Subgroups — #27387620

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810670	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283277	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810648	C₂	1	1	yes	maximal, characteristic
5	#20283132	C₂	1	1	yes	maximal, characteristic
6	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
8	#20275489	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275489	C₂	1	1	yes	maximal, characteristic, complemented
10	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
11	#20275487	C₂	1	1	yes	maximal, characteristic, complemented
12	#20275487	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274440	C₂	1	1	yes	maximal, characteristic
14	#341×D₅₃	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#11719912	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₂×#13133174	C₂²	1	1	yes	characteristic
23	#16810670	—	2	4	no	—
24	C₂×#13133153	C₂²	1	1	yes	characteristic
25	C₂×#13133153	C₂²	1	1	yes	characteristic
26	C₂×#13133152	C₂²	1	1	yes	characteristic
27	C₂×#13133152	C₂²	1	1	yes	characteristic
28	C₂×#13133152	C₂²	1	1	yes	characteristic
29	C₂×#13133152	C₂²	1	1	yes	characteristic
30	#108×D₅₃	—	2	4	no	—
31	#16810648	—	2	1	no	—
32	#16810648	—	2	1	no	—
33	#16810648	—	2	1	no	—
34	#16810648	—	2	1	no	—
35	#16810648	C₄	1	2	yes	—
36	#16810648	C₄	1	2	yes	—
37	#16810648	C₄	1	2	yes	—
38	#16810648	C₄	1	2	yes	—
39	#16810647	C₂²	1	1	yes	characteristic
40	#16810646	C₂²	1	1	yes	characteristic
41	#16810646	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
43	C₂×#13133146	C₂²	1	1	yes	characteristic
44	C₂×#13133146	C₂²	1	1	yes	characteristic
45	#16810580	—	2	1	no	—
46	#16810580	—	2	1	no	—
47	#16810577	—	2	1	no	—
48	#16810577	—	2	1	no	—
49	C₄×#11719917	C₂²	1	1	yes	characteristic
50	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
51	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
52	#16810388	C₂²	1	1	yes	characteristic
53	#16810388	C₂²	1	1	yes	characteristic
54	#16810387	C₂²	1	1	yes	characteristic
55	#16810387	C₂²	1	1	yes	characteristic
56	#16810387	C₂²	1	1	yes	characteristic
57	#16810387	C₂²	1	1	yes	characteristic
58	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
60	C₅₃×#423	C₂²	1	1	yes	characteristic
61	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
63	C₅₃×#341	C₂²	1	1	yes	characteristic
64	C₅₃×#341	C₂²	1	1	yes	characteristic
65	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
66	C₂×#11719917	C₂³	1	1	yes	characteristic
67	C₂×#11719917	C₂³	1	1	yes	characteristic
68	C₂×#11719917	C₂³	1	1	yes	characteristic
69	C₂×#11719917	C₂³	1	1	yes	characteristic
70	C₂×#11719912	—	2	1	no	—
71	C₂×#11719912	—	2	1	no	—
72	C₂×#11719912	—	2	2	no	—
73	C₂×#11719912	—	2	1	no	—
74	C₂×#11719912	—	2	1	no	—
75	C₂×#11719912	—	2	2	no	—
76	C₂×#11719912	C₂×C₄	1	2	yes	—
77	C₂×#11719912	C₂×C₄	1	2	yes	—
78	M₁₆×D₅₃	—	2	2	no	—
79	M₁₆×D₅₃	—	2	2	no	—
80	M₁₆×D₅₃	—	2	2	no	—
81	M₁₆×D₅₃	—	2	2	no	—
82	M₁₆×D₅₃	—	2	2	no	—
83	M₁₆×D₅₃	—	2	2	no	—
84	M₁₆×D₅₃	D₄	1	4	yes	—
85	M₁₆×D₅₃	Q₈	1	4	yes	—
86	C₂²×#88322	C₂³	1	1	yes	characteristic
87	C₂²×#88322	C₂³	1	1	yes	characteristic
88	C₂×C₈×D₅₃	—	2	1	no	—
89	C₂×C₈×D₅₃	—	2	1	no	—
