—
235	D₄×C₅₃	—	2	1	no	—
236	D₄×C₅₃	C₂×C₄²	1	2	yes	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
249	C₂²⋊D₅₃	—	2	2	no	—
250	C₂²⋊D₅₃	—	2	2	no	—
251	C₂×Dic₅₃	—	2	1	no	—
252	C₂×Dic₅₃	—	2	4	no	—
253	C₂×Dic₅₃	—	2	2	no	—
254	C₂×Dic₅₃	—	2	2	no	—
255	C₂×Dic₅₃	—	4	2	no	—
256	C₂×Dic₅₃	—	4	4	no	—
257	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
258	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
259	D₂₁₂	—	2	1	no	—
260	D₂₁₂	C₂×C₄²	1	2	yes	—
261	C₄×D₅₃	—	2	2	no	—
262	C₄×D₅₃	—	2	2	no	—
263	C₄×D₅₃	—	4	4	no	—
264	Dic₁₀₆	—	2	1	no	—
265	Dic₁₀₆	C₂×C₄²	1	2	yes	—
266	C₈×C₅₃	—	2	2	no	—
267	C₈×C₅₃	—	2	2	no	—
268	C₈×C₅₃	—	2	2	no	—
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	C₂×C₄²	1	2	yes	—
274	#88322	C₂×C₄²	1	2	yes	—
275	#88322	C₄×D₄	1	2	yes	—
276	#88322	C₄×D₄	1	2	yes	—
277	C₄×#444	—	53	1	no	maximal, Sylow
278	C₂²×C₅₃	—	2	2	no	—
279	C₂²×C₅₃	—	2	2	no	—
280	C₂²×C₅₃	—	4	2	no	—
281	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
282	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
283	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
284	C₂×D₅₃	—	2	2	no	—
285	C₂×D₅₃	—	4	2	no	—
286	C₄×C₅₃	—	2	2	no	—
287	C₄×C₅₃	—	4	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	—	2	2	no	—
295	Dic₅₃	—	4	2	no	—
296	Dic₅₃	—	4	2	no	—
297	Dic₅₃	#444	1	4	yes	complemented
298	C₂×#444	—	53	1	no	—
299	C₂×#444	—	53	1	no	—
300	C₄×#135	—	53	1	no	—
301	C₄×#131	—	53	1	no	—
302	C₄×#131	—	53	1	no	—
303	C₄×#108	—	53	1	no	—
304	#1773	—	53	1	no	—
305	#1773	—	53	1	no	—
306	#1765	—	53	1	no	—
307	#1765	—	53	1	no	—
308	#1765	—	53	1	no	—
309	#1765	D₅₃	1	1	yes	characteristic, complemented
310	C₄×#104	—	53	1	no	—
311	C₄×#104	—	53	1	no	—
312	C₄×#97	—	53	1	no	—
313	C₂×C₅₃	—	2	1	no	—
314	C₂×C₅₃	—	2	1	no	—
315	C₂×C₅₃	—	4	2	no	—
316	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
317	C₂×C₅₃	C₂×#444	1	2	yes	—
318	D₅₃	—	4	2	no	—
319	C₂×#135	—	53	1	no	—
320	C₂×#131	—	53	1	no	—
321	C₂×#131	—	53	1	no	—
322	C₂×#131	—	53	1	no	—
323	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
324	C₄×#41	—	53	1	no	—
325	C₄×#41	—	53	1	no	—
326	#444	—	53	8	no	—
327	#444	—	53	8	no	—
328	C₄×Q₁₆	—	53	2	no	—
329	C₄×SD₁₆	—	53	2	no	—
330	C₄×SD₁₆	—	53	2	no	—
331	C₄×D₈	—	53	2	no	—
332	C₂×#108	—	53	1	no	—
333	C₂×#108	—	53	1	no	—
334	#427	—	53	1	no	—
335	C₂×#104	—	53	1	no	—
336	C₂×#104	—	53	1	no	—
337	C₂×#104	—	53	1	no	—
338	C₂×#104	C₂×D₅₃	1	1	yes	characteristic
339	#416	—	53	1	no	—
340	#416	C₂×D₅₃	1	1	yes	characteristic
341	#405	—	53	1	no	—
342	#405	—	53	1	no	—
343	#405	—	53	1	no	—
344	#405	—	106	1	no	—
345	#405	—	106	1	no	—
346	#405	—	106	1	no	—
347	#405	—	106	1	no	—
348	#405	C₂×D₅₃	1	1	yes	characteristic
349	C₄×M₁₆	—	53	1	no	—
