Subgroups — #27394714

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
3	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274921	C₂	1	2	yes	maximal
8	#20274918	C₂	1	2	yes	maximal
9	#20274885	C₂	1	2	yes	maximal, complemented
10	#20274851	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810678	—	2	1	no	—
15	#16810678	—	2	1	no	—
16	#16810675	C₄	1	2	yes	—
17	#16810675	C₄	1	2	yes	—
18	#16810675	C₄	1	2	yes	—
19	#16810675	C₄	1	2	yes	—
20	#16810674	—	2	1	no	—
21	#16810674	—	2	1	no	—
22	C₂×#13133153	C₂²	1	2	yes	—
23	C₂×#13133152	C₂²	1	2	yes	—
24	C₂×#13133152	C₂²	1	2	yes	—
25	#16810647	C₂²	1	1	yes	characteristic
26	#16810646	C₂²	1	2	yes	—
27	#16810646	C₂²	1	1	yes	characteristic
28	#16810646	C₂²	1	1	yes	characteristic
29	#16810645	—	2	1	no	—
30	#16810645	—	2	1	no	—
31	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×D₅₃	—	2	1	no	—
33	C₄⋊C₈×D₅₃	—	2	1	no	—
34	#16810586	C₂²	1	1	yes	characteristic
35	#16810586	C₂²	1	1	yes	characteristic
36	C₂×#13133129	C₂²	1	2	yes	—
37	C₄×#11719917	C₂²	1	1	yes	characteristic
38	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
39	#16810388	C₂²	1	2	yes	—
40	#16810387	C₂²	1	2	yes	—
41	#16810387	C₂²	1	2	yes	—
42	C₂×#108×C₅₃	C₂²	1	2	yes	—
43	C₅₃×#423	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#341	C₂²	1	2	yes	—
47	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	1	yes	characteristic
52	C₂×#11719917	C₂³	1	2	yes	—
53	C₂×#11719912	—	2	2	no	—
54	M₁₆×D₅₃	—	4	2	no	—
55	M₁₆×D₅₃	—	4	2	no	—
56	C₂²×#88322	C₂³	1	2	yes	—
57	C₂×C₈×D₅₃	—	2	2	no	—
58	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	D₄	1	1	yes	characteristic
66	C₄²×D₅₃	D₄	1	1	yes	characteristic
67	C₄²×D₅₃	D₄	1	1	yes	characteristic
68	C₄²×D₅₃	D₄	1	1	yes	characteristic
69	#13133153	—	4	2	no	—
70	#13133152	—	4	2	no	—
71	#13133152	—	4	2	no	—
72	C₄⋊#88322	—	2	1	no	—
73	C₄⋊#88322	—	2	1	no	—
74	C₄⋊#88322	C₂×C₄	1	2	yes	—
75	C₄⋊#88322	C₂×C₄	1	2	yes	—
76	#13133146	—	2	2	no	—
77	#13133146	—	2	2	no	—
78	#13133146	—	2	1	no	—
79	#13133146	—	2	1	no	—
80	#13133146	—	2	1	no	—
81	#13133146	—	2	1	no	—
82	#13133129	—	2	1	no	—
83	#13133129	—	2	1	no	—
84	#13133129	—	2	1	no	—
85	#13133129	—	2	1	no	—
86	#13133129	C₂×C₄	1	4	yes	—
87	#13133129	C₂×C₄	1	4	yes	—
88	#13133128	—	2	1	no	—
89	C₄×#88322	—	2	1	no	—
90	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
91	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
92	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	#108×C₅₃	—	4	2	no	—
98	C₄⋊C₈×C₅₃	—	2	1	no	—
99	C₄⋊C₈×C₅₃	—	2	1	no	—
100	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
101	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
102	#97×C₅₃	—	2	1	no	—
103	C₄×C₈×C₅₃	—	2	1	no	—
104	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₄×D₅₃	—	2	1	no	—
112	C₂×C₄×D₅₃	—	2	1	no	—
113	C₂×C₄×D₅₃	—	2	1	no	—
114	C₂×C₄×D₅₃	—	2	1	no	—
115	C₂×C₄×D₅₃	—	2	4	no	—
116	C₂×C₄×D₅₃	—	4	2	no	—
117	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
129	M₁₆×C₅₃	—	2	1	no	—
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
141	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
