Subgroups — #27355572

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11719913	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20273619	C₂	1	2	yes	maximal, complemented
5	#20273595	C₂	1	2	yes	maximal, complemented
6	#20273574	C₂	1	2	yes	maximal, complemented
7	#20273529	C₂	1	2	yes	maximal, complemented
8	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273145	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273141	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719913	C₄	1	1	yes	characteristic
13	C₂×M₁₆×D₅₃	—	2	1	no	—
14	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	#16810753	C₂²	1	2	yes	complemented
17	#16810751	—	2	2	no	—
18	C₈×Dic₁₀₆	C₂²	1	2	yes	complemented
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	C₄	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	C₄×#11719913	C₂²	1	1	yes	characteristic
27	#16810702	C₂²	1	2	yes	—
28	#16810696	—	2	2	no	—
29	C₈×D₂₁₂	C₂²	1	2	yes	—
30	C₂×#13133149	C₄	1	1	yes	characteristic
31	#16810647	C₂²	1	1	yes	characteristic
32	#16810647	C₂²	1	1	yes	characteristic
33	#16810646	—	2	1	no	—
34	C₂×#13133142	C₄	1	1	yes	characteristic
35	C₂×#13133139	C₄	1	1	yes	characteristic
36	#16810586	—	2	1	no	—
37	M₁₆×Dic₅₃	—	2	1	no	—
38	#16810376	C₂²	1	2	yes	complemented
39	#16810376	C₂²	1	2	yes	complemented
40	#16810355	C₂²	1	2	yes	complemented
41	#16810355	C₂²	1	2	yes	complemented
42	#16810352	C₂²	1	2	yes	—
43	#16810352	C₂²	1	2	yes	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#322	C₂²	1	2	yes	complemented
48	C₅₃×#322	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×#11719917	—	2	1	no	—
59	C₂×#11719912	—	2	2	no	—
60	M₁₆×D₅₃	—	4	4	no	—
61	C₂×C₈×D₅₃	—	2	2	no	—
62	C₄×Dic₁₀₆	—	2	1	no	—
63	C₄×Dic₁₀₆	—	2	1	no	—
64	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
68	C₄×D₂₁₂	—	2	1	no	—
69	C₄×D₂₁₂	—	2	1	no	—
70	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
71	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
72	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
73	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×#11719919	—	2	1	no	—
77	C₂×#11719921	—	2	1	no	—
78	C₂×C₄×Dic₅₃	—	2	1	no	—
79	#13133149	C₂×C₄	1	1	yes	characteristic
80	#13133149	C₂×C₄	1	1	yes	characteristic
81	#13133149	C₂×C₄	1	1	yes	characteristic
82	#13133149	C₂×C₄	1	1	yes	characteristic
83	C₄⋊#88322	—	2	1	no	—
84	C₄⋊#88322	—	2	1	no	—
85	C₄⋊#88322	C₂³	1	2	yes	—
86	C₄⋊#88322	C₂³	1	2	yes	—
87	#13133146	—	2	2	no	—
88	#13133146	—	2	2	no	—
89	#13133142	C₂×C₄	1	1	yes	characteristic
90	#13133142	C₂×C₄	1	1	yes	characteristic
91	#13133142	C₂×C₄	1	1	yes	characteristic
92	#13133142	C₂×C₄	1	1	yes	characteristic
93	#13133139	C₂×C₄	1	1	yes	characteristic
94	#13133139	C₂×C₄	1	1	yes	characteristic
95	#13133139	C₂×C₄	1	1	yes	characteristic
96	#13133139	C₂×C₄	1	1	yes	characteristic
97	#13133129	—	2	2	no	—
98	#13133129	—	2	2	no	—
99	#13133127	—	2	2	no	—
100	C₈×Dic₅₃	—	2	2	no	—
101	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
102	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	—	2	1	no	—