350	C₄×M₁₆	—	53	1	no	—
351	C₄×M₁₆	—	106	1	no	—
352	C₄×M₁₆	—	106	1	no	—
353	C₂×#97	—	53	1	no	—
354	C₂×#97	—	53	1	no	—
355	C₂×#97	—	53	1	no	—
356	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
357	C₂×C₄×C₈	—	53	1	no	—
358	C₂×C₄×C₈	—	53	1	no	—
359	C₂×C₄×C₈	—	106	1	no	—
360	C₂×C₄×C₈	—	106	1	no	—
361	C₄³	—	53	1	no	—
362	#341	—	53	1	no	—
363	#341	C₂×D₅₃	1	1	yes	characteristic
364	#339	—	53	1	no	—
365	#339	—	53	1	no	—
366	#339	—	53	1	no	—
367	#339	C₂×D₅₃	1	1	yes	characteristic
368	C₅₃	C₄×#444	1	1	yes	characteristic, complemented, Sylow
369	C₂×#41	—	53	1	no	—
370	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
371	#135	—	53	4	no	—
372	#135	—	53	4	no	—
373	C₂×Q₁₆	—	53	1	no	—
374	C₂×SD₁₆	—	53	1	no	—
375	C₂×SD₁₆	—	53	1	no	—
376	C₂×D₈	—	53	1	no	—
377	#131	—	2	2	no	—
378	#131	—	53	4	no	—
379	#131	—	53	4	no	—
380	#131	—	53	4	no	—
381	#131	—	106	2	no	—
382	#131	—	106	2	no	—
383	#131	—	106	2	no	—
384	#131	C₄×D₅₃	1	4	yes	complemented
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	—	106	1	no	—
391	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
392	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
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	—	106	1	no	—
399	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
400	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
401	C₄×Q₈	—	53	2	no	—
402	C₄×Q₈	—	53	2	no	—
403	C₄×D₄	—	53	2	no	—
404	C₄×D₄	—	53	2	no	—
405	C₄×D₄	—	106	2	no	—
406	C₄×D₄	—	106	2	no	—
407	#117	—	53	1	no	—
408	#117	—	106	1	no	—
409	#117	—	106	1	no	—
410	#117	C₂²×D₅₃	1	1	yes	characteristic
411	C₂×C₄²	—	53	1	no	—
412	C₂×C₄²	—	53	1	no	—
413	C₂×C₄²	—	106	1	no	—
414	C₂×C₄²	—	106	1	no	—
415	C₂×C₄²	—	106	2	no	—
416	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
417	#108	—	53	4	no	—
418	#108	—	53	4	no	—
419	#107	—	53	2	no	—
420	#106	—	53	2	no	—
421	#104	—	53	4	no	—
422	#104	—	53	4	no	—
423	#104	—	53	4	no	—
424	#104	C₄×D₅₃	1	4	yes	complemented
425	#103	—	53	2	no	—
426	#103	C₄×D₅₃	1	2	yes	—
427	#102	—	53	2	no	—
428	#102	C₄×D₅₃	1	2	yes	—
429	#97	—	2	1	no	—
430	#97	—	2	2	no	—
431	#97	—	53	4	no	—
432	#97	—	53	4	no	—
433	#97	—	106	1	no	—
434	#97	—	106	1	no	—
435	#97	—	106	1	no	—
436	#97	—	106	2	no	—
437	#97	—	106	1	no	—
438	#97	—	106	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	—	53	2	no	—
441	C₄×C₈	—	53	2	no	—
442	C₄×C₈	—	53	1	no	—
443	C₄×C₈	—	53	1	no	—
444	C₄×C₈	—	53	1	no	—
445	C₄×C₈	—	53	1	no	—
446	C₄×C₈	—	106	1	no	—
447	C₄×C₈	—	106	1	no	—
448	C₄×C₈	—	106	1	no	—
449	C₄×C₈	—	106	1	no	—
450	C₄×C₈	—	106	1	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	—	53	2	no	—
454	#41	—	53	2	no	—
455	#41	—	106	1	no	—
456	#41	—	106	1	no	—
457	#41	C₂×C₄×D₅₃	1	2	yes	—