157	#11719917	—	2	1	no	—
158	#11719917	—	2	2	no	—
159	#11719917	—	2	1	no	—
160	#11719917	—	2	2	no	—
161	#11719917	—	2	1	no	—
162	#11719917	—	2	1	no	—
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	2	1	no	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	C₂²×C₄	1	4	yes	—
169	#11719912	—	4	4	no	—
170	C₈×D₅₃	—	4	4	no	—
171	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
172	C₂²×D₅₃	—	2	1	no	—
173	C₂²×D₅₃	—	2	1	no	—
174	C₂²×D₅₃	—	2	2	no	—
175	C₂²×D₅₃	—	4	1	no	—
176	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×Dic₅₃	—	2	1	no	—
191	C₂×Dic₅₃	—	2	1	no	—
192	C₂×Dic₅₃	—	2	2	no	—
193	C₂×Dic₅₃	—	2	2	no	—
194	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₄×D₅₃	—	2	2	no	—
201	C₄×D₅₃	—	2	1	no	—
202	C₄×D₅₃	—	2	1	no	—
203	C₄×D₅₃	—	2	1	no	—
204	C₄×D₅₃	—	2	1	no	—
205	C₄×D₅₃	—	2	1	no	—
206	C₄×D₅₃	—	2	1	no	—
207	C₄×D₅₃	—	2	1	no	—
208	C₄×D₅₃	—	2	1	no	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	2	2	no	—
212	C₄×D₅₃	—	2	2	no	—
213	C₄×D₅₃	—	4	1	no	—
214	C₄×D₅₃	—	4	1	no	—
215	C₄×D₅₃	#108	1	4	yes	complemented
216	C₈×C₅₃	—	2	4	no	—
217	C₈×C₅₃	—	4	2	no	—
218	C₈×C₅₃	—	4	1	no	—
219	C₈×C₅₃	—	4	1	no	—
220	#88322	—	2	4	no	—
221	#88322	—	4	2	no	—
222	#88322	—	4	1	no	—
223	#88322	—	4	1	no	—
224	#21993	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
228	C₂²×C₅₃	#544	1	1	yes	characteristic
229	C₂²×C₅₃	#544	1	1	yes	characteristic
230	C₂×D₅₃	—	2	1	no	—
231	C₂×D₅₃	—	2	1	no	—
232	C₂×D₅₃	—	2	1	no	—
233	C₂×D₅₃	—	2	1	no	—
234	C₂×D₅₃	—	2	1	no	—
235	C₂×D₅₃	—	4	2	no	—
236	C₂×D₅₃	—	4	2	no	—
237	C₂×D₅₃	C₂×#108	1	2	yes	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	—	2	1	no	—
241	C₄×C₅₃	C₂×#108	1	2	yes	—
242	C₄×C₅₃	#520	1	1	yes	characteristic
243	C₄×C₅₃	#520	1	1	yes	characteristic
244	C₄×C₅₃	#535	1	1	yes	characteristic
245	C₄×C₅₃	#536	1	1	yes	characteristic
246	Dic₅₃	—	2	1	no	—
247	Dic₅₃	—	2	1	no	—
248	Dic₅₃	—	2	1	no	—
249	Dic₅₃	—	2	1	no	—
250	Dic₅₃	C₂×#108	1	2	yes	—
251	Dic₅₃	#430	1	2	yes	—
252	#2920	—	53	1	no	—
253	C₂×#431	—	53	1	no	—
254	C₂×#423	—	53	1	no	—
255	#1950	—	53	1	no	—
256	#1950	—	53	1	no	—
257	#1950	—	53	1	no	—
258	#1950	D₅₃	1	1	yes	characteristic, complemented
259	#1855	—	53	2	no	—
260	#1855	—	53	2	no	—
261	#1848	—	53	2	no	—
262	#1837	—	53	2	no	—
263	C₂×C₅₃	—	2	1	no	—
264	C₂×C₅₃	—	2	1	no	—
265	C₂×C₅₃	#2317	1	1	yes	characteristic
266	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
267	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
268	D₅₃	—	2	2	no	—
269	D₅₃	—	4	1	no	—
270	C₂²×M₁₆	—	53	2	no	—
271	C₂²×C₄²	—	53	1	no	—
272	#432	—	106	1	no	—
273	#432	—	106	1	no	—
274	#431	—	53	2	no	—
275	#431	—	53	2	no	—
276	#431	—	53	2	no	—
277	#431	—	53	2	no	—
278	#431	—	106	1	no	—
279	#431	—	106	1	no	—
280	C₂×#108	—	53	2	no	—
281	C₂×#108	—	53	2	no	—
282	C₂×#108	—	53	2	no	—
283	C₂×#108	C₂×D₅₃	1	2	yes	—
284	#423	—	53	1	no	—
285	#423	—	53	2	no	—
286	#423	—	53	1	no	—
287	#423	—	53	1	no	—
288	#423	—	106	1	no	—
289	#423	—	106	1	no	—
290	#423	C₂×D₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₈	—	53	1	no	—
292	C₂×C₄⋊C₈	—	106	1	no	—
293	C₂×C₄⋊C₈	—	106	1	no	—
