Subgroups — #27353948

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283833	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283832	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133237	C₂	1	1	yes	maximal, characteristic
4	#20273626	C₂	1	2	yes	maximal, complemented
5	#20273621	C₂	1	2	yes	maximal, complemented
6	#20273608	C₂	1	2	yes	maximal, complemented
7	#20273602	C₂	1	2	yes	maximal, complemented
8	C₂×#16810373	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273103	C₂	1	1	yes	maximal, characteristic, complemented
10	#20272999	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1477	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	#16810750	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	#16810700	C₂²	1	2	yes	—
26	#16810695	—	2	2	no	—
27	#16810694	—	2	2	no	—
28	#16810680	—	2	1	no	—
29	#16810679	—	2	1	no	—
30	#16810676	C₂²	1	1	yes	characteristic
31	#16810659	C₄	1	1	yes	characteristic
32	#16810647	—	2	1	no	—
33	#16810645	—	2	1	no	—
34	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
35	#16810619	C₄	1	1	yes	characteristic
36	#16810617	C₄	1	1	yes	characteristic
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	#16810373	C₂²	1	2	yes	—
39	#16810373	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	#16810345	C₂²	1	2	yes	—
43	C₈×#88322	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	C₂×#11719912	—	2	2	no	—
57	C₂²×#88322	—	2	1	no	—
58	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
59	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
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	1	no	—
74	C₄⋊#88322	—	2	2	no	—
75	C₄⋊#88322	D₄	1	1	yes	characteristic
76	C₄⋊#88322	D₄	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	2	yes	—
78	#13133146	—	2	2	no	—
79	#13133141	C₂×C₄	1	1	yes	characteristic
80	#13133141	C₂×C₄	1	1	yes	characteristic
81	#13133140	C₂×C₄	1	1	yes	characteristic
82	#13133140	C₂×C₄	1	1	yes	characteristic
83	#13133138	C₂×C₄	1	1	yes	characteristic
84	#13133138	C₂×C₄	1	1	yes	characteristic
85	#13133137	C₂×C₄	1	1	yes	characteristic
86	#13133137	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊#88322	—	2	2	no	—
88	C₂²⋊#88322	D₄	1	2	yes	—
89	#13133129	—	2	2	no	—
90	#13133128	—	2	2	no	—
91	#13133127	—	2	2	no	—
92	C₄×#88322	C₂×C₄	1	1	yes	characteristic
93	C₄×#88322	C₂×C₄	1	1	yes	characteristic
94	C₄×#88322	C₂³	1	1	yes	characteristic
95	C₄×#88322	C₂³	1	1	yes	characteristic
96	C₄×#88322	C₂³	1	2	yes	—
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	—	2	1	no	—
110	C₄⋊C₈×C₅₃	—	2	1	no	—
111	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
112	C₂²⋊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	—	2	2	no	—
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	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	8	1	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	—	8	2	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	8	1	no	—
168	C₂×#88322	M₁₆	1	4	yes	complemented