107	C₂²⋊C₈×C₅₃	—	2	1	no	—
108	#97×C₅₃	D₄	1	2	yes	—
109	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₅₃	C₂³	1	2	yes	—
112	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₄×C₈×C₅₃	C₂³	1	2	yes	—
115	C₂³×D₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
119	C₂²×Dic₅₃	—	2	1	no	—
120	C₂²×Dic₅₃	—	2	1	no	—
121	C₂²×Dic₅₃	—	2	1	no	—
122	C₂×D₂₁₂	—	2	2	no	—
123	C₂×D₂₁₂	—	2	1	no	—
124	C₂×D₂₁₂	—	4	1	no	—
125	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
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×C₄×D₅₃	—	2	1	no	—
131	C₂×C₄×D₅₃	—	4	2	no	—
132	C₂×C₄×D₅₃	—	4	2	no	—
133	C₂×Dic₁₀₆	—	2	2	no	—
134	C₂×Dic₁₀₆	—	2	1	no	—
135	C₂×Dic₁₀₆	—	4	1	no	—
136	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
137	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
138	M₁₆×C₅₃	—	2	2	no	—
139	M₁₆×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	1	no	—
143	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
148	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
149	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₅₃	#41	1	1	yes	characteristic
152	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
153	#11719921	—	2	1	no	—
154	#11719921	—	2	1	no	—
155	#11719921	—	2	1	no	—
156	#11719921	—	2	1	no	—
157	C₄⋊Dic₅₃	—	2	1	no	—
158	C₄⋊Dic₅₃	—	2	1	no	—
159	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
163	#11719919	—	2	1	no	—
164	#11719919	—	2	1	no	—
165	#11719919	—	2	1	no	—
166	#11719919	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₄×Dic₅₃	—	2	1	no	—
169	C₄×Dic₅₃	—	2	1	no	—
170	C₄×Dic₅₃	—	2	1	no	—
171	#11719917	—	4	2	no	—
172	#11719917	—	4	2	no	—
173	C₂×#88322	—	2	2	no	—
174	C₂×#88322	—	2	2	no	—
175	#11719913	—	2	2	no	—
176	#11719913	—	2	2	no	—
177	#11719913	—	2	2	no	—
178	#11719913	M₁₆	1	4	yes	complemented
179	#11719912	—	4	4	no	—
180	C₈×D₅₃	—	4	4	no	—
181	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂²×D₅₃	—	2	1	no	—
183	C₂²×D₅₃	—	2	1	no	—
184	C₂²×D₅₃	—	4	2	no	—
185	C₂²×D₅₃	—	4	2	no	—
186	C₂²×D₅₃	—	8	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	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	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₄×C₅₃	#137	1	2	yes	—
195	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×Dic₅₃	—	2	1	no	—
199	C₂×Dic₅₃	—	2	1	no	—
200	C₂×Dic₅₃	—	2	1	no	—
201	C₂×Dic₅₃	—	2	1	no	—
202	C₂×Dic₅₃	—	2	1	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	4	2	no	—
205	C₂×Dic₅₃	—	4	2	no	—
206	C₂×Dic₅₃	—	8	1	no	—
207	D₂₁₂	—	2	1	no	—
208	D₂₁₂	—	2	1	no	—
209	D₂₁₂	—	2	1	no	—
210	D₂₁₂	—	4	2	no	—
211	D₂₁₂	C₂×M₁₆	1	2	yes	—
212	C₄×D₅₃	—	4	1	no	—
213	C₄×D₅₃	—	4	1	no	—
214	C₄×D₅₃	—	4	1	no	—
215	C₄×D₅₃	—	4	1	no	—
216	C₄×D₅₃	—	4	2	no	—
217	C₄×D₅₃	—	4	2	no	—
218	Dic₁₀₆	—	2	1	no	—
219	Dic₁₀₆	—	2	1	no	—
220	Dic₁₀₆	—	2	1	no	—
221	Dic₁₀₆	—	4	2	no	—
222	Dic₁₀₆	C₂×M₁₆	1	2	yes	complemented
223	C₈×C₅₃	—	2	1	no	—
224	C₈×C₅₃	—	2	1	no	—
225	C₈×C₅₃	—	2	1	no	—
226	C₈×C₅₃	—	2	2	no	—
227	C₈×C₅₃	—	2	2	no	—
228	C₈×C₅₃	—	4	2	no	—
229	C₈×C₅₃	C₂×M₁₆	1	2	yes	complemented
230	#88322	—	4	2	no	—