90	C₂×C₈×D₅₃	—	2	2	no	—
91	C₂×C₈×D₅₃	—	2	1	no	—
92	C₂×C₈×D₅₃	—	2	1	no	—
93	C₂×C₈×D₅₃	—	2	2	no	—
94	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
95	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
96	#13133174	C₂×C₄	1	1	yes	characteristic
97	#13133174	C₂×C₄	1	1	yes	characteristic
98	#13133174	C₂×C₄	1	1	yes	characteristic
99	#13133174	C₂×C₄	1	1	yes	characteristic
100	#13133174	D₄	1	1	yes	characteristic
101	#13133174	D₄	1	1	yes	characteristic
102	#13133174	D₄	1	1	yes	characteristic
103	#13133174	D₄	1	1	yes	characteristic
104	C₂×#11719919	Q₈	1	1	yes	characteristic
105	C₂×#11719919	Q₈	1	1	yes	characteristic
106	C₂×#11719921	Q₈	1	1	yes	characteristic
107	C₂×#11719921	Q₈	1	1	yes	characteristic
108	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
109	#13133153	—	2	2	no	—
110	#13133153	—	2	2	no	—
111	#13133152	—	2	2	no	—
112	#13133152	—	2	2	no	—
113	#13133152	—	2	2	no	—
114	#13133152	—	2	2	no	—
115	C₄⋊#88322	—	2	1	no	—
116	C₄⋊#88322	—	2	1	no	—
117	C₄⋊#88322	C₂×C₄	1	2	yes	—
118	C₄⋊#88322	C₂×C₄	1	2	yes	—
119	#13133146	—	2	1	no	—
120	#13133146	—	2	1	no	—
121	#13133146	—	2	1	no	—
122	#13133146	—	2	1	no	—
123	#13133146	C₂×C₄	1	2	yes	—
124	#13133146	C₂×C₄	1	2	yes	—
125	#13133146	C₂×C₄	1	2	yes	—
126	#13133146	C₂×C₄	1	2	yes	—
127	#13133128	—	2	1	no	—
128	#13133127	—	2	1	no	—
129	#13133127	—	2	1	no	—
130	C₄×#88322	—	2	1	no	—
131	C₈×Dic₅₃	—	2	1	no	—
132	C₈×Dic₅₃	—	2	1	no	—
133	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
134	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
135	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
136	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
137	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
138	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
140	#108×C₅₃	—	2	2	no	—
141	#108×C₅₃	—	2	2	no	—
142	C₄⋊C₈×C₅₃	—	2	1	no	—
143	C₄⋊C₈×C₅₃	—	2	1	no	—
144	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
145	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
146	#97×C₅₃	—	2	1	no	—
147	C₄×C₈×C₅₃	—	2	1	no	—
148	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
154	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
158	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
161	C₂×C₄×D₅₃	#41	1	2	yes	—
162	C₂×C₄×D₅₃	#41	1	2	yes	—
163	M₁₆×C₅₃	—	2	1	no	—
164	M₁₆×C₅₃	—	2	1	no	—
165	M₁₆×C₅₃	—	2	1	no	—
166	M₁₆×C₅₃	—	2	1	no	—
167	M₁₆×C₅₃	—	2	1	no	—
168	M₁₆×C₅₃	—	2	1	no	—
169	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
170	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
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	1	no	—
179	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
180	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
181	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
185	#11719921	C₄⋊C₄	1	1	yes	characteristic
186	#11719921	C₄⋊C₄	1	1	yes	characteristic
187	#11719921	C₄⋊C₄	1	1	yes	characteristic
188	#11719921	C₄⋊C₄	1	1	yes	characteristic
189	#11719921	C₄⋊C₄	1	1	yes	characteristic
190	#11719921	C₄⋊C₄	1	1	yes	characteristic
191	#11719921	C₄⋊C₄	1	1	yes	characteristic
192	#11719921	C₄⋊C₄	1	1	yes	characteristic
