Subgroups — #27355801

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	#20283804	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283752	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719924	C₂	1	1	yes	maximal, characteristic
4	#20273620	C₂	1	2	yes	maximal, complemented
5	#20273555	C₂	1	2	yes	maximal, complemented
6	#20273554	C₂	1	2	yes	maximal, complemented
7	#20273530	C₂	1	2	yes	maximal, complemented
8	#20273476	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273159	C₂	1	1	yes	maximal, characteristic
10	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719924	C₄	1	1	yes	characteristic
13	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
14	#16810754	—	2	2	no	—
15	#16810754	C₂²	1	2	yes	complemented
16	Q₈×#88322	C₂²	1	2	yes	complemented
17	C₄×#11719924	C₄	1	1	yes	characteristic
18	C₄×#11719924	C₄	1	1	yes	characteristic
19	C₄×#11719924	C₄	1	1	yes	characteristic
20	C₄×#11719924	C₄	1	1	yes	characteristic
21	C₄×#11719924	C₂²	1	1	yes	characteristic
22	C₄×#11719924	C₂²	1	1	yes	characteristic
23	C₄×#11719924	C₂²	1	1	yes	characteristic
24	C₄×#11719924	C₂²	1	1	yes	characteristic
25	#16810700	C₂²	1	2	yes	—
26	#16810696	—	2	2	no	—
27	#16810688	C₂²	1	2	yes	—
28	#16810680	—	2	1	no	—
29	#16810680	C₂²	1	1	yes	characteristic
30	#16810676	—	2	1	no	—
31	#16810676	C₂²	1	1	yes	characteristic
32	C₂×#13133148	C₄	1	1	yes	characteristic
33	C₂×#13133141	C₄	1	1	yes	characteristic
34	C₂×#13133137	C₄	1	1	yes	characteristic
35	#16810585	—	2	1	no	—
36	C₂²⋊C₈×D₅₃	—	2	1	no	—
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	#16810375	C₂²	1	2	yes	complemented
39	#16810373	C₂²	1	2	yes	complemented
40	#16810352	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	#16810345	C₂²	1	2	yes	complemented
43	#16810345	C₂²	1	2	yes	complemented
44	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	—	2	1	no	—
51	C₂×#11719924	—	2	4	no	—
52	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719912	—	2	2	no	—
57	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
58	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅₃	—	2	2	no	—
60	C₄×D₂₁₂	—	2	1	no	—
61	C₄×D₂₁₂	—	2	1	no	—
62	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
65	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×#11719921	—	2	1	no	—
68	#13133148	C₂×C₄	1	1	yes	characteristic
69	#13133148	C₂×C₄	1	1	yes	characteristic
70	#13133148	C₂×C₄	1	1	yes	characteristic
71	#13133148	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#88322	—	2	2	no	—
73	C₄⋊#88322	—	2	2	no	—
74	C₄⋊#88322	D₄	1	2	yes	—
75	C₄⋊#88322	C₂³	1	2	yes	—
76	#13133141	C₂×C₄	1	1	yes	characteristic
77	#13133141	C₂×C₄	1	1	yes	characteristic
78	#13133141	C₂×C₄	1	1	yes	characteristic
79	#13133141	C₂×C₄	1	1	yes	characteristic
80	#13133137	C₂×C₄	1	1	yes	characteristic
81	#13133137	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	1	no	—
85	C₂²⋊#88322	D₄	1	1	yes	characteristic
86	C₂²⋊#88322	D₄	1	1	yes	characteristic
87	#13133129	—	2	2	no	—
88	#13133129	—	2	2	no	—
89	#13133128	—	2	1	no	—
90	#13133128	—	2	2	no	—
91	C₄×#88322	—	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	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×M₁₆×C₅₃	—	2	1	no	—
99	C₄×Q₈×C₅₃	—	2	1	no	—
100	C₄×Q₈×C₅₃	—	2	1	no	—
101	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
106	C₂×C₄⋊C₄×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	#97×C₅₃	—	2	1	no	—
112	#97×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	C₂³	1	2	yes	—