231	#88322	—	4	2	no	—
232	#18040	—	53	1	no	maximal, Sylow
233	C₂²×C₅₃	—	2	1	no	—
234	C₂²×C₅₃	—	2	1	no	—
235	C₂²×C₅₃	#440	1	2	yes	—
236	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂×D₅₃	—	2	2	no	—
238	C₂×D₅₃	—	4	1	no	—
239	C₂×D₅₃	—	4	1	no	—
240	C₂×D₅₃	—	4	1	no	—
241	C₂×D₅₃	—	4	1	no	—
242	C₂×D₅₃	—	4	1	no	—
243	C₂×D₅₃	—	8	4	no	—
244	C₄×C₅₃	—	2	1	no	—
245	C₄×C₅₃	—	2	1	no	—
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	#440	1	2	yes	—
248	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
249	C₄×C₅₃	#568	1	1	yes	characteristic
250	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
252	Dic₅₃	—	2	2	no	—
253	Dic₅₃	—	4	1	no	—
254	Dic₅₃	—	4	1	no	—
255	Dic₅₃	—	4	1	no	—
256	Dic₅₃	—	4	1	no	—
257	Dic₅₃	—	4	1	no	—
258	Q₈×M₁₆	—	53	1	no	—
259	C₂×C₄×SD₁₆	—	53	1	no	—
260	D₄×M₁₆	—	53	1	no	—
261	#1592	—	53	2	no	—
262	#1588	—	53	2	no	—
263	#1585	—	53	2	no	—
264	SD₁₆⋊C₈	—	53	2	no	—
265	#1567	—	53	1	no	—
266	#1487	—	53	1	no	—
267	#1486	—	53	1	no	—
268	#1451	D₅₃	1	1	yes	characteristic, complemented
269	C₂×C₅₃	—	2	2	no	—
270	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
271	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
272	C₂×C₅₃	#2957	1	1	yes	characteristic
273	D₅₃	—	4	2	no	—
274	D₅₃	—	8	1	no	—
275	C₂²×SD₁₆	—	53	1	no	—
276	C₂²×M₁₆	—	106	1	no	—
277	C₂×C₄×Q₈	—	53	1	no	—
278	C₂×C₄×D₄	—	53	1	no	—
279	#446	—	53	2	no	—
280	#446	—	106	2	no	—
281	C₈×Q₈	—	53	2	no	—
282	C₄×SD₁₆	—	53	1	no	—
283	C₄×SD₁₆	—	53	1	no	—
284	C₄×SD₁₆	—	53	1	no	—
285	C₄×SD₁₆	—	53	1	no	—
286	C₄×SD₁₆	—	53	1	no	—
287	C₄×SD₁₆	—	53	1	no	—
288	C₄×SD₁₆	—	53	1	no	—
289	C₄×SD₁₆	—	53	1	no	—
290	#436	—	53	2	no	—
291	#435	—	106	2	no	—
292	C₈×D₄	—	53	2	no	—
293	#431	C₂×D₅₃	1	1	yes	characteristic
294	C₂×#106	—	53	1	no	—
295	#423	—	53	1	no	—
296	#423	—	53	1	no	—
297	#423	—	106	1	no	—
298	C₂×#103	—	53	1	no	—
299	C₂×#102	—	53	1	no	—
300	#407	—	106	1	no	—
301	C₄×M₁₆	—	106	1	no	—
302	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
303	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
304	#335	—	53	2	no	—
305	#335	—	53	2	no	—
306	Q₈⋊C₈	—	53	2	no	—
307	Q₈⋊C₈	—	53	2	no	—
308	D₄⋊C₈	—	53	2	no	—
309	D₄⋊C₈	—	53	2	no	—
310	#322	C₂×D₅₃	1	2	yes	complemented
311	#322	C₂×D₅₃	1	2	yes	complemented
312	C₅₃	#18040	1	1	yes	characteristic, complemented, Sylow
313	C₂²×Q₈	—	53	1	no	—
314	C₂²×D₄	—	53	1	no	—
315	C₂³×C₄	—	106	1	no	—
316	C₂×SD₁₆	—	53	1	no	—
317	C₂×SD₁₆	—	53	1	no	—
318	C₂×SD₁₆	—	53	1	no	—
319	C₂×SD₁₆	—	53	1	no	—
320	C₂×SD₁₆	—	106	4	no	—
321	C₂×M₁₆	—	106	1	no	—
322	C₂×M₁₆	—	106	1	no	—
323	C₂×M₁₆	—	106	2	no	—
324	C₂×M₁₆	—	212	4	no	—
325	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
326	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
327	C₂²×C₈	—	106	2	no	—
328	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
329	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
330	C₄×Q₈	—	53	1	no	—
331	C₄×Q₈	—	53	1	no	—
332	C₄×Q₈	—	53	1	no	—
333	C₄×Q₈	—	53	1	no	—
334	C₄×Q₈	—	106	1	no	—
335	C₄×Q₈	—	106	1	no	—
336	C₄×D₄	—	53	1	no	—
337	C₄×D₄	—	53	1	no	—
338	C₄×D₄	—	53	1	no	—