193	#11719919	C₄⋊C₄	1	1	yes	characteristic
194	#11719919	C₄⋊C₄	1	1	yes	characteristic
195	#11719919	C₄⋊C₄	1	1	yes	characteristic
196	#11719919	C₄⋊C₄	1	1	yes	characteristic
197	#11719919	C₄⋊C₄	1	1	yes	characteristic
198	#11719919	C₄⋊C₄	1	1	yes	characteristic
199	#11719919	C₄⋊C₄	1	1	yes	characteristic
200	#11719919	C₄⋊C₄	1	1	yes	characteristic
201	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
202	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
203	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
204	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
205	#11719917	—	2	1	no	—
206	#11719917	—	2	1	no	—
207	#11719917	—	2	1	no	—
208	#11719917	—	2	1	no	—
209	#11719917	—	2	1	no	—
210	#11719917	—	2	1	no	—
211	#11719917	C₂×D₄	1	2	yes	—
212	#11719917	C₂×Q₈	1	2	yes	—
213	C₂×#88322	—	2	1	no	—
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	—	2	1	no	—
219	C₂×#88322	—	2	1	no	—
220	C₂×#88322	—	2	1	no	—
221	C₂×#88322	C₂²×C₄	1	2	yes	—
222	C₂×#88322	C₂²×C₄	1	2	yes	—
223	#11719912	—	2	4	no	—
224	#11719912	—	2	4	no	—
225	#11719912	—	2	4	no	—
226	#11719912	—	2	4	no	—
227	#11719912	—	4	2	no	—
228	#11719912	—	4	2	no	—
229	C₈×D₅₃	—	2	4	no	—
230	C₈×D₅₃	—	2	4	no	—
231	C₈×D₅₃	—	2	4	no	—
232	C₈×D₅₃	—	2	4	no	—
233	C₈×D₅₃	—	4	2	no	—
234	C₈×D₅₃	—	4	2	no	—
235	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
236	C₂²×D₅₃	—	4	1	no	—
237	C₂²×D₅₃	—	4	1	no	—
238	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂²×D₅₃	#125	1	2	yes	—
241	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
242	C₂×C₄×C₅₃	—	4	1	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
254	C₂×Dic₅₃	—	4	1	no	—
255	C₂×Dic₅₃	—	4	1	no	—
256	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
259	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
260	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
261	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
262	C₂×Dic₅₃	#125	1	2	yes	—
263	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
264	C₄×D₅₃	C₄×D₄	1	2	yes	—
265	C₄×D₅₃	C₄×D₄	1	2	yes	—
266	C₄×D₅₃	C₄×Q₈	1	2	yes	—
267	C₄×D₅₃	C₄×Q₈	1	2	yes	—
268	C₄×D₅₃	#121	1	2	yes	—
269	C₄×D₅₃	#121	1	2	yes	—
270	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
271	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
272	C₈×C₅₃	—	2	2	no	—
273	C₈×C₅₃	—	2	2	no	—
274	C₈×C₅₃	—	2	2	no	—
275	C₈×C₅₃	—	2	2	no	—
276	C₈×C₅₃	—	4	1	no	—
277	C₈×C₅₃	—	4	1	no	—
278	#88322	—	2	2	no	—
279	#88322	—	2	2	no	—
280	#88322	—	2	2	no	—
281	#88322	—	2	2	no	—
282	#88322	—	4	1	no	—
283	#88322	—	4	1	no	—
284	#21178	—	53	1	no	maximal, Sylow
285	C₂²×C₅₃	—	2	1	no	—
286	C₂²×C₅₃	—	2	1	no	—
287	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
288	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
289	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
290	C₂×D₅₃	—	2	1	no	—
291	C₂×D₅₃	—	2	1	no	—
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	2	1	no	—
294	C₂×D₅₃	—	4	2	no	—
295	C₂×D₅₃	—	4	2	no	—
296	C₂×D₅₃	#389	1	2	yes	—
297	C₂×D₅₃	#389	1	2	yes	—
298	C₄×C₅₃	—	2	1	no	—
299	C₄×C₅₃	—	2	1	no	—
300	C₄×C₅₃	—	2	1	no	—
301	C₄×C₅₃	—	2	1	no	—
302	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