114	C₂³×D₅₃	—	2	1	no	—
115	C₂×Q₈×C₅₃	—	2	1	no	—
116	C₂×Q₈×C₅₃	—	2	2	no	—
117	C₂×Q₈×C₅₃	—	4	1	no	—
118	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²×Dic₅₃	—	2	1	no	—
126	C₂×D₂₁₂	—	2	2	no	—
127	C₂×D₂₁₂	—	2	1	no	—
128	C₂×D₂₁₂	—	4	1	no	—
129	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×D₅₃	—	2	1	no	—
132	C₂×C₄×D₅₃	—	2	1	no	—
133	C₂×C₄×D₅₃	—	2	1	no	—
134	C₂×C₄×D₅₃	—	2	1	no	—
135	C₂×C₄×D₅₃	—	8	2	no	—
136	M₁₆×C₅₃	—	4	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₅₃	—	2	1	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	1	no	—
150	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	#41	1	1	yes	characteristic
153	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
154	#11719924	—	2	2	no	—
155	#11719924	—	2	2	no	—
156	#11719924	—	2	2	no	—
157	#11719924	M₁₆	1	4	yes	complemented
158	#11719921	—	2	1	no	—
159	#11719921	—	2	1	no	—
160	#11719921	—	2	1	no	—
161	#11719921	—	2	1	no	—
162	C₄⋊Dic₅₃	—	2	1	no	—
163	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	4	1	no	—
168	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
173	C₂×#88322	C₂×D₄	1	2	yes	—
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	Q₈×C₅₃	—	2	1	no	—
183	Q₈×C₅₃	—	2	1	no	—
184	Q₈×C₅₃	—	2	1	no	—
185	Q₈×C₅₃	—	4	2	no	—
186	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	2	1	no	—
205	C₂×Dic₅₃	—	8	1	no	—
206	D₂₁₂	—	2	1	no	—
207	D₂₁₂	—	2	1	no	—
208	D₂₁₂	—	2	1	no	—
209	D₂₁₂	—	4	2	no	—
210	D₂₁₂	C₂×M₁₆	1	2	yes	—
211	C₄×D₅₃	—	4	2	no	—
212	C₄×D₅₃	—	4	2	no	—
213	C₄×D₅₃	—	8	1	no	—
214	C₄×D₅₃	—	8	1	no	—
215	C₈×C₅₃	—	4	2	no	—
216	C₈×C₅₃	—	4	2	no	—
217	#88322	—	2	1	no	—
218	#88322	—	2	1	no	—
219	#88322	—	2	1	no	—
220	#88322	—	2	2	no	—
221	#88322	—	2	2	no	—
222	#88322	—	4	2	no	—
223	#88322	C₂×M₁₆	1	2	yes	—
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	2	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	—	2	1	no	—
241	C₄×C₅₃	—	4	1	no	—
242	C₄×C₅₃	—	4	1	no	—
243	C₄×C₅₃	—	4	1	no	—
244	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
245	C₄×C₅₃	#568	1	1	yes	characteristic
246	C₄×C₅₃	#577	1	2	yes	—
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	—	53	1	no	—
257	#1491	D₅₃	1	1	yes	characteristic, complemented
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	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
284	#432	—	106	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
294	C₂×C₄×C₈	—	53	1	no	—
295	#335	—	53	2	no	—
296	#334	—	53	2	no	—
297	Q₈⋊C₈	C₂×D₅₃	1	2	yes	complemented
298	Q₈⋊C₈	C₂×D₅₃	1	2	yes	complemented
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	C₄×D₅₃	1	1	yes	characteristic
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	—	2	1	no	—
313	C₂×M₁₆	—	106	2	no	—
314	C₂²×C₈	—	53	1	no	—
315	C₂²×C₈	—	53	1	no	—
316	C₂²×C₈	—	106	2	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
320	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
321	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
322	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
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	—	2	1	no	—
330	C₂×C₄⋊C₄	—	53	1	no	—
331	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	106	1	no	—
333	C₂×C₄²	—	2	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	—	53	2	no	—
341	C₄⋊C₈	—	106	2	no	—
342	C₄⋊C₈	—	106	2	no	—
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	—	53	1	no	—
354	C₂²⋊C₈	—	53	1	no	—
355	C₂²⋊C₈	—	106	2	no	—
356	C₂²⋊C₈	—	106	2	no	—
357	C₂²⋊C₈	—	106	1	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	1	no	—