339	C₄×D₄	—	53	1	no	—
340	C₄×D₄	—	106	1	no	—
341	C₄×D₄	—	106	1	no	—
342	C₂×C₄⋊C₄	—	53	1	no	—
343	C₂×C₄⋊C₄	—	53	1	no	—
344	C₂×C₄⋊C₄	—	106	1	no	—
345	C₂×C₂²⋊C₄	—	106	1	no	—
346	C₂×C₄²	—	106	1	no	—
347	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
348	#106	—	53	1	no	—
349	#106	—	53	1	no	—
350	#106	—	53	1	no	—
351	#106	—	53	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	53	2	no	—
354	C₄⋊C₈	—	53	2	no	—
355	C₄⋊C₈	—	106	2	no	—
356	C₄⋊C₈	—	106	2	no	—
357	C₄⋊C₈	—	106	1	no	—
358	C₄⋊C₈	—	106	1	no	—
359	#103	—	53	1	no	—
360	#103	—	53	1	no	—
361	#103	—	53	1	no	—
362	#103	—	53	1	no	—
363	#102	—	53	1	no	—
364	#102	—	53	1	no	—
365	#102	—	53	1	no	—
366	#102	—	53	1	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	C₂²⋊C₈	—	106	2	no	—
369	C₂²⋊C₈	—	106	2	no	—
370	#97	—	106	2	no	—
371	#97	D₂₁₂	1	2	yes	—
372	C₄×C₈	—	106	2	no	—
373	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
374	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
375	C₄×C₈	C₂²×D₅₃	1	2	yes	—
376	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
377	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
378	C₄×C₈	C₂²×D₅₃	1	2	yes	—
379	C₂⁴	—	106	1	no	—
380	C₂×Q₈	—	53	1	no	—
381	C₂×Q₈	—	53	1	no	—
382	C₂×Q₈	—	106	2	no	—
383	C₂×Q₈	—	106	1	no	—
384	C₂×Q₈	—	212	1	no	—
385	C₂×D₄	—	53	1	no	—
386	C₂×D₄	—	53	1	no	—
387	C₂×D₄	—	106	2	no	—
388	C₂×D₄	—	106	1	no	—
389	C₂×D₄	—	212	1	no	—
390	C₂²×C₄	—	106	1	no	—
391	C₂²×C₄	—	106	1	no	—
392	C₂²×C₄	—	106	1	no	—
393	C₂²×C₄	—	106	1	no	—
394	C₂²×C₄	—	106	1	no	—
395	C₂²×C₄	—	106	1	no	—
396	C₂²×C₄	—	106	1	no	—
397	C₂²×C₄	—	212	2	no	—
398	C₂²×C₄	—	212	2	no	—
399	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
401	C₂²×C₄	#11719945	1	1	yes	characteristic
402	SD₁₆	—	53	4	no	—
403	SD₁₆	—	106	2	no	—
404	SD₁₆	—	106	2	no	—
405	SD₁₆	—	106	2	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	212	4	no	—
409	M₁₆	—	212	2	no	—
410	M₁₆	—	212	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	4	1	no	—
414	C₂×C₈	—	106	2	no	—
415	C₂×C₈	—	106	2	no	—
416	C₂×C₈	—	212	4	no	—
417	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
418	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
419	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
420	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
421	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
422	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
423	C₄⋊C₄	—	53	1	no	—
424	C₄⋊C₄	—	53	1	no	—
425	C₄⋊C₄	—	53	1	no	—
426	C₄⋊C₄	—	53	1	no	—
427	C₄⋊C₄	—	106	1	no	—
428	C₄⋊C₄	—	106	1	no	—
429	C₄⋊C₄	—	106	1	no	—
430	C₄⋊C₄	—	106	1	no	—
431	C₄⋊C₄	—	106	1	no	—
432	C₄⋊C₄	—	106	1	no	—
433	C₂²⋊C₄	—	106	1	no	—
434	C₂²⋊C₄	—	106	1	no	—
435	C₂²⋊C₄	—	106	1	no	—
436	C₂²⋊C₄	—	106	1	no	—
437	C₄²	—	106	1	no	—
438	C₄²	—	106	1	no	—
439	C₄²	—	106	1	no	—
440	C₄²	—	106	1	no	—
441	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
442	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
443	C₄²	#11719945	1	1	yes	characteristic
444	C₄²	C₂³×D₅₃	1	1	yes	characteristic
445	C₂³	—	106	1	no	—