303	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
304	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
305	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
306	Dic₅₃	—	2	1	no	—
307	Dic₅₃	—	2	1	no	—
308	Dic₅₃	—	2	1	no	—
309	Dic₅₃	—	2	1	no	—
310	Dic₅₃	#389	1	2	yes	—
311	Dic₅₃	#389	1	2	yes	—
312	C₂×#430	—	53	1	no	—
313	C₂²×#108	—	53	1	no	—
314	#2905	—	53	1	no	—
315	C₂×#424	—	53	1	no	—
316	#2873	—	53	1	no	—
317	#1950	—	53	1	no	—
318	#1950	—	53	1	no	—
319	#1950	—	53	1	no	—
320	#1950	—	53	1	no	—
321	#1950	—	53	1	no	—
322	#1950	—	53	1	no	—
323	#1950	—	53	1	no	—
324	#1950	D₅₃	1	1	yes	characteristic, complemented
325	#1738	—	53	1	no	—
326	C₂×#341	—	53	1	no	—
327	C₂×C₅₃	—	2	1	no	—
328	C₂×C₅₃	—	2	1	no	—
329	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
330	C₂×C₅₃	#2916	1	1	yes	characteristic
331	C₂×C₅₃	#2916	1	1	yes	characteristic
332	D₅₃	—	2	2	no	—
333	D₅₃	—	2	2	no	—
334	C₂²×M₁₆	—	53	1	no	—
335	C₂²×M₁₆	—	53	1	no	—
336	C₂²×M₁₆	—	53	1	no	—
337	C₂²×M₁₆	—	53	1	no	—
338	C₂²×M₁₆	—	53	1	no	—
339	C₂³×C₈	—	53	1	no	—
340	C₂×#117	—	53	1	no	—
341	#430	—	106	4	no	—
342	C₂×#108	—	53	1	no	—
343	C₂×#108	—	53	1	no	—
344	C₂×#108	—	53	1	no	—
345	C₂×#108	—	53	1	no	—
346	C₂×#108	—	53	1	no	—
347	C₂×#108	—	53	1	no	—
348	C₂×#108	—	106	4	no	—
349	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
350	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
351	#424	—	53	2	no	—
352	#424	—	53	2	no	—
353	#424	—	53	2	no	—
354	#424	—	53	2	no	—
355	#424	—	106	1	no	—
356	#424	—	106	1	no	—
357	#424	—	106	1	no	—
358	#424	—	106	1	no	—
359	#423	—	53	1	no	—
360	#423	—	53	1	no	—
361	#423	—	53	1	no	—
362	#423	C₂×D₅₃	1	1	yes	characteristic
363	C₂×C₄⋊C₈	—	53	1	no	—
364	C₂×C₄⋊C₈	—	53	1	no	—
365	C₂×C₄⋊C₈	—	53	1	no	—
366	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
367	#405	—	106	1	no	—
368	#405	—	106	1	no	—
369	#405	—	106	1	no	—
370	#405	—	106	1	no	—
371	C₄×M₁₆	—	53	1	no	—
372	C₄×M₁₆	—	53	1	no	—
373	C₄×M₁₆	—	53	1	no	—
374	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
375	#341	—	53	1	no	—
376	#341	—	53	1	no	—
377	#341	—	53	1	no	—
378	#341	—	53	1	no	—
379	#341	—	53	1	no	—
380	#341	—	53	1	no	—
381	#341	C₂×D₅₃	1	1	yes	characteristic
382	#341	C₂×D₅₃	1	1	yes	characteristic
383	C₅₃	#21178	1	1	yes	characteristic, complemented, Sylow
384	C₂³×C₄	—	53	1	no	—
385	C₂×M₁₆	—	53	2	no	—
386	C₂×M₁₆	—	53	2	no	—
387	C₂×M₁₆	—	53	1	no	—
388	C₂×M₁₆	—	53	1	no	—
389	C₂×M₁₆	—	53	4	no	—
390	C₂×M₁₆	—	53	4	no	—
391	C₂×M₁₆	—	53	1	no	—
392	C₂×M₁₆	—	53	1	no	—
393	C₂×M₁₆	—	106	1	no	—
394	C₂×M₁₆	—	106	1	no	—
395	C₂×M₁₆	—	106	2	no	—
396	C₂×M₁₆	—	106	2	no	—
397	C₂×M₁₆	—	106	1	no	—
398	C₂×M₁₆	—	106	1	no	—
399	C₂×M₁₆	—	106	2	no	—
400	C₂×M₁₆	—	106	2	no	—
401	C₂×M₁₆	—	106	2	no	—
402	C₂×M₁₆	—	106	2	no	—
403	C₂×M₁₆	—	106	2	no	—
404	C₂×M₁₆	—	106	2	no	—
405	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
406	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
407	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
408	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
409	C₂²×C₈	—	53	1	no	—
410	C₂²×C₈	—	53	1	no	—
411	C₂²×C₈	—	53	2	no	—
412	C₂²×C₈	—	53	2	no	—
413	C₂²×C₈	—	106	2	no	—
