Subgroups — #27353669

Order 13568.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283682	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283671	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133237	C₂	1	1	yes	maximal, characteristic
4	#20273596	C₂	1	2	yes	maximal, complemented
5	#20273589	C₂	1	2	yes	maximal, complemented
6	#20273582	C₂	1	2	yes	maximal, complemented
7	#20273575	C₂	1	2	yes	maximal, complemented
8	#20273461	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273098	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₄	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	#16810754	C₂²	1	2	yes	—
15	#16810747	C₂²	1	2	yes	—
16	C₄×#11719925	C₄	1	1	yes	characteristic
17	C₄×#11719925	C₂²	1	1	yes	characteristic
18	C₄×#11719924	C₄	1	1	yes	characteristic
19	C₄×#11719924	C₂²	1	1	yes	characteristic
20	C₄×#11719923	C₄	1	1	yes	characteristic
21	C₄×#11719923	C₂²	1	1	yes	characteristic
22	C₄×#11719922	C₄	1	1	yes	characteristic
23	C₄×#11719922	C₂²	1	1	yes	characteristic
24	#16810701	C₂²	1	2	yes	—
25	#16810695	—	2	2	no	—
26	#16810688	C₂²	1	2	yes	—
27	#16810686	—	2	2	no	—
28	#16810676	—	2	1	no	—
29	#16810676	C₂²	1	1	yes	characteristic
30	#16810674	—	2	1	no	—
31	#16810659	C₄	1	1	yes	characteristic
32	#16810619	C₄	1	1	yes	characteristic
33	#16810617	C₄	1	1	yes	characteristic
34	C₄×#11719917	—	2	1	no	—
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
37	C₄×C₈×D₅₃	—	2	1	no	—
38	#16810375	C₂²	1	2	yes	—
39	#16810373	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	#16810344	C₂²	1	2	yes	—
43	#16810344	C₂²	1	2	yes	—
44	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#431	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	#13133237	—	2	4	no	—
51	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
52	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719917	—	2	2	no	—
56	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
57	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
58	C₂²×#88322	D₄	1	2	yes	—
59	C₂×C₈×D₅₃	—	2	2	no	—
60	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
62	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
64	C₄×C₂²⋊D₅₃	—	2	1	no	—
65	C₄×C₂²⋊D₅₃	—	2	1	no	—
66	#13133176	C₂×C₄	1	1	yes	characteristic
67	#13133174	—	2	1	no	—
68	C₄²×D₅₃	—	2	1	no	—
69	#13133150	C₂×C₄	1	1	yes	characteristic
70	#13133150	C₂×C₄	1	1	yes	characteristic
71	#13133148	C₂×C₄	1	1	yes	characteristic
72	#13133148	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#88322	—	2	2	no	—
74	C₄⋊#88322	C₂³	1	2	yes	—
75	#13133146	—	2	2	no	—
76	#13133141	C₂×C₄	1	1	yes	characteristic
77	#13133141	C₂×C₄	1	1	yes	characteristic
78	#13133140	C₂×C₄	1	1	yes	characteristic
79	#13133140	C₂×C₄	1	1	yes	characteristic
80	#13133138	C₂×C₄	1	1	yes	characteristic
81	#13133138	C₂×C₄	1	1	yes	characteristic
82	#13133137	C₂×C₄	1	1	yes	characteristic
83	#13133137	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊#88322	—	2	2	no	—
85	C₂²⋊#88322	—	2	1	no	—
86	#13133129	—	2	2	no	—
87	#13133128	—	2	2	no	—
88	C₄×#88322	—	2	1	no	—
89	C₄×#88322	C₂×C₄	1	1	yes	characteristic
90	C₄×#88322	C₂×C₄	1	1	yes	characteristic
91	C₄×#88322	D₄	1	1	yes	characteristic
92	C₄×#88322	D₄	1	1	yes	characteristic
93	C₄×#88322	C₂³	1	1	yes	characteristic
94	C₄×#88322	C₂³	1	1	yes	characteristic
95	C₄×#88322	C₂³	1	2	yes	—
96	C₈×Dic₅₃	—	2	2	no	—