446	C₂³	—	106	1	no	—
447	C₂³	—	424	1	no	—
448	C₂³	—	212	2	no	—
449	C₂³	—	212	2	no	—
450	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
451	Q₈	—	53	2	no	—
452	Q₈	—	106	1	no	—
453	Q₈	—	106	1	no	—
454	Q₈	—	106	1	no	—
455	Q₈	—	212	2	no	—
456	D₄	—	53	2	no	—
457	D₄	—	106	1	no	—
458	D₄	—	106	1	no	—
459	D₄	—	106	1	no	—
460	D₄	—	212	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	106	1	no	—
465	C₂×C₄	—	106	1	no	—
466	C₂×C₄	—	106	1	no	—
467	C₂×C₄	—	106	1	no	—
468	C₂×C₄	—	106	1	no	—
469	C₂×C₄	—	106	1	no	—
470	C₂×C₄	—	424	1	no	—
471	C₂×C₄	—	212	2	no	—
472	C₂×C₄	—	212	2	no	—
473	C₂×C₄	—	212	1	no	—
474	C₂×C₄	—	212	1	no	—
475	C₂×C₄	—	212	1	no	—
476	C₂×C₄	—	212	1	no	—
477	C₂×C₄	—	212	2	no	—
478	C₂×C₄	—	212	2	no	—
479	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
480	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
481	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
482	C₂×C₄	#13133245	1	2	yes	—
483	C₂×C₄	#13133250	1	2	yes	—
484	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
486	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
487	C₈	—	2	1	no	—
488	C₈	—	2	1	no	—
489	C₈	—	2	1	no	—
490	C₈	—	2	2	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	2	no	—
493	C₈	—	212	2	no	—
494	C₈	—	212	2	no	—
495	C₈	M₁₆×D₅₃	1	2	yes	complemented
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	106	2	no	—
499	C₂²	—	424	4	no	—
500	C₂²	—	212	1	no	—
501	C₂²	—	212	1	no	—
502	C₂²	—	212	1	no	—
503	C₂²	—	212	1	no	—
504	C₂²	—	212	1	no	—
505	C₂²	#16810725	1	2	yes	—
506	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	106	2	no	—
511	C₄	—	212	1	no	—
512	C₄	—	212	1	no	—
513	C₄	—	212	1	no	—
514	C₄	—	212	1	no	—
515	C₄	—	212	1	no	—
516	C₄	#16810725	1	2	yes	—
517	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
518	C₄	#16811389	1	1	yes	characteristic
519	C₄	C₂×#13133245	1	1	yes	characteristic
520	C₄	C₂×#13133250	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	424	1	no	—
523	C₂	—	212	2	no	—
524	C₂	M₁₆×D₂₁₂	1	1	yes	characteristic
525	C₂	C₂×#16810725	1	1	yes	characteristic
526	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18040
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
1. M₁₆×Dic₁₀₆	C₂	semidirect
4. #20273619	C₂	semidirect
5. #20273595	C₂	semidirect
6. #20273574	C₂	semidirect
7. #20273529	C₂	semidirect
8. #20273474	C₂	semidirect
9. #20273145	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
16. #16810753	C₂²	semidirect
18. C₈×Dic₁₀₆	C₂²	semidirect
38. #16810376	C₂²	semidirect
39. #16810376	C₂²	semidirect
40. #16810355	C₂²	semidirect
41. #16810355	C₂²	semidirect
47. C₅₃×#322	C₂²	semidirect
48. C₅₃×#322	C₂²	semidirect
178. #11719913	M₁₆	semidirect
222. Dic₁₀₆	C₂×M₁₆	semidirect
229. C₈×C₅₃	C₂×M₁₆	semidirect
268. #1451	D₅₃	semidirect
310. #322	C₂×D₅₃	semidirect
311. #322	C₂×D₅₃	semidirect
312. C₅₃	#18040	semidirect
495. C₈	M₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×C₄×#11719913	2
M₁₆×D₂₁₂	2
#20273619	2
#20273595	2
#20273574	2
#20273529	2
#20273474	2
#20273145	2
#20273141	2
C₅₃×#1451	2
#18040	53