Subgroups — #27391916

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284497	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
3	#20275660	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275311	C₂	1	2	yes	maximal, complemented
6	#20274989	C₂	1	2	yes	maximal
7	#20274732	C₂	1	1	yes	maximal, characteristic
8	#20274595	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274554	C₂	1	2	yes	maximal
11	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₂²	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	#16810927	C₄	1	2	yes	complemented
16	#16810920	C₄	1	2	yes	complemented
17	#16810915	C₄	1	1	yes	characteristic, complemented
18	#16810910	C₄	1	1	yes	characteristic, complemented
19	#16810905	C₄	1	1	yes	characteristic, complemented
20	#16810900	C₄	1	1	yes	characteristic, complemented
21	C₂×#13133152	C₂²	1	2	yes	—
22	#16810659	C₂²	1	2	yes	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×#13133143	C₂²	1	2	yes	—
25	#16810617	C₂²	1	2	yes	—
26	#16810578	C₂²	1	1	yes	characteristic
27	#16810578	C₂²	1	1	yes	characteristic
28	#16810577	—	2	1	no	—
29	#16810577	—	2	1	no	—
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	C₄×#11719912	—	2	1	no	—
32	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
33	C₄×C₈×D₅₃	—	2	1	no	—
34	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
35	#16810387	C₂²	1	2	yes	—
36	#16810384	C₂²	1	2	yes	—
37	#16810382	C₂²	1	2	yes	—
38	#16810382	C₂²	1	2	yes	—
39	#16810377	C₂²	1	2	yes	—
40	C₅₃×#533	C₂²	1	1	yes	characteristic
41	C₅₃×#446	—	2	1	no	—
42	C₅₃×#446	—	2	1	no	—
43	C₈×Q₈×C₅₃	—	2	1	no	—
44	C₈×Q₈×C₅₃	—	2	1	no	—
45	C₅₃×#424	C₂²	1	2	yes	—
46	C₅₃×#423	C₂²	1	1	yes	characteristic
47	C₅₃×#405	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719945	C₂³	1	1	yes	characteristic
50	#13133223	—	2	2	no	—
51	#13133223	—	2	2	no	—
52	#13133223	—	2	2	no	—
53	#13133223	Q₈	1	4	yes	complemented
54	C₂×#11719917	C₂³	1	1	yes	characteristic
55	C₂×#11719917	C₂³	1	1	yes	characteristic
56	C₂×#11719912	—	2	1	no	—
57	C₂×#11719912	—	2	1	no	—
58	C₂²×#88322	C₂³	1	2	yes	—
59	C₂×C₈×D₅₃	—	2	1	no	—
60	C₂×C₈×D₅₃	—	2	1	no	—
61	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
65	C₄×C₂²⋊D₅₃	—	2	1	no	—
66	C₄×C₂²⋊D₅₃	—	2	1	no	—
67	#13133176	C₂³	1	1	yes	characteristic
68	#13133174	—	2	1	no	—
69	C₂×C₄⋊Dic₅₃	—	2	1	no	—
70	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
71	C₄²×D₅₃	—	2	1	no	—
72	#13133152	—	2	4	no	—
73	#13133150	C₂×C₄	1	2	yes	—
74	#13133150	C₂×C₄	1	2	yes	—
75	#13133148	C₂×C₄	1	2	yes	—
76	#13133148	C₂×C₄	1	2	yes	—
77	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
78	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
79	C₄⋊#88322	C₂×C₄	1	2	yes	—
80	#13133143	—	2	4	no	—
81	#13133140	C₂×C₄	1	2	yes	—
82	#13133140	C₂×C₄	1	2	yes	—
83	#13133137	C₂×C₄	1	2	yes	—
84	#13133137	C₂×C₄	1	2	yes	—
85	#13133128	—	2	1	no	—
86	#13133127	—	2	1	no	—
87	#13133127	—	2	1	no	—
88	C₄×#88322	—	2	1	no	—
89	C₈×Dic₅₃	—	2	1	no	—
90	C₈×Dic₅₃	—	2	1	no	—
91	C₈×Dic₅₃	—	2	4	no	—
92	#13133124	—	2	1	no	—
93	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	#125×C₅₃	C₂×C₄	1	2	yes	—
99	C₂²⋊Q₈×C₅₃	—	2	1	no	—
100	C₂²⋊Q₈×C₅₃	—	2	1	no	—
101	C₄×Q₈×C₅₃	—	2	1	no	—
102	C₄×Q₈×C₅₃	—	2	1	no	—
103	#117×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₈×C₅₃	—	2	2	no	—
106	C₄⋊C₈×C₅₃	—	2	2	no	—
107	C₄⋊C₈×C₅₃	—	2	1	no	—
108	C₄⋊C₈×C₅₃	—	2	1	no	—
109	#97×C₅₃	—	2	2	no	—
110	C₄×C₈×C₅₃	—	2	2	no	—
111	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