97	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂²×C₈×C₅₃	—	2	1	no	—
99	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
105	#117×C₅₃	—	2	1	no	—
106	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	—	2	1	no	—
108	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
110	C₂²⋊C₈×C₅₃	—	2	1	no	—
111	C₄×C₈×C₅₃	—	2	1	no	—
112	C₄×C₈×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	C₂³	1	2	yes	—
114	#41×C₅₃	—	2	2	no	—
115	#41×C₅₃	—	2	2	no	—
116	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
125	C₂×C₂²⋊D₅₃	—	2	1	no	—
126	#11719945	—	2	2	no	—
127	#11719945	M₁₆	1	4	yes	complemented
128	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	—	2	1	no	—
130	C₂×C₄×D₅₃	—	2	1	no	—
131	C₂×C₄×D₅₃	—	2	1	no	—
132	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₅₃	—	2	4	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	#41	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂²⋊Dic₅₃	—	2	1	no	—
151	#11719925	—	2	2	no	—
152	#11719924	—	2	2	no	—
153	#11719923	—	2	2	no	—
154	#11719922	—	2	2	no	—
155	#11719921	—	2	1	no	—
156	#11719921	—	2	1	no	—
157	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
159	#11719919	—	2	1	no	—
160	#11719919	—	2	1	no	—
161	C₄×Dic₅₃	—	2	1	no	—
162	C₄×Dic₅₃	—	2	1	no	—
163	C₄×Dic₅₃	—	2	1	no	—
164	#11719917	—	4	4	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	4	2	no	—
169	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
170	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
171	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
172	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
173	C₂×#88322	C₂×D₄	1	2	yes	—
174	C₂×#88322	C₂×D₄	1	2	yes	—
175	C₈×D₅₃	—	4	4	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
178	C₂²×D₅₃	—	2	1	no	—
179	Q₈×C₅₃	—	2	1	no	—
180	Q₈×C₅₃	—	2	1	no	—
181	D₄×C₅₃	—	2	1	no	—
182	D₄×C₅₃	—	2	1	no	—
183	D₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
196	C₂×C₄×C₅₃	#131	1	2	yes	—
197	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
200	C₂²⋊D₅₃	—	2	4	no	—
201	C₂×Dic₅₃	—	2	1	no	—
202	C₂×Dic₅₃	—	2	1	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	D₂₁₂	—	2	1	no	—
205	D₂₁₂	C₂×M₁₆	1	2	yes	—
206	C₄×D₅₃	—	2	4	no	—
207	C₄×D₅₃	—	4	2	no	—
208	C₄×D₅₃	—	4	2	no	—
209	Dic₁₀₆	—	2	1	no	—
210	Dic₁₀₆	C₂×M₁₆	1	2	yes	—
211	C₈×C₅₃	—	2	4	no	—
212	C₈×C₅₃	—	4	2	no	—
213	#88322	—	2	1	no	—
214	#88322	—	2	1	no	—
215	#88322	—	2	1	no	—
216	#88322	—	2	1	no	—
217	#88322	—	2	2	no	—
218	#88322	—	2	2	no	—
219	#88322	—	4	2	no	—
220	#17868	—	53	1	no	maximal, Sylow
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	—	4	1	no	—
224	C₂²×C₅₃	—	4	1	no	—
225	C₂²×C₅₃	—	4	1	no	—
226	C₂²×C₅₃	#435	1	2	yes	—
227	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂×D₅₃	—	2	2	no	—
229	C₂×D₅₃	—	4	1	no	—
230	C₂×D₅₃	—	4	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	—	4	1	no	—
236	C₄×C₅₃	#435	1	2	yes	—
237	C₄×C₅₃	#443	1	2	yes	—
238	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
239	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
240	C₄×C₅₃	#575	1	2	yes	—
241	Dic₅₃	—	2	2	no	—
242	Dic₅₃	—	4	1	no	—
243	Dic₅₃	—	4	1	no	—
244	#2965	—	53	1	no	—
245	C₈×#41	—	53	1	no	—
246	C₄×#135	—	53	1	no	—
247	#1588	—	53	2	no	—
248	#1587	—	53	2	no	—
249	#1586	—	53	2	no	—
250	#1585	—	53	2	no	—
251	#1564	—	53	1	no	—
252	#1476	—	53	1	no	—
253	#1476	D₅₃	1	1	yes	characteristic, complemented
254	C₂×#322	—	53	1	no	—
