Subgroups — #27355796

Order 13568.

668 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283805	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283752	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719913	C₂	1	1	yes	maximal, characteristic
4	#20273616	C₂	1	2	yes	maximal, complemented
5	#20273557	C₂	1	2	yes	maximal, complemented
6	#20273550	C₂	1	2	yes	maximal, complemented
7	#20273526	C₂	1	2	yes	maximal, complemented
8	#20273161	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273157	C₂	1	1	yes	maximal, characteristic
10	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719913	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	#16810751	—	2	2	no	—
16	#16810750	C₂²	1	2	yes	complemented
17	#16810747	C₂²	1	2	yes	complemented
18	C₄×#11719913	C₄	1	1	yes	characteristic
19	C₄×#11719913	C₄	1	1	yes	characteristic
20	C₄×#11719913	C₄	1	1	yes	characteristic
21	C₄×#11719913	C₄	1	1	yes	characteristic
22	C₄×#11719913	C₂²	1	1	yes	characteristic
23	C₄×#11719913	C₂²	1	1	yes	characteristic
24	C₄×#11719913	C₂²	1	1	yes	characteristic
25	C₄×#11719913	C₂²	1	1	yes	characteristic
26	#16810700	C₂²	1	2	yes	—
27	#16810696	—	2	2	no	—
28	#16810688	C₂²	1	2	yes	—
29	#16810677	—	2	1	no	—
30	#16810676	C₂²	1	1	yes	characteristic
31	#16810673	—	2	1	no	—
32	C₂×#13133149	C₄	1	1	yes	characteristic
33	C₂×#13133142	C₄	1	1	yes	characteristic
34	C₂×#13133139	C₄	1	1	yes	characteristic
35	#16810585	—	2	1	no	—
36	C₂²⋊C₈×D₅₃	—	2	1	no	—
37	C₄×#11719917	C₂²	1	1	yes	characteristic
38	#16810353	C₂²	1	2	yes	complemented
39	#16810353	C₂²	1	2	yes	complemented
40	#16810350	C₂²	1	2	yes	—
41	#16810350	C₂²	1	2	yes	—
42	#16810345	C₂²	1	2	yes	complemented
43	#16810344	C₂²	1	2	yes	complemented
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#335	C₂²	1	2	yes	complemented
48	C₅₃×#334	C₂²	1	2	yes	complemented
49	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	—	2	1	no	—
52	C₂×#11719913	—	2	4	no	—
53	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
57	C₂×#11719917	—	2	1	no	—
58	C₂×#11719912	—	2	2	no	—
59	C₂×C₈×D₅₃	—	2	2	no	—
60	C₄×Dic₁₀₆	—	2	1	no	—
61	C₄×Dic₁₀₆	—	2	1	no	—
62	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
66	C₄×D₂₁₂	—	2	1	no	—
67	C₄×D₂₁₂	—	2	1	no	—
68	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
69	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
70	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
71	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719919	—	2	1	no	—
75	C₂×#11719921	—	2	1	no	—
76	C₂×C₄×Dic₅₃	—	2	1	no	—
77	#13133149	C₂×C₄	1	1	yes	characteristic
78	#13133149	C₂×C₄	1	1	yes	characteristic
79	#13133149	C₂×C₄	1	1	yes	characteristic
80	#13133149	C₂×C₄	1	1	yes	characteristic
81	C₄⋊#88322	D₄	1	2	yes	—
82	#13133146	—	2	2	no	—
83	#13133146	—	2	2	no	—
84	#13133142	C₂×C₄	1	1	yes	characteristic
85	#13133142	C₂×C₄	1	1	yes	characteristic