114	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
115	C₂×Q₈×C₅₃	—	2	1	no	—
116	C₂×Q₈×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₂²⋊D₅₃	—	2	1	no	—
121	C₂²×Dic₅₃	—	2	2	no	—
122	#11719945	—	2	1	no	—
123	#11719945	—	2	1	no	—
124	#11719945	C₂×Q₈	1	2	yes	—
125	#11719945	#41	1	2	yes	—
126	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	—	2	1	no	—
128	C₂×C₄×D₅₃	—	2	1	no	—
129	C₂×C₄×D₅₃	—	2	1	no	—
130	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	—	2	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	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
141	C₂×C₈×C₅₃	#41	1	2	yes	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
148	C₂²⋊C₄×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂²⋊Dic₅₃	—	2	1	no	—
155	#11719921	—	2	1	no	—
156	#11719921	—	2	1	no	—
157	C₄⋊Dic₅₃	—	2	4	no	—
158	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
160	#11719919	—	2	1	no	—
161	#11719919	—	2	1	no	—
162	C₄×Dic₅₃	—	2	1	no	—
163	C₄×Dic₅₃	—	2	1	no	—
164	C₄×Dic₅₃	—	2	1	no	—
165	C₄×Dic₅₃	—	2	2	no	—
166	C₄×Dic₅₃	—	2	2	no	—
167	#11719917	—	2	1	no	—
168	#11719917	—	2	1	no	—
169	#11719917	C₂×Q₈	1	2	yes	—
170	#11719917	#41	1	2	yes	—
171	C₂×#88322	—	2	1	no	—
172	C₂×#88322	—	2	1	no	—
173	C₂×#88322	—	2	2	no	—
174	C₂×#88322	—	2	2	no	—
175	C₂×#88322	C₂²×C₄	1	2	yes	—
176	C₂×#88322	C₂²×C₄	1	2	yes	—
177	#11719912	—	2	4	no	—
178	#11719912	—	2	4	no	—
179	C₈×D₅₃	—	2	4	no	—
180	C₈×D₅₃	—	2	4	no	—
181	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂²×D₅₃	—	2	1	no	—
183	Q₈×C₅₃	—	4	2	no	—
184	Q₈×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂²⋊D₅₃	—	2	2	no	—
200	C₂²⋊D₅₃	—	2	2	no	—
201	C₂×Dic₅₃	—	2	1	no	—
202	C₂×Dic₅₃	—	2	4	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	2	1	no	—
205	C₂×Dic₅₃	—	4	2	no	—
206	C₂×Dic₅₃	—	4	2	no	—
207	D₂₁₂	—	2	1	no	—
208	D₂₁₂	C₄×Q₈	1	2	yes	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	4	2	no	—
212	C₄×D₅₃	—	4	2	no	—
213	Dic₁₀₆	—	2	1	no	—
214	Dic₁₀₆	C₄×Q₈	1	2	yes	—
215	C₈×C₅₃	—	2	1	no	—
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	—	4	2	no	—
218	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
219	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
220	#88322	—	2	2	no	—
221	#88322	—	2	2	no	—
222	#88322	—	2	2	no	—
223	#88322	—	2	2	no	—
224	#21627	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂²×C₅₃	#529	1	1	yes	characteristic
229	C₂²×C₅₃	#533	1	1	yes	characteristic
230	C₂×D₅₃	—	2	2	no	—
231	C₂×D₅₃	—	4	1	no	—
232	C₂×D₅₃	—	4	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	4	2	no	—
237	C₄×C₅₃	—	4	2	no	—
238	C₄×C₅₃	#443	1	2	yes	—
239	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
240	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
241	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
242	C₄×C₅₃	#525	1	1	yes	characteristic
243	Dic₅₃	—	2	2	no	—
244	Dic₅₃	—	4	4	no	—
245	Dic₅₃	—	4	1	no	—
246	Dic₅₃	—	4	1	no	—
247	#3076	—	53	1	no	—
248	#2960	D₅₃	1	1	yes	characteristic, complemented
249	C₄×#131	—	53	1	no	—
250	#1986	—	53	1	no	—
251	#1950	—	53	1	no	—
252	#1922	—	53	2	no	—
253	#1866	—	53	2	no	—
254	#1806	—	53	1	no	—
255	#1772	—	53	2	no	—
256	C₈×C₄⋊C₄	—	53	1	no	—
257	#1764	—	53	2	no	—
258	C₂×C₅₃	—	2	1	no	—
259	C₂×C₅₃	—	2	1	no	—
260	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
261	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
262	C₂×C₅₃	#2955	1	1	yes	characteristic
263	D₅₃	—	4	2	no	—
264	C₂×#131	—	53	1	no	—
265	C₂×#131	—	53	1	no	—
266	#533	C₂×D₅₃	1	1	yes	characteristic
267	C₄×#41	—	53	1	no	—