414	C₂²×C₈	—	106	2	no	—
415	C₂²×C₈	—	106	1	no	—
416	C₂²×C₈	—	106	1	no	—
417	C₂²×C₈	—	106	1	no	—
418	C₂²×C₈	—	106	1	no	—
419	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
420	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
421	#117	—	53	1	no	—
422	#117	—	53	1	no	—
423	#117	—	53	1	no	—
424	#117	—	53	1	no	—
425	#117	—	53	1	no	—
426	#117	—	53	1	no	—
427	#117	—	53	1	no	—
428	#117	—	53	1	no	—
429	C₂×C₄⋊C₄	—	53	1	no	—
430	C₂×C₄⋊C₄	—	53	1	no	—
431	C₂×C₂²⋊C₄	—	53	1	no	—
432	C₂×C₂²⋊C₄	—	53	1	no	—
433	C₂×C₄²	—	53	1	no	—
434	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
435	#108	—	2	2	no	—
436	#108	—	2	2	no	—
437	#108	—	106	2	no	—
438	#108	—	106	2	no	—
439	#108	—	106	2	no	—
440	#108	—	106	2	no	—
441	#108	—	106	2	no	—
442	#108	—	106	2	no	—
443	C₄⋊C₈	—	2	1	no	—
444	C₄⋊C₈	—	2	1	no	—
445	C₄⋊C₈	—	53	2	no	—
446	C₄⋊C₈	—	53	2	no	—
447	C₄⋊C₈	—	53	2	no	—
448	C₄⋊C₈	—	53	2	no	—
449	C₄⋊C₈	—	53	2	no	—
450	C₄⋊C₈	—	53	2	no	—
451	C₄⋊C₈	—	106	1	no	—
452	C₄⋊C₈	—	106	1	no	—
453	C₄⋊C₈	—	106	1	no	—
454	C₄⋊C₈	—	106	1	no	—
455	C₄⋊C₈	—	106	1	no	—
456	C₄⋊C₈	—	106	1	no	—
457	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
458	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
459	#97	—	2	1	no	—
460	#97	—	106	1	no	—
461	#97	—	106	1	no	—
462	#97	—	106	1	no	—
463	C₄×C₈	—	2	1	no	—
464	C₄×C₈	—	106	1	no	—
465	C₄×C₈	—	106	1	no	—
466	C₄×C₈	—	106	1	no	—
467	C₂⁴	—	53	1	no	—
468	C₂²×C₄	—	53	1	no	—
469	C₂²×C₄	—	53	1	no	—
470	C₂²×C₄	—	53	1	no	—
471	C₂²×C₄	—	53	2	no	—
472	C₂²×C₄	—	53	1	no	—
473	C₂²×C₄	—	53	1	no	—
474	C₂²×C₄	—	53	2	no	—
475	C₂²×C₄	—	53	2	no	—
476	C₂²×C₄	—	53	2	no	—
477	C₂²×C₄	—	53	1	no	—
478	C₂²×C₄	—	53	1	no	—
479	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
480	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
482	M₁₆	—	2	1	no	—
483	M₁₆	—	2	1	no	—
484	M₁₆	—	2	1	no	—
485	M₁₆	—	2	1	no	—
486	M₁₆	—	2	1	no	—
487	M₁₆	—	2	1	no	—
488	M₁₆	—	53	2	no	—
489	M₁₆	—	53	2	no	—
490	M₁₆	—	106	4	no	—
491	M₁₆	—	106	4	no	—
492	M₁₆	—	106	4	no	—
493	M₁₆	—	106	4	no	—
494	M₁₆	—	106	1	no	—
495	M₁₆	—	106	1	no	—
496	M₁₆	—	106	1	no	—
497	M₁₆	—	106	1	no	—
498	M₁₆	—	106	1	no	—
499	M₁₆	—	106	1	no	—
500	M₁₆	—	212	2	no	—
501	M₁₆	—	212	2	no	—
502	M₁₆	D₄×D₅₃	1	2	yes	—
503	M₁₆	Q₈×D₅₃	1	2	yes	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	53	2	no	—
513	C₂×C₈	—	53	2	no	—
514	C₂×C₈	—	106	4	no	—
515	C₂×C₈	—	106	4	no	—
516	C₂×C₈	—	106	4	no	—
517	C₂×C₈	—	106	4	no	—
518	C₂×C₈	—	106	1	no	—
519	C₂×C₈	—	106	1	no	—
520	C₂×C₈	—	106	1	no	—
521	C₂×C₈	—	106	1	no	—
522	C₂×C₈	—	106	1	no	—
523	C₂×C₈	—	106	1	no	—
524	C₂×C₈	—	106	1	no	—
525	C₂×C₈	—	106	1	no	—
526	C₂×C₈	—	212	2	no	—
527	C₂×C₈	—	212	2	no	—
528	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
529	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
530	C₄⋊C₄	—	53	1	no	—
531	C₄⋊C₄	—	53	1	no	—
532	C₄⋊C₄	—	53	1	no	—
533	C₄⋊C₄	—	53	1	no	—
534	C₄⋊C₄	—	53	1	no	—
535	C₄⋊C₄	—	53	1	no	—
536	C₄⋊C₄	—	53	1	no	—
537	C₄⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	1	no	—
539	C₂²⋊C₄	—	53	1	no	—
540	C₂²⋊C₄	—	53	1	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	1	no	—
543	C₂²⋊C₄	—	53	1	no	—