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	#11719912	—	8	2	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	1	no	—
185	C₂×C₄×C₅₃	—	2	1	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	1	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	8	1	no	—
194	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
198	C₂×C₄×C₅₃	#131	1	2	yes	—
199	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
202	C₂²⋊D₅₃	—	4	2	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	2	1	no	—
205	C₂×Dic₅₃	—	2	1	no	—
206	D₂₁₂	—	2	1	no	—
207	D₂₁₂	—	2	1	no	—
208	C₄×D₅₃	—	4	1	no	—
209	C₄×D₅₃	—	4	1	no	—
210	C₄×D₅₃	—	4	2	no	—
211	C₄×D₅₃	—	8	1	no	—
212	Dic₁₀₆	—	2	1	no	—
213	Dic₁₀₆	—	2	1	no	—
214	C₈×C₅₃	—	4	2	no	—
215	C₈×C₅₃	—	4	2	no	—
216	#88322	—	2	1	no	—
217	#88322	—	2	1	no	—
218	#88322	—	2	2	no	—
219	#88322	—	2	2	no	—
220	#88322	—	4	2	no	—
221	#88322	C₂×M₁₆	1	2	yes	—
222	#88322	C₂×M₁₆	1	2	yes	—
223	#17899	—	53	1	no	maximal, Sylow
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	—	4	1	no	—
228	C₂²×C₅₃	—	4	1	no	—
229	C₂²×C₅₃	#436	1	2	yes	—
230	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂×D₅₃	—	4	1	no	—
232	C₂×D₅₃	—	4	1	no	—
233	C₂×D₅₃	—	4	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	2	1	no	—
237	C₄×C₅₃	—	4	1	no	—
238	C₄×C₅₃	—	4	1	no	—
239	C₄×C₅₃	—	4	1	no	—
240	C₄×C₅₃	#436	1	2	yes	—
241	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
242	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
243	C₄×C₅₃	#577	1	1	yes	characteristic
244	C₄×C₅₃	#577	1	1	yes	characteristic
245	Dic₅₃	—	4	1	no	—
246	Dic₅₃	—	4	1	no	—
247	Dic₅₃	—	4	1	no	—
248	#2989	—	53	1	no	—
249	#2989	—	53	1	no	—
250	C₄×#135	—	53	1	no	—
251	#1594	—	53	2	no	—
252	#1593	—	53	2	no	—
253	#1590	—	53	2	no	—
254	#1590	—	53	2	no	—
255	C₂×#334	—	53	1	no	—
256	#1477	—	53	1	no	—
257	#1477	D₅₃	1	1	yes	characteristic, complemented
258	#1450	—	53	1	no	—
259	C₂×C₅₃	—	2	2	no	—
260	C₂×C₅₃	—	8	1	no	—
261	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
262	C₂×C₅₃	#2952	1	1	yes	characteristic
263	C₂×C₅₃	#2957	1	1	yes	characteristic
264	D₅₃	—	8	1	no	—
265	C₂×#135	—	53	1	no	—
266	C₄×#41	—	53	1	no	—
267	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
268	#446	—	53	2	no	—
269	#446	—	53	2	no	—
270	C₄×Q₁₆	—	53	1	no	—
271	C₄×Q₁₆	—	53	1	no	—
272	C₄×SD₁₆	—	53	1	no	—
273	C₄×SD₁₆	—	53	1	no	—
274	C₄×SD₁₆	—	53	1	no	—
275	C₄×SD₁₆	—	53	1	no	—
276	C₄×D₈	—	53	1	no	—
277	C₄×D₈	—	53	1	no	—
278	#436	—	53	2	no	—
279	#436	—	53	2	no	—
280	#435	—	106	2	no	—
281	#435	—	106	2	no	—
282	#432	—	106	1	no	—
283	#432	—	106	1	no	—
284	#431	—	53	1	no	—
285	#431	C₂×D₅₃	1	1	yes	characteristic
286	#427	—	53	1	no	—
287	#423	—	106	1	no	—
288	#423	—	106	1	no	—
289	C₂×C₄⋊C₈	—	53	1	no	—
290	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
291	#416	—	53	1	no	—
292	#416	—	53	1	no	—
293	C₂×C₄×C₈	—	53	1	no	—
294	#334	—	53	2	no	—
295	#334	—	53	2	no	—
296	Q₈⋊C₈	—	53	2	no	—
297	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