268	#479	—	53	2	no	—
269	#478	—	53	2	no	—
270	#477	—	53	1	no	—
271	#476	—	53	1	no	—
272	#475	—	53	1	no	—
273	#474	—	53	1	no	—
274	#446	—	2	1	no	—
275	#446	—	2	1	no	—
276	C₈×Q₈	—	2	1	no	—
277	C₈×Q₈	—	2	1	no	—
278	C₂×#108	—	53	2	no	—
279	#427	—	53	2	no	—
280	#424	C₂×D₅₃	1	2	yes	—
281	#423	C₂×D₅₃	1	1	yes	characteristic
282	C₂×C₄⋊C₈	—	53	1	no	—
283	C₂×#104	—	53	2	no	—
284	#416	—	53	2	no	—
285	#405	—	53	1	no	—
286	#405	—	53	1	no	—
287	#405	—	106	1	no	—
288	#405	—	106	1	no	—
289	#405	C₂×D₅₃	1	2	yes	—
290	C₄×M₁₆	—	53	1	no	—
291	C₄×M₁₆	—	106	1	no	—
292	C₂×C₄×C₈	—	53	2	no	—
293	C₂×C₄×C₈	—	106	1	no	—
294	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
295	C₄×C₄⋊C₄	—	53	1	no	—
296	#341	—	53	2	no	—
297	#339	—	53	2	no	—
298	#339	—	53	2	no	—
299	#339	—	53	2	no	—
300	#336	—	53	2	no	—
301	C₅₃	#21627	1	1	yes	characteristic, complemented, Sylow
302	C₂×#41	—	53	1	no	—
303	#131	—	53	4	no	—
304	#131	—	106	2	no	—
305	#131	—	106	2	no	—
306	#131	—	106	2	no	—
307	C₂×M₁₆	—	53	1	no	—
308	C₂×M₁₆	—	53	1	no	—
309	C₂×M₁₆	—	106	1	no	—
310	C₂×M₁₆	—	106	1	no	—
311	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
312	C₂²×C₈	—	53	2	no	—
313	C₂²×C₈	—	106	1	no	—
314	C₂²×C₈	—	106	1	no	—
315	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
316	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
317	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
318	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
319	#125	C₄×D₅₃	1	2	yes	—
320	C₂²⋊Q₈	—	2	1	no	—
321	C₂²⋊Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	—	53	1	no	—
325	C₄×Q₈	—	53	1	no	—
326	C₄×D₄	—	53	1	no	—
327	C₄×D₄	—	53	1	no	—
328	C₄×D₄	—	106	1	no	—
329	C₄×D₄	—	106	1	no	—
330	#117	—	53	1	no	—
331	#117	—	106	1	no	—
332	#117	C₂²×D₅₃	1	2	yes	—
333	C₂×C₄⋊C₄	—	106	1	no	—
334	C₂×C₄²	—	53	2	no	—
335	C₂×C₄²	—	106	1	no	—
336	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
337	#108	—	106	4	no	—
338	#107	—	53	2	no	—
339	#107	—	53	2	no	—
340	#106	—	53	2	no	—
341	#106	—	53	2	no	—
342	C₄⋊C₈	—	2	2	no	—
343	C₄⋊C₈	—	2	2	no	—
344	C₄⋊C₈	—	2	1	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	—	53	1	no	—
347	C₄⋊C₈	—	53	1	no	—
348	C₄⋊C₈	—	53	2	no	—
349	#104	—	106	4	no	—
350	#103	—	53	2	no	—
351	#103	—	53	2	no	—
352	#102	—	53	2	no	—
353	#102	—	53	2	no	—
354	#97	—	2	2	no	—
355	#97	—	106	1	no	—
356	#97	—	106	1	no	—
357	#97	—	106	1	no	—
358	C₄×C₈	—	2	2	no	—
359	C₄×C₈	—	106	4	no	—
360	C₄×C₈	—	106	1	no	—
361	C₄×C₈	—	106	1	no	—
362	C₄×C₈	—	106	1	no	—
363	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
364	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
365	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
366	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
367	#95	—	106	1	no	—
368	#41	—	53	2	no	—
369	#41	—	53	2	no	—
370	#41	—	106	1	no	—
371	#41	—	106	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	53	1	no	—
375	C₂×D₄	—	53	1	no	—
376	C₂×D₄	—	106	1	no	—
377	C₂²×C₄	—	106	1	no	—
378	C₂²×C₄	—	106	2	no	—
379	C₂²×C₄	—	106	1	no	—
380	C₂²×C₄	—	106	1	no	—
381	C₂²×C₄	#11719945	1	1	yes	characteristic
382	C₂²×C₄	#11719945	1	1	yes	characteristic
383	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	53	2	no	—
387	M₁₆	—	53	2	no	—
388	M₁₆	—	106	4	no	—
389	M₁₆	—	106	1	no	—
390	M₁₆	—	106	1	no	—
391	M₁₆	—	106	4	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	53	2	no	—
396	C₂×C₈	—	53	2	no	—
397	C₂×C₈	—	106	4	no	—
398	C₂×C₈	—	106	1	no	—
399	C₂×C₈	—	106	1	no	—
400	C₂×C₈	—	106	2	no	—
401	C₂×C₈	—	106	2	no	—