86	#13133142	C₂×C₄	1	1	yes	characteristic
87	#13133142	C₂×C₄	1	1	yes	characteristic
88	#13133139	C₂×C₄	1	1	yes	characteristic
89	#13133139	C₂×C₄	1	1	yes	characteristic
90	#13133139	C₂×C₄	1	1	yes	characteristic
91	#13133139	C₂×C₄	1	1	yes	characteristic
92	C₂²⋊#88322	D₄	1	1	yes	characteristic
93	C₂²⋊#88322	D₄	1	1	yes	characteristic
94	#13133129	—	2	2	no	—
95	#13133129	—	2	2	no	—
96	#13133128	—	2	1	no	—
97	#13133127	—	2	2	no	—
98	C₄×#88322	C₂³	1	2	yes	—
99	C₄×#88322	C₂³	1	2	yes	—
100	C₈×Dic₅₃	—	2	2	no	—
101	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
105	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
106	C₂²⋊C₈×C₅₃	—	2	1	no	—
107	C₂²⋊C₈×C₅₃	—	2	1	no	—
108	#97×C₅₃	—	2	1	no	—
109	#97×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂³×D₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×Dic₅₃	—	2	1	no	—
119	C₂²×Dic₅₃	—	2	1	no	—
120	C₂²×Dic₅₃	—	2	1	no	—
121	C₂×D₂₁₂	—	2	2	no	—
122	C₂×D₂₁₂	—	2	1	no	—
123	C₂×D₂₁₂	—	4	1	no	—
124	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	2	1	no	—
128	C₂×C₄×D₅₃	—	2	1	no	—
129	C₂×C₄×D₅₃	—	2	1	no	—
130	C₂×C₄×D₅₃	—	8	2	no	—
131	C₂×Dic₁₀₆	—	2	2	no	—
132	C₂×Dic₁₀₆	—	2	1	no	—
133	C₂×Dic₁₀₆	—	4	1	no	—
134	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
135	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	4	1	no	—
140	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	#41	1	1	yes	characteristic
147	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
148	#11719921	—	2	1	no	—
149	#11719921	—	2	1	no	—
150	#11719921	—	2	1	no	—
151	#11719921	—	2	1	no	—
152	C₄⋊Dic₅₃	—	2	1	no	—
153	C₄⋊Dic₅₃	—	2	1	no	—
154	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	#11719919	—	2	1	no	—
159	#11719919	—	2	1	no	—
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	1	no	—
166	#11719917	—	4	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	C₂×D₄	1	2	yes	—
169	C₂×#88322	C₂×D₄	1	2	yes	—
170	#11719913	—	2	2	no	—
171	#11719913	—	2	2	no	—
172	#11719913	—	2	2	no	—
173	#11719913	M₁₆	1	4	yes	complemented
174	#11719912	—	8	2	no	—
175	C₈×D₅₃	—	8	2	no	—
176	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₅₃	—	2	1	no	—
178	C₂²×D₅₃	—	2	1	no	—
179	C₂²×D₅₃	—	4	2	no	—
180	C₂²×D₅₃	—	4	2	no	—
181	C₂²×D₅₃	—	8	1	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
192	C₂×Dic₅₃	—	2	1	no	—
193	C₂×Dic₅₃	—	2	1	no	—
194	C₂×Dic₅₃	—	2	1	no	—
195	C₂×Dic₅₃	—	2	1	no	—
196	C₂×Dic₅₃	—	2	1	no	—
197	C₂×Dic₅₃	—	2	1	no	—
198	C₂×Dic₅₃	—	4	2	no	—
199	C₂×Dic₅₃	—	4	2	no	—
200	C₂×Dic₅₃	—	8	1	no	—
201	D₂₁₂	—	2	1	no	—
202	D₂₁₂	—	2	1	no	—
203	D₂₁₂	—	2	1	no	—
204	D₂₁₂	—	4	2	no	—
205	D₂₁₂	C₂×M₁₆	1	2	yes	—
206	C₄×D₅₃	—	4	2	no	—
207	C₄×D₅₃	—	4	2	no	—
208	C₄×D₅₃	—	8	1	no	—
209	C₄×D₅₃	—	8	1	no	—
210	Dic₁₀₆	—	2	1	no	—
211	Dic₁₀₆	—	2	1	no	—
212	Dic₁₀₆	—	2	1	no	—
213	Dic₁₀₆	—	4	2	no	—
214	Dic₁₀₆	C₂×M₁₆	1	2	yes	—
215	C₈×C₅₃	—	2	1	no	—