298	D₄⋊C₈	—	53	2	no	—
299	D₄⋊C₈	C₂×D₅₃	1	2	yes	—
300	#322	—	53	2	no	—
301	C₈²	—	53	2	no	—
302	C₅₃	#17899	1	1	yes	characteristic, complemented, Sylow
303	C₂×#41	—	53	1	no	—
304	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
305	#135	—	106	4	no	—
306	C₂×Q₁₆	—	53	1	no	—
307	C₂×SD₁₆	—	53	1	no	—
308	C₂×SD₁₆	—	53	1	no	—
309	C₂×D₈	—	53	1	no	—
310	C₂×M₁₆	—	106	2	no	—
311	C₂×M₁₆	—	106	2	no	—
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	—	53	1	no	—
314	C₂²×C₈	—	53	1	no	—
315	C₂²×C₈	—	106	1	no	—
316	C₄×Q₈	—	53	1	no	—
317	C₄×Q₈	—	53	1	no	—
318	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
319	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
320	C₄×D₄	—	2	1	no	—
321	C₄×D₄	—	2	1	no	—
322	C₄×D₄	—	53	1	no	—
323	C₄×D₄	—	53	1	no	—
324	C₄×D₄	—	106	1	no	—
325	C₄×D₄	—	106	1	no	—
326	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
327	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
328	#117	—	2	1	no	—
329	#117	—	53	1	no	—
330	#117	—	106	1	no	—
331	#117	C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	106	1	no	—
334	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
335	#107	—	53	1	no	—
336	#107	—	53	1	no	—
337	#106	—	53	1	no	—
338	#106	—	53	1	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	—	53	1	no	—
342	C₄⋊C₈	—	53	1	no	—
343	C₄⋊C₈	—	53	2	no	—
344	C₄⋊C₈	—	106	2	no	—
345	C₄⋊C₈	—	106	2	no	—
346	C₄⋊C₈	—	106	1	no	—
347	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
348	#103	—	53	1	no	—
349	#103	—	53	1	no	—
350	#103	—	53	1	no	—
351	#103	—	53	1	no	—
352	#102	—	53	1	no	—
353	#102	—	53	1	no	—
354	#102	—	53	1	no	—
355	#102	—	53	1	no	—
356	C₂²⋊C₈	—	2	1	no	—
357	C₂²⋊C₈	—	53	2	no	—
358	C₂²⋊C₈	—	106	2	no	—
359	C₂²⋊C₈	—	106	2	no	—
360	#97	—	106	2	no	—
361	#97	—	106	2	no	—
362	C₄×C₈	—	53	1	no	—
363	C₄×C₈	—	53	1	no	—
364	C₄×C₈	—	53	1	no	—
365	C₄×C₈	—	53	2	no	—
366	C₄×C₈	—	53	1	no	—
367	C₄×C₈	C₂²×D₅₃	1	2	yes	—
368	#41	—	2	2	no	—
369	#41	—	2	2	no	—
370	#41	—	106	2	no	—
371	#41	—	106	2	no	—
372	C₂×Q₈	—	53	1	no	—
373	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	53	1	no	—
376	C₂×D₄	—	106	1	no	—
377	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	106	1	no	—
382	C₂²×C₄	—	106	1	no	—
383	C₂²×C₄	—	106	1	no	—
384	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
386	C₂²×C₄	#11719947	1	1	yes	characteristic
387	Q₁₆	—	106	2	no	—
388	SD₁₆	—	106	2	no	—
389	SD₁₆	—	106	2	no	—
390	D₈	—	106	2	no	—
391	M₁₆	—	424	2	no	—
392	M₁₆	—	424	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	8	1	no	—
396	C₂×C₈	—	53	4	no	—
397	C₂×C₈	—	53	1	no	—
398	C₂×C₈	—	53	1	no	—
399	C₂×C₈	—	53	1	no	—
400	C₂×C₈	—	53	1	no	—
401	C₂×C₈	—	53	2	no	—
402	C₂×C₈	—	53	2	no	—
403	C₂×C₈	—	106	2	no	—
404	C₂×C₈	—	106	2	no	—
405	C₂×C₈	—	424	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	53	1	no	—
409	C₄⋊C₄	—	53	1	no	—
410	C₄⋊C₄	—	106	1	no	—
411	C₄⋊C₄	—	106	1	no	—
412	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