363	C₄×C₈	—	53	1	no	—
364	C₄×C₈	—	53	1	no	—
365	C₄×C₈	—	53	1	no	—
366	C₄×C₈	—	53	2	no	—
367	C₄×C₈	—	106	2	no	—
368	C₄×C₈	C₂²×D₅₃	1	2	yes	—
369	C₂⁴	—	106	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
374	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
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	—	2	1	no	—
381	C₂²×C₄	—	2	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	#11719947	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	—	4	2	no	—
396	M₁₆	—	424	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	53	1	no	—
400	C₂×C₈	—	53	1	no	—
401	C₂×C₈	—	53	1	no	—
402	C₂×C₈	—	53	1	no	—
403	C₂×C₈	—	53	2	no	—
404	C₂×C₈	—	53	2	no	—
405	C₂×C₈	—	106	2	no	—
406	C₂×C₈	—	106	2	no	—
407	C₂×C₈	—	424	2	no	—
408	C₂×C₈	—	212	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	53	1	no	—
415	C₄⋊C₄	—	53	1	no	—
416	C₄⋊C₄	—	106	1	no	—
417	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
418	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
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	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
428	C₄²	D₄×D₅₃	1	1	yes	characteristic
429	C₄²	#11719947	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	#13133177	1	1	yes	characteristic
437	Q₈	—	2	1	no	—
438	Q₈	—	2	1	no	—
439	Q₈	—	2	1	no	—
440	Q₈	—	4	2	no	—
441	Q₈	M₁₆×D₅₃	1	2	yes	—
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	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	106	1	no	—
458	C₂×C₄	—	106	1	no	—
459	C₂×C₄	—	424	1	no	—
460	C₂×C₄	—	424	1	no	—
461	C₂×C₄	—	424	1	no	—
462	C₂×C₄	—	212	2	no	—
463	C₂×C₄	—	212	2	no	—
464	C₂×C₄	#13133177	1	1	yes	characteristic
465	C₂×C₄	#13133177	1	1	yes	characteristic
466	C₂×C₄	#13133177	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×#11719947	1	1	yes	characteristic
470	C₈	—	4	2	no	—
471	C₈	—	4	2	no	—
472	C₈	—	53	2	no	—
473	C₈	—	106	1	no	—
474	C₈	—	106	1	no	—
475	C₈	—	106	1	no	—
476	C₈	—	106	2	no	—
477	C₈	—	106	2	no	—
478	C₈	—	212	2	no	—
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	#16810742	1	2	yes	—
489	C₂²	C₂×#13133177	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	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	#16811432	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	424	1	no	—
505	C₂	—	212	2	no	—
506	C₂	#20283447	1	1	yes	characteristic
507	C₂	#20283570	1	1	yes	characteristic
508	C₂	C₂×#16810742	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×#1491	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1491	D₅₃
hypocenter	C₅₃	#18060
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1491

Complements

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

N	H ≅ G/N	type
1. #20283804	C₂	semidirect
4. #20273620	C₂	semidirect
5. #20273555	C₂	semidirect
6. #20273554	C₂	semidirect
7. #20273530	C₂	semidirect
8. #20273476	C₂	semidirect
10. #20273002	C₂	semidirect
11. C₅₃×#1491	C₂	semidirect
15. #16810754	C₂²	semidirect
16. Q₈×#88322	C₂²	semidirect
38. #16810375	C₂²	semidirect
39. #16810373	C₂²	semidirect
42. #16810345	C₂²	semidirect
43. #16810345	C₂²	semidirect
47. C₅₃×Q₈⋊C₈	C₂²	semidirect
48. C₅₃×Q₈⋊C₈	C₂²	semidirect
157. #11719924	M₁₆	semidirect
257. #1491	D₅₃	semidirect
297. Q₈⋊C₈	C₂×D₅₃	semidirect
298. Q₈⋊C₈	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
#20283804	2
#20283752	2
C₂×C₄×#11719924	2
#20273620	2
#20273555	2
#20273554	2
#20273530	2
#20273476	2
#20273159	2
#20273002	2
C₅₃×#1491	2
#18060	53