402	C₂×C₈	—	106	4	no	—
403	C₂×C₈	Q₈×D₅₃	1	2	yes	—
404	C₂×C₈	#11719949	1	2	yes	—
405	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
406	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
407	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	53	1	no	—
414	C₄⋊C₄	—	53	1	no	—
415	C₄⋊C₄	—	106	4	no	—
416	C₄⋊C₄	—	106	1	no	—
417	C₄⋊C₄	—	106	1	no	—
418	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
419	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
420	C₂²⋊C₄	—	2	2	no	—
421	C₂²⋊C₄	—	106	1	no	—
422	C₂²⋊C₄	—	106	1	no	—
423	C₂²⋊C₄	—	106	1	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	106	1	no	—
426	C₄²	—	106	1	no	—
427	C₄²	—	106	1	no	—
428	C₄²	—	106	2	no	—
429	C₄²	—	106	2	no	—
430	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
432	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
433	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
434	C₂³	—	106	1	no	—
435	C₂³	C₂×#11719945	1	1	yes	characteristic
436	Q₈	—	4	2	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	53	2	no	—
439	Q₈	—	106	1	no	—
440	D₄	—	53	2	no	—
441	D₄	—	106	2	no	—
442	D₄	—	106	2	no	—
443	D₄	—	106	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	106	2	no	—
450	C₂×C₄	—	106	2	no	—
451	C₂×C₄	—	106	4	no	—
452	C₂×C₄	—	106	1	no	—
453	C₂×C₄	—	106	1	no	—
454	C₂×C₄	—	212	2	no	—
455	C₂×C₄	—	212	2	no	—
456	C₂×C₄	—	212	2	no	—
457	C₂×C₄	—	212	2	no	—
458	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
460	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
461	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
464	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
465	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
466	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
467	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
468	C₈	—	2	1	no	—
469	C₈	—	2	1	no	—
470	C₈	—	4	2	no	—
471	C₈	—	106	2	no	—
472	C₈	—	106	2	no	—
473	C₈	—	106	2	no	—
474	C₈	—	106	2	no	—
475	C₈	#13133193	1	2	yes	—
476	C₈	#13133193	1	2	yes	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	106	2	no	—
480	C₂²	—	212	1	no	—
481	C₂²	—	212	1	no	—
482	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
483	C₂²	#16811160	1	1	yes	characteristic
484	C₂²	#16811177	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	106	2	no	—
491	C₄	—	212	4	no	—
492	C₄	—	212	1	no	—
493	C₄	—	212	1	no	—
494	C₄	#16810738	1	2	yes	—
495	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
496	C₄	C₄×#11719949	1	1	yes	characteristic
497	C₄	C₂×#13133193	1	1	yes	characteristic
498	C₄	#16811130	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	212	2	no	—
502	C₂	C₄×#13133193	1	1	yes	characteristic
503	C₂	C₂×#16810738	1	1	yes	characteristic
504	C₂	#20283616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#21627
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #20275660	C₂	semidirect
4. #20275488	C₂	semidirect
5. #20275311	C₂	semidirect
8. #20274595	C₂	semidirect
9. C₈×C₄⋊Dic₅₃	C₂	semidirect
11. C₅₃×#2960	C₂	semidirect
15. #16810927	C₄	semidirect
16. #16810920	C₄	semidirect
17. #16810915	C₄	semidirect
18. #16810910	C₄	semidirect
19. #16810905	C₄	semidirect
20. #16810900	C₄	semidirect
53. #13133223	Q₈	semidirect
248. #2960	D₅₃	semidirect
301. C₅₃	#21627	semidirect

Maximal subgroups

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

subgroup	index
#20284497	2
C₄×#13133223	2
#20275660	2
#20275488	2
#20275311	2
#20274989	2
#20274732	2
#20274595	2
C₈×C₄⋊Dic₅₃	2
#20274554	2
C₅₃×#2960	2
#21627	53