458	#41	C₂×C₄×D₅₃	1	2	yes	—
459	C₂×Q₈	—	53	1	no	—
460	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	53	1	no	—
463	C₂×D₄	—	106	1	no	—
464	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	53	2	no	—
468	C₂²×C₄	—	106	2	no	—
469	C₂²×C₄	—	106	2	no	—
470	C₂²×C₄	—	106	1	no	—
471	C₂²×C₄	—	106	1	no	—
472	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
473	Q₁₆	—	53	4	no	—
474	SD₁₆	—	53	4	no	—
475	SD₁₆	—	53	4	no	—
476	D₈	—	53	4	no	—
477	M₁₆	—	2	1	no	—
478	M₁₆	—	2	1	no	—
479	M₁₆	—	2	4	no	—
480	M₁₆	—	53	2	no	—
481	M₁₆	—	53	2	no	—
482	M₁₆	—	106	4	no	—
483	M₁₆	—	106	4	no	—
484	M₁₆	—	106	1	no	—
485	M₁₆	—	106	1	no	—
486	M₁₆	—	106	4	no	—
487	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
488	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	4	no	—
494	C₂×C₈	—	53	1	no	—
495	C₂×C₈	—	53	1	no	—
496	C₂×C₈	—	53	2	no	—
497	C₂×C₈	—	53	2	no	—
498	C₂×C₈	—	53	2	no	—
499	C₂×C₈	—	53	1	no	—
500	C₂×C₈	—	53	1	no	—
501	C₂×C₈	—	53	2	no	—
502	C₂×C₈	—	106	4	no	—
503	C₂×C₈	—	106	4	no	—
504	C₂×C₈	—	106	1	no	—
505	C₂×C₈	—	106	1	no	—
506	C₂×C₈	—	106	4	no	—
507	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
508	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
509	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
510	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
511	C₂×C₈	#11719947	1	2	yes	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	106	2	no	—
514	C₄⋊C₄	—	106	2	no	—
515	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	106	2	no	—
518	C₂²⋊C₄	—	106	2	no	—
519	C₂²⋊C₄	—	106	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	53	2	no	—
523	C₄²	—	53	1	no	—
524	C₄²	—	53	1	no	—
525	C₄²	—	53	2	no	—
526	C₄²	—	106	2	no	—
527	C₄²	—	106	2	no	—
528	C₄²	—	106	1	no	—
529	C₄²	—	106	1	no	—
530	C₄²	—	212	4	no	—
531	C₄²	C₂×C₄×D₅₃	1	2	yes	—
532	C₂³	—	2	1	no	—
533	C₂³	—	106	1	no	—
534	C₂³	#41×D₅₃	1	1	yes	characteristic
535	Q₈	—	2	1	no	—
536	Q₈	—	53	2	no	—
537	Q₈	—	106	1	no	—
538	Q₈	C₄²×D₅₃	1	2	yes	—
539	D₄	—	2	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	1	no	—
542	D₄	—	53	2	no	—
543	D₄	—	106	2	no	—
544	D₄	—	106	2	no	—
545	D₄	—	106	1	no	—
546	D₄	C₄²×D₅₃	1	2	yes	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	53	2	no	—
554	C₂×C₄	—	53	2	no	—
555	C₂×C₄	—	106	4	no	—
556	C₂×C₄	—	106	1	no	—
557	C₂×C₄	—	106	2	no	—
558	C₂×C₄	—	106	2	no	—
559	C₂×C₄	—	106	2	no	—
560	C₂×C₄	—	106	2	no	—
561	C₂×C₄	—	212	4	no	—
562	C₂×C₄	—	212	4	no	—
563	C₂×C₄	—	212	2	no	—
564	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
565	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
566	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
568	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