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	—	2	1	no	—
218	C₈×C₅₃	—	4	2	no	—
219	C₈×C₅₃	—	4	2	no	—
220	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
221	#88322	—	2	2	no	—
222	#88322	—	2	2	no	—
223	#88322	—	4	2	no	—
224	#18060	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	#444	1	2	yes	—
228	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
229	C₂×D₅₃	—	2	2	no	—
230	C₂×D₅₃	—	4	1	no	—
231	C₂×D₅₃	—	4	1	no	—
232	C₂×D₅₃	—	4	1	no	—
233	C₂×D₅₃	—	4	1	no	—
234	C₂×D₅₃	—	4	1	no	—
235	C₂×D₅₃	—	8	4	no	—
236	C₄×C₅₃	—	2	1	no	—
237	C₄×C₅₃	—	2	1	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
241	C₄×C₅₃	#568	1	1	yes	characteristic
242	C₄×C₅₃	#577	1	2	yes	—
243	Dic₅₃	—	2	2	no	—
244	Dic₅₃	—	4	1	no	—
245	Dic₅₃	—	4	1	no	—
246	Dic₅₃	—	4	1	no	—
247	Dic₅₃	—	4	1	no	—
248	Dic₅₃	—	4	1	no	—
249	#2984	—	53	1	no	—
250	#2974	—	53	1	no	—
251	C₂×C₄×SD₁₆	—	53	1	no	—
252	#1592	—	53	2	no	—
253	#1583	—	53	2	no	—
254	#1582	—	53	2	no	—
255	SD₁₆⋊C₈	—	53	2	no	—
256	#1568	D₅₃	1	1	yes	characteristic, complemented
257	#1491	—	53	1	no	—
258	#1490	—	53	1	no	—
259	#1451	—	53	1	no	—
260	C₂×C₅₃	—	2	2	no	—
261	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
262	C₂×C₅₃	#2949	1	1	yes	characteristic
263	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
264	D₅₃	—	4	2	no	—
265	D₅₃	—	8	1	no	—
266	C₂²×SD₁₆	—	53	1	no	—
267	C₂×C₄×Q₈	—	53	1	no	—
268	C₂×C₄×D₄	—	53	1	no	—
269	#446	—	53	2	no	—
270	#446	—	106	2	no	—
271	C₈×Q₈	—	53	2	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×SD₁₆	—	53	1	no	—
277	C₄×SD₁₆	—	53	1	no	—
278	C₄×SD₁₆	—	53	1	no	—
279	C₄×SD₁₆	—	53	1	no	—
280	#436	—	53	2	no	—
281	#435	—	106	2	no	—
282	C₈×D₄	—	53	2	no	—
283	#432	—	106	1	no	—
284	#432	C₂×D₅₃	1	1	yes	characteristic
285	#432	C₂×D₅₃	1	1	yes	characteristic
286	#431	—	53	1	no	—
287	#431	—	106	1	no	—
288	C₂×#106	—	53	1	no	—
289	C₂×#103	—	53	1	no	—
290	C₂×#102	—	53	1	no	—
291	#407	—	106	1	no	—
292	C₂×C₂²⋊C₈	—	106	1	no	—
293	C₄×M₁₆	—	53	1	no	—
294	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
295	#335	C₂×D₅₃	1	2	yes	complemented
296	#334	C₂×D₅₃	1	2	yes	complemented
297	Q₈⋊C₈	—	53	2	no	—
298	Q₈⋊C₈	—	53	2	no	—
299	D₄⋊C₈	—	53	2	no	—
300	D₄⋊C₈	—	53	2	no	—
301	#322	—	53	2	no	—
302	#322	—	53	2	no	—
303	C₅₃	#18060	1	1	yes	characteristic, complemented, Sylow
304	C₂²×Q₈	—	53	1	no	—
305	C₂²×D₄	—	53	1	no	—
306	C₂³×C₄	—	106	1	no	—
307	C₂×SD₁₆	—	53	1	no	—
308	C₂×SD₁₆	—	53	1	no	—
309	C₂×SD₁₆	—	53	1	no	—
310	C₂×SD₁₆	—	53	1	no	—
311	C₂×SD₁₆	—	106	4	no	—
312	C₂×M₁₆	—	106	2	no	—
313	C₂×M₁₆	—	106	1	no	—
314	C₂²×C₈	—	106	2	no	—
315	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
316	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
317	C₄×Q₈	—	53	1	no	—
318	C₄×Q₈	—	53	1	no	—
319	C₄×Q₈	—	53	1	no	—
320	C₄×Q₈	—	53	1	no	—
321	C₄×Q₈	—	106	1	no	—
322	C₄×Q₈	—	106	1	no	—
323	C₄×D₄	—	53	1	no	—