413	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	106	1	no	—
418	C₂²⋊C₄	—	106	1	no	—
419	C₂²⋊C₄	—	106	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	106	1	no	—
424	C₄²	—	106	1	no	—
425	C₄²	—	106	1	no	—
426	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₄²	D₄×D₅₃	1	1	yes	characteristic
428	C₄²	#11719947	1	1	yes	characteristic
429	C₄²	C₂³×D₅₃	1	1	yes	characteristic
430	C₂³	—	2	1	no	—
431	C₂³	—	106	1	no	—
432	C₂³	#13133177	1	1	yes	characteristic
433	Q₈	—	2	1	no	—
434	Q₈	—	2	1	no	—
435	Q₈	—	106	1	no	—
436	Q₈	—	106	1	no	—
437	D₄	—	2	1	no	—
438	D₄	—	2	1	no	—
439	D₄	—	4	2	no	—
440	D₄	—	106	1	no	—
441	D₄	—	106	1	no	—
442	D₄	—	212	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	8	1	no	—
453	C₂×C₄	—	106	1	no	—
454	C₂×C₄	—	106	1	no	—
455	C₂×C₄	—	106	1	no	—
456	C₂×C₄	—	424	1	no	—
457	C₂×C₄	—	212	1	no	—
458	C₂×C₄	—	212	1	no	—
459	C₂×C₄	—	212	2	no	—
460	C₂×C₄	#13133177	1	1	yes	characteristic
461	C₂×C₄	#13133177	1	1	yes	characteristic
462	C₂×C₄	#13133177	1	1	yes	characteristic
463	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
464	C₂×C₄	#13133223	1	2	yes	—
465	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
467	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
468	C₈	—	4	2	no	—
469	C₈	—	4	2	no	—
470	C₈	—	53	2	no	—
471	C₈	—	53	2	no	—
472	C₈	—	106	1	no	—
473	C₈	—	106	1	no	—
474	C₈	—	106	2	no	—
475	C₈	—	106	2	no	—
476	C₈	—	212	2	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	212	1	no	—
483	C₂²	—	212	1	no	—
484	C₂²	—	212	1	no	—
485	C₂²	#16810698	1	2	yes	—
486	C₂²	C₂×#13133177	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	4	1	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	212	1	no	—
494	C₄	—	212	1	no	—
495	C₄	—	212	1	no	—
496	C₄	#16810698	1	2	yes	—
497	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
498	C₄	C₂×#13133223	1	1	yes	characteristic
499	C₄	#16811432	1	1	yes	characteristic
500	C₄	#16811433	1	1	yes	characteristic
501	C₂	—	2	2	no	—
502	C₂	—	8	1	no	—
503	C₂	—	424	1	no	—
504	C₂	C₂×#16810698	1	1	yes	characteristic
505	C₂	#20283594	1	1	yes	characteristic
506	C₂	#20283628	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1477	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1477	D₅₃
hypocenter	C₅₃	#17899
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1477

Complements

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

N	H ≅ G/N	type
1. #20283833	C₂	semidirect
2. #20283832	C₂	semidirect
4. #20273626	C₂	semidirect
5. #20273621	C₂	semidirect
6. #20273608	C₂	semidirect
7. #20273602	C₂	semidirect
8. C₂×#16810373	C₂	semidirect
9. #20273103	C₂	semidirect
10. #20272999	C₂	semidirect
11. C₅₃×#1477	C₂	semidirect
168. C₂×#88322	M₁₆	semidirect
257. #1477	D₅₃	semidirect
302. C₅₃	#17899	semidirect

Maximal subgroups

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

subgroup	index
#20283833	2
#20283832	2
C₄×#13133237	2
#20273626	2
#20273621	2
#20273608	2
#20273602	2
C₂×#16810373	2
#20273103	2
#20272999	2
C₅₃×#1477	2
#17899	53