569	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₈	—	2	2	no	—
573	C₈	—	53	2	no	—
574	C₈	—	53	2	no	—
575	C₈	—	53	2	no	—
576	C₈	—	53	2	no	—
577	C₈	—	106	2	no	—
578	C₈	—	106	2	no	—
579	C₈	#13133177	1	2	yes	—
580	C₈	#13133177	1	2	yes	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	106	2	no	—
585	C₂²	—	212	2	no	—
586	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
587	C₂²	C₄×#11719947	1	1	yes	characteristic
588	C₂²	#16811091	1	1	yes	characteristic
589	C₄	—	2	2	no	—
590	C₄	—	4	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	53	4	no	—
593	C₄	—	106	2	no	—
594	C₄	—	106	2	no	—
595	C₄	—	212	2	no	—
596	C₄	—	212	2	no	—
597	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
598	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
599	C₄	C₂×#13133177	1	1	yes	characteristic
600	C₄	C₄×#11719947	1	1	yes	characteristic
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	4	2	no	—
604	C₂	—	212	2	no	—
605	C₂	C₄×#13133177	1	1	yes	characteristic
606	C₂	#444×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1765	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1765	D₅₃
hypocenter	C₅₃	C₄×#444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1765

Complements

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

N	H ≅ G/N	type
4. #20283137	C₂	semidirect
5. #131×Dic₅₃	C₂	semidirect
6. #20274616	C₂	semidirect
7. #20274600	C₂	semidirect
8. #20274598	C₂	semidirect
9. #20274562	C₂	semidirect
10. #20274559	C₂	semidirect
11. #20274558	C₂	semidirect
12. #104×Dic₅₃	C₂	semidirect
13. #20274519	C₂	semidirect
14. #97×Dic₅₃	C₂	semidirect
15. C₅₃×#1765	C₂	semidirect
22. #16810743	C₄	semidirect
23. #16810742	C₄	semidirect
24. #16810715	C₄	semidirect
25. #16810714	C₄	semidirect
26. #16810708	C₄	semidirect
27. #16810703	C₄	semidirect
73. #13133225	C₂×C₄	semidirect
75. #13133224	C₂×C₄	semidirect
77. #13133223	C₂×C₄	semidirect
102. #13133152	C₂×C₄	semidirect
103. #13133152	C₂×C₄	semidirect
106. #13133145	C₂×C₄	semidirect
107. #13133144	C₂×C₄	semidirect
108. #13133143	C₂×C₄	semidirect
112. #13133128	C₂×C₄	semidirect
118. #13133127	C₂×C₄	semidirect
132. #131×C₅₃	C₂×C₄	semidirect
141. #104×C₅₃	C₂×C₄	semidirect
191. #11719925	C₄²	semidirect
192. #11719924	C₄²	semidirect
193. #11719923	C₄²	semidirect
194. #11719922	C₄²	semidirect
297. Dic₅₃	#444	semidirect
309. #1765	D₅₃	semidirect
368. C₅₃	C₄×#444	semidirect
384. #131	C₄×D₅₃	semidirect
424. #104	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×#16810742	2
#20283483	2
#20283137	2
#131×Dic₅₃	2
#20274616	2
#20274600	2
#20274598	2
#20274562	2
#20274559	2
#20274558	2
#104×Dic₅₃	2
#20274519	2
#97×Dic₅₃	2
C₅₃×#1765	2
C₄×#444	53