255	C₂×C₅₃	—	2	2	no	—
256	C₂×C₅₃	—	8	1	no	—
257	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
258	C₂×C₅₃	#2943	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
260	D₅₃	—	4	2	no	—
261	C₂×#135	—	53	1	no	—
262	C₄×#41	—	53	1	no	—
263	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
264	#446	—	53	2	no	—
265	C₈×Q₈	—	53	2	no	—
266	C₄×Q₁₆	—	53	1	no	—
267	C₄×Q₁₆	—	53	1	no	—
268	C₄×SD₁₆	—	53	1	no	—
269	C₄×SD₁₆	—	53	1	no	—
270	C₄×SD₁₆	—	53	1	no	—
271	C₄×SD₁₆	—	53	1	no	—
272	C₄×D₈	—	53	1	no	—
273	C₄×D₈	—	53	1	no	—
274	#436	—	53	2	no	—
275	#435	—	106	2	no	—
276	C₈×D₄	—	53	2	no	—
277	C₈×D₄	—	106	2	no	—
278	#431	—	53	1	no	—
279	#431	—	106	1	no	—
280	#431	—	106	1	no	—
281	#431	C₂×D₅₃	1	1	yes	characteristic
282	#427	—	53	1	no	—
283	#416	—	53	1	no	—
284	#416	—	53	1	no	—
285	C₄×M₁₆	—	106	1	no	—
286	C₂×C₄×C₈	—	53	1	no	—
287	C₂×C₄×C₈	—	53	1	no	—
288	C₂×C₄×C₈	—	106	1	no	—
289	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
290	#335	—	53	2	no	—
291	#334	—	53	2	no	—
292	Q₈⋊C₈	—	53	2	no	—
293	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
294	D₄⋊C₈	—	53	2	no	—
295	D₄⋊C₈	C₂×D₅₃	1	2	yes	—
296	#322	—	53	2	no	—
297	#322	—	53	2	no	—
298	C₅₃	#17868	1	1	yes	characteristic, complemented, Sylow
299	C₂×#41	—	53	1	no	—
300	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
301	#135	—	106	4	no	—
302	C₂×Q₁₆	—	53	1	no	—
303	C₂×SD₁₆	—	53	1	no	—
304	C₂×SD₁₆	—	53	1	no	—
305	C₂×D₈	—	53	1	no	—
306	C₂×M₁₆	—	106	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	53	1	no	—
309	C₂²×C₈	—	53	1	no	—
310	C₂²×C₈	—	53	2	no	—
311	C₂²×C₈	—	106	2	no	—
312	C₄×Q₈	—	53	1	no	—
313	C₄×Q₈	—	53	1	no	—
314	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
315	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	53	1	no	—
319	C₄×D₄	—	53	1	no	—
320	C₄×D₄	—	106	1	no	—
321	C₄×D₄	—	106	1	no	—
322	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
323	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
324	#117	—	2	1	no	—
325	#117	—	53	1	no	—
326	#117	—	106	1	no	—
327	#117	C₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	106	1	no	—
330	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
331	#107	—	53	1	no	—
332	#107	—	53	1	no	—
333	#106	—	53	1	no	—
334	#106	—	53	1	no	—
335	C₄⋊C₈	—	53	2	no	—
336	C₄⋊C₈	—	106	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
339	#103	—	53	1	no	—
340	#103	—	53	1	no	—
341	#103	—	53	1	no	—
342	#103	—	53	1	no	—
343	#102	—	53	1	no	—
344	#102	—	53	1	no	—
345	#102	—	53	1	no	—
346	#102	—	53	1	no	—
347	C₂²⋊C₈	—	2	1	no	—
348	C₂²⋊C₈	—	106	2	no	—
349	C₂²⋊C₈	—	106	2	no	—
350	C₂²⋊C₈	—	106	1	no	—
351	#97	—	106	2	no	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	—	53	1	no	—
355	C₄×C₈	—	53	1	no	—
356	C₄×C₈	—	53	1	no	—
357	C₄×C₈	—	53	1	no	—
358	C₄×C₈	—	53	1	no	—
359	C₄×C₈	—	53	1	no	—
360	C₄×C₈	—	53	2	no	—
361	C₄×C₈	—	106	2	no	—
362	C₄×C₈	—	106	1	no	—
363	C₄×C₈	C₂²×D₅₃	1	2	yes	—
364	#41	—	2	2	no	—
365	#41	—	2	2	no	—
366	#41	—	53	4	no	—
367	#41	—	106	2	no	—
368	C₂×Q₈	—	53	1	no	—
369	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	53	1	no	—
372	C₂×D₄	—	106	1	no	—
373	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	106	1	no	—
378	C₂²×C₄	—	106	1	no	—
379	C₂²×C₄	—	106	1	no	—
380	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
381	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
382	C₂²×C₄	#11719947	1	1	yes	characteristic