324	C₄×D₄	—	53	1	no	—
325	C₄×D₄	—	53	1	no	—
326	C₄×D₄	—	53	1	no	—
327	C₄×D₄	—	106	1	no	—
328	C₄×D₄	—	106	1	no	—
329	C₂×C₄⋊C₄	—	53	1	no	—
330	C₂×C₄⋊C₄	—	53	1	no	—
331	C₂×C₄⋊C₄	—	106	1	no	—
332	C₂×C₂²⋊C₄	—	106	1	no	—
333	C₂×C₄²	—	106	1	no	—
334	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
335	#106	—	53	1	no	—
336	#106	—	53	1	no	—
337	#106	—	53	1	no	—
338	#106	—	53	1	no	—
339	C₄⋊C₈	—	53	2	no	—
340	C₄⋊C₈	—	106	2	no	—
341	C₄⋊C₈	—	106	2	no	—
342	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
343	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
344	#103	—	53	1	no	—
345	#103	—	53	1	no	—
346	#103	—	53	1	no	—
347	#103	—	53	1	no	—
348	#102	—	53	1	no	—
349	#102	—	53	1	no	—
350	#102	—	53	1	no	—
351	#102	—	53	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	2	1	no	—
354	C₂²⋊C₈	—	53	1	no	—
355	C₂²⋊C₈	—	53	1	no	—
356	C₂²⋊C₈	—	106	2	no	—
357	C₂²⋊C₈	—	106	2	no	—
358	#97	—	2	1	no	—
359	#97	—	2	1	no	—
360	#97	—	106	2	no	—
361	#97	—	106	1	no	—
362	C₄×C₈	—	53	2	no	—
363	C₄×C₈	—	53	2	no	—
364	C₄×C₈	—	106	2	no	—
365	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
366	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
367	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
368	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
369	C₂⁴	—	106	1	no	—
370	C₂×Q₈	—	53	1	no	—
371	C₂×Q₈	—	53	1	no	—
372	C₂×Q₈	—	106	2	no	—
373	C₂×Q₈	—	106	1	no	—
374	C₂×Q₈	—	212	1	no	—
375	C₂×D₄	—	53	1	no	—
376	C₂×D₄	—	53	1	no	—
377	C₂×D₄	—	106	2	no	—
378	C₂×D₄	—	106	1	no	—
379	C₂×D₄	—	212	1	no	—
380	C₂²×C₄	—	106	1	no	—
381	C₂²×C₄	—	106	1	no	—
382	C₂²×C₄	—	106	1	no	—
383	C₂²×C₄	—	106	1	no	—
384	C₂²×C₄	—	106	1	no	—
385	C₂²×C₄	—	106	1	no	—
386	C₂²×C₄	—	106	1	no	—
387	C₂²×C₄	—	424	2	no	—
388	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
390	C₂²×C₄	#11719949	1	1	yes	characteristic
391	SD₁₆	—	53	4	no	—
392	SD₁₆	—	106	2	no	—
393	SD₁₆	—	106	2	no	—
394	SD₁₆	—	106	2	no	—
395	M₁₆	—	424	2	no	—
396	M₁₆	—	212	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	4	1	no	—
401	C₂×C₈	—	53	2	no	—
402	C₂×C₈	—	53	2	no	—
403	C₂×C₈	—	106	2	no	—
404	C₂×C₈	—	424	2	no	—
405	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
406	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
407	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
408	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
409	C₄⋊C₄	—	53	1	no	—
410	C₄⋊C₄	—	53	1	no	—
411	C₄⋊C₄	—	53	1	no	—
412	C₄⋊C₄	—	53	1	no	—
413	C₄⋊C₄	—	106	1	no	—
414	C₄⋊C₄	—	106	1	no	—
415	C₄⋊C₄	—	106	1	no	—
416	C₄⋊C₄	—	106	1	no	—
417	C₄⋊C₄	—	106	1	no	—
418	C₄⋊C₄	—	106	1	no	—
419	C₂²⋊C₄	—	106	1	no	—
420	C₂²⋊C₄	—	106	1	no	—
421	C₂²⋊C₄	—	106	1	no	—
422	C₂²⋊C₄	—	106	1	no	—
423	C₄²	—	106	1	no	—
424	C₄²	—	106	1	no	—
425	C₄²	—	106	1	no	—
426	C₄²	—	106	1	no	—
427	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
428	C₄²	D₄×D₅₃	1	1	yes	characteristic