544	C₂²⋊C₄	—	53	1	no	—
545	C₂²⋊C₄	—	53	1	no	—
546	C₄²	—	53	1	no	—
547	C₄²	—	53	1	no	—
548	C₄²	—	53	1	no	—
549	C₄²	—	53	1	no	—
550	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
551	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
552	C₄²	D₄×D₅₃	1	1	yes	characteristic
553	C₄²	D₄×D₅₃	1	1	yes	characteristic
554	C₂³	—	53	1	no	—
555	C₂³	—	53	1	no	—
556	C₂³	—	53	2	no	—
557	C₂³	—	53	2	no	—
558	C₂³	—	212	1	no	—
559	C₂³	—	212	1	no	—
560	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	53	1	no	—
564	C₂×C₄	—	53	1	no	—
565	C₂×C₄	—	53	2	no	—
566	C₂×C₄	—	53	2	no	—
567	C₂×C₄	—	53	2	no	—
568	C₂×C₄	—	53	2	no	—
569	C₂×C₄	—	53	2	no	—
570	C₂×C₄	—	53	2	no	—
571	C₂×C₄	—	53	2	no	—
572	C₂×C₄	—	53	2	no	—
573	C₂×C₄	—	53	2	no	—
574	C₂×C₄	—	53	2	no	—
575	C₂×C₄	—	53	1	no	—
576	C₂×C₄	—	53	1	no	—
577	C₂×C₄	—	53	1	no	—
578	C₂×C₄	—	53	1	no	—
579	C₂×C₄	—	212	1	no	—
580	C₂×C₄	—	212	1	no	—
581	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
582	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
583	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
584	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
585	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
586	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
587	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
588	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
589	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
590	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
591	C₈	—	2	2	no	—
592	C₈	—	2	2	no	—
593	C₈	—	2	2	no	—
594	C₈	—	2	2	no	—
595	C₈	—	4	1	no	—
596	C₈	—	4	1	no	—
597	C₈	—	106	2	no	—
598	C₈	—	106	2	no	—
599	C₈	—	106	2	no	—
600	C₈	—	106	2	no	—
601	C₈	—	212	1	no	—
602	C₈	—	212	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	53	2	no	—
606	C₂²	—	53	2	no	—
607	C₂²	—	106	1	no	—
608	C₂²	—	106	1	no	—
609	C₂²	—	106	1	no	—
610	C₂²	—	106	1	no	—
611	C₂²	—	212	2	no	—
612	C₂²	—	212	2	no	—
613	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
614	C₂²	#125×D₅₃	1	1	yes	characteristic
615	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	53	2	no	—
621	C₄	—	53	2	no	—
622	C₄	—	106	1	no	—
623	C₄	—	106	1	no	—
624	C₄	—	106	1	no	—
625	C₄	—	106	1	no	—
626	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
627	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
628	C₄	#121×D₅₃	1	1	yes	characteristic
629	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	106	2	no	—
633	C₂	—	106	2	no	—
634	C₂	#389×D₅₃	1	1	yes	characteristic
635	C₂	#20283323	1	1	yes	characteristic
636	C₂	#20283325	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	#21178
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
6. #20275491	C₂	semidirect
7. #20275490	C₂	semidirect
8. #20275489	C₂	semidirect
9. #20275489	C₂	semidirect
10. #20275488	C₂	semidirect
11. #20275487	C₂	semidirect
12. #20275487	C₂	semidirect
15. C₅₃×#1950	C₂	semidirect
324. #1950	D₅₃	semidirect
383. C₅₃	#21178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810670	2
C₂×#108×D₅₃	2
#20283277	2
C₂×#16810648	2
#20283132	2
#20275491	2
#20275490	2
#20275489	2
#20275488	2
#20275487	2
#20274440	2
#341×D₅₃	2
C₅₃×#1950	2
#21178	53