383	Q₁₆	—	106	2	no	—
384	SD₁₆	—	106	2	no	—
385	SD₁₆	—	106	2	no	—
386	D₈	—	106	2	no	—
387	M₁₆	—	212	4	no	—
388	C₂×C₈	—	2	4	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	53	1	no	—
392	C₂×C₈	—	53	1	no	—
393	C₂×C₈	—	53	1	no	—
394	C₂×C₈	—	53	1	no	—
395	C₂×C₈	—	53	2	no	—
396	C₂×C₈	—	53	2	no	—
397	C₂×C₈	—	106	2	no	—
398	C₂×C₈	—	106	2	no	—
399	C₂×C₈	—	106	2	no	—
400	C₂×C₈	—	212	4	no	—
401	C₂×C₈	—	212	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	53	1	no	—
405	C₄⋊C₄	—	53	1	no	—
406	C₄⋊C₄	—	106	1	no	—
407	C₄⋊C₄	—	106	1	no	—
408	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
409	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	106	1	no	—
414	C₂²⋊C₄	—	106	1	no	—
415	C₂²⋊C₄	—	106	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	106	1	no	—
420	C₄²	—	106	1	no	—
421	C₄²	—	106	1	no	—
422	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
423	C₄²	D₄×D₅₃	1	1	yes	characteristic
424	C₄²	#11719947	1	1	yes	characteristic
425	C₄²	C₂³×D₅₃	1	1	yes	characteristic
426	C₂³	—	2	1	no	—
427	C₂³	—	106	1	no	—
428	C₂³	#13133177	1	1	yes	characteristic
429	Q₈	—	2	1	no	—
430	Q₈	—	2	1	no	—
431	Q₈	—	53	2	no	—
432	Q₈	—	106	1	no	—
433	D₄	—	2	1	no	—
434	D₄	—	2	1	no	—
435	D₄	—	4	2	no	—
436	D₄	—	53	2	no	—
437	D₄	—	106	4	no	—
438	D₄	—	106	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	106	1	no	—
448	C₂×C₄	—	106	4	no	—
449	C₂×C₄	—	106	1	no	—
450	C₂×C₄	—	106	1	no	—
451	C₂×C₄	—	212	2	no	—
452	C₂×C₄	—	212	2	no	—
453	C₂×C₄	#13133177	1	1	yes	characteristic
454	C₂×C₄	#13133177	1	1	yes	characteristic
455	C₂×C₄	#13133177	1	1	yes	characteristic
456	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
457	C₂×C₄	#13133223	1	2	yes	—
458	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
461	C₈	—	2	4	no	—
462	C₈	—	4	2	no	—
463	C₈	—	106	1	no	—
464	C₈	—	106	1	no	—
465	C₈	—	106	1	no	—
466	C₈	—	106	1	no	—
467	C₈	—	106	2	no	—
468	C₈	—	106	2	no	—
469	C₈	—	212	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	106	2	no	—
476	C₂²	—	212	1	no	—
477	C₂²	—	212	1	no	—
478	C₂²	#16810697	1	2	yes	—
479	C₂²	C₂×#13133177	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	106	2	no	—
486	C₄	—	212	1	no	—
487	C₄	—	212	1	no	—
488	C₄	#16810697	1	2	yes	—
489	C₄	#16810737	1	2	yes	—
490	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
491	C₄	C₂×#13133223	1	1	yes	characteristic
492	C₄	#16811420	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	8	1	no	—
495	C₂	—	212	2	no	—
496	C₂	C₂×#16810697	1	1	yes	characteristic
497	C₂	#20283511	1	1	yes	characteristic
498	C₂	C₂×#16810737	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1476	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1476	D₅₃
hypocenter	C₅₃	#17868
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
1. #20283682	C₂	semidirect
2. #20283671	C₂	semidirect
4. #20273596	C₂	semidirect
5. #20273589	C₂	semidirect
6. #20273582	C₂	semidirect
7. #20273575	C₂	semidirect
8. #20273461	C₂	semidirect
9. #20273098	C₂	semidirect
10. C₂×#16810344	C₂	semidirect
11. C₅₃×#1476	C₂	semidirect
127. #11719945	M₁₆	semidirect
253. #1476	D₅₃	semidirect
298. C₅₃	#17868	semidirect

Maximal subgroups

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

subgroup	index
#20283682	2
#20283671	2
C₄×#13133237	2
#20273596	2
#20273589	2
#20273582	2
#20273575	2
#20273461	2
#20273098	2
C₂×#16810344	2
C₅₃×#1476	2
#17868	53