429	C₄²	#11719949	1	1	yes	characteristic
430	C₄²	C₂³×D₅₃	1	1	yes	characteristic
431	C₂³	—	106	1	no	—
432	C₂³	—	106	1	no	—
433	C₂³	—	424	1	no	—
434	C₂³	—	212	2	no	—
435	C₂³	—	212	2	no	—
436	C₂³	#13133179	1	1	yes	characteristic
437	Q₈	—	53	2	no	—
438	Q₈	—	106	1	no	—
439	Q₈	—	106	1	no	—
440	Q₈	—	106	1	no	—
441	Q₈	—	212	2	no	—
442	D₄	—	53	2	no	—
443	D₄	—	106	1	no	—
444	D₄	—	106	1	no	—
445	D₄	—	106	1	no	—
446	D₄	—	212	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	106	1	no	—
452	C₂×C₄	—	106	1	no	—
453	C₂×C₄	—	106	1	no	—
454	C₂×C₄	—	106	1	no	—
455	C₂×C₄	—	106	1	no	—
456	C₂×C₄	—	106	1	no	—
457	C₂×C₄	—	424	1	no	—
458	C₂×C₄	—	424	1	no	—
459	C₂×C₄	—	424	1	no	—
460	C₂×C₄	—	212	2	no	—
461	C₂×C₄	—	212	2	no	—
462	C₂×C₄	—	212	2	no	—
463	C₂×C₄	—	212	2	no	—
464	C₂×C₄	#13133179	1	1	yes	characteristic
465	C₂×C₄	#13133179	1	1	yes	characteristic
466	C₂×C₄	#13133179	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
468	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	4	2	no	—
474	C₈	—	4	2	no	—
475	C₈	—	106	2	no	—
476	C₈	—	106	2	no	—
477	C₈	—	212	2	no	—
478	C₈	M₁₆×D₅₃	1	2	yes	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	106	2	no	—
482	C₂²	—	424	4	no	—
483	C₂²	—	212	1	no	—
484	C₂²	—	212	1	no	—
485	C₂²	—	212	1	no	—
486	C₂²	—	212	1	no	—
487	C₂²	—	212	1	no	—
488	C₂²	#16810744	1	2	yes	—
489	C₂²	C₂×#13133179	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	106	2	no	—
495	C₄	—	212	1	no	—
496	C₄	—	212	1	no	—
497	C₄	—	212	1	no	—
498	C₄	—	212	1	no	—
499	C₄	—	212	1	no	—
500	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
501	C₄	#16811389	1	1	yes	characteristic
502	C₄	#16811430	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	424	1	no	—
505	C₂	—	212	2	no	—
506	C₂	#20283450	1	1	yes	characteristic
507	C₂	#20283573	1	1	yes	characteristic
508	C₂	C₂×#16810744	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×#1568	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#18060
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #20283805	C₂	semidirect
4. #20273616	C₂	semidirect
5. #20273557	C₂	semidirect
6. #20273550	C₂	semidirect
7. #20273526	C₂	semidirect
8. #20273161	C₂	semidirect
10. #20273004	C₂	semidirect
11. C₅₃×#1568	C₂	semidirect
16. #16810750	C₂²	semidirect
17. #16810747	C₂²	semidirect
38. #16810353	C₂²	semidirect
39. #16810353	C₂²	semidirect
42. #16810345	C₂²	semidirect
43. #16810344	C₂²	semidirect
47. C₅₃×#335	C₂²	semidirect
48. C₅₃×#334	C₂²	semidirect
173. #11719913	M₁₆	semidirect
256. #1568	D₅₃	semidirect
295. #335	C₂×D₅₃	semidirect
296. #334	C₂×D₅₃	semidirect
303. C₅₃	#18060	semidirect

Maximal subgroups

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

subgroup	index
#20283805	2
#20283752	2
C₂×C₄×#11719913	2
#20273616	2
#20273557	2
#20273550	2
#20273526	2
#20273161	2
#20273157	2
#20273004	2
C₅₃×#1568	2
#18060	53