294	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
295	#407	—	53	1	no	—
296	#407	—	53	1	no	—
297	C₂×C₂²⋊C₈	—	53	2	no	—
298	C₄×M₁₆	—	53	1	no	—
299	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
300	C₂×C₄×C₈	—	53	2	no	—
301	#341	—	53	2	no	—
302	#341	—	53	2	no	—
303	#341	—	53	2	no	—
304	#341	C₂×D₅₃	1	2	yes	—
305	C₅₃	#21993	1	1	yes	characteristic, complemented, Sylow
306	C₂³×C₄	—	53	1	no	—
307	C₂³×C₄	—	53	1	no	—
308	C₂³×C₄	—	53	1	no	—
309	C₂×M₁₆	—	53	2	no	—
310	C₂×M₁₆	—	53	1	no	—
311	C₂×M₁₆	—	53	1	no	—
312	C₂×M₁₆	—	106	2	no	—
313	C₂×M₁₆	—	212	2	no	—
314	C₂×M₁₆	—	212	2	no	—
315	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
316	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
317	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
318	C₂²×C₈	—	53	2	no	—
319	C₂²×C₈	—	106	2	no	—
320	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	1	no	—
323	C₂×C₄²	—	53	1	no	—
324	C₂×C₄²	—	53	1	no	—
325	C₂×C₄²	—	53	1	no	—
326	C₂×C₄²	—	53	1	no	—
327	C₂×C₄²	—	53	1	no	—
328	C₂×C₄²	—	53	1	no	—
329	C₂×C₄²	—	53	1	no	—
330	C₂×C₄²	—	53	1	no	—
331	C₂×C₄²	—	53	1	no	—
332	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
333	#108	—	4	2	no	—
334	#108	—	212	2	no	—
335	#108	—	212	2	no	—
336	#108	—	212	2	no	—
337	C₄⋊C₈	—	2	1	no	—
338	C₄⋊C₈	—	2	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	—	106	1	no	—
344	C₄⋊C₈	—	106	1	no	—
345	C₄⋊C₈	—	106	1	no	—
346	C₄⋊C₈	—	106	1	no	—
347	C₄⋊C₈	—	106	1	no	—
348	C₄⋊C₈	—	106	1	no	—
349	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
350	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
351	C₂²⋊C₈	—	53	4	no	—
352	C₂²⋊C₈	—	53	4	no	—
353	C₂²⋊C₈	—	106	1	no	—
354	C₂²⋊C₈	—	106	1	no	—
355	C₂²⋊C₈	—	106	1	no	—
356	C₂²⋊C₈	—	106	1	no	—
357	#97	—	2	1	no	—
358	#97	—	106	1	no	—
359	C₄×C₈	—	2	1	no	—
360	C₄×C₈	—	53	4	no	—
361	C₄×C₈	—	53	4	no	—
362	C₄×C₈	—	106	1	no	—
363	C₂⁴	—	53	1	no	—
364	C₂²×C₄	—	53	1	no	—
365	C₂²×C₄	—	53	1	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	1	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	53	1	no	—
375	C₂²×C₄	—	53	1	no	—
376	C₂²×C₄	—	53	1	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	53	1	no	—
379	C₂²×C₄	—	106	4	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	—	212	2	no	—
385	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
386	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
388	M₁₆	—	2	1	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	2	1	no	—
391	M₁₆	—	2	2	no	—
392	M₁₆	—	2	1	no	—
393	M₁₆	—	2	1	no	—
394	M₁₆	—	106	1	no	—
395	M₁₆	—	106	2	no	—
396	M₁₆	—	106	1	no	—
397	M₁₆	—	106	2	no	—
398	M₁₆	—	106	1	no	—
399	M₁₆	—	106	1	no	—
400	M₁₆	—	212	4	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	53	4	no	—
407	C₂×C₈	—	106	2	no	—
408	C₂×C₈	—	106	2	no	—
409	C₂×C₈	—	106	1	no	—
410	C₂×C₈	—	106	1	no	—
411	C₂×C₈	—	106	2	no	—
412	C₂×C₈	—	212	4	no	—
413	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
414	C₄²	—	53	1	no	—
415	C₄²	—	53	1	no	—
416	C₄²	—	53	1	no	—
417	C₄²	—	53	1	no	—
418	C₄²	—	53	1	no	—
419	C₄²	—	53	1	no	—
420	C₄²	—	53	1	no	—
421	C₄²	—	53	1	no	—
422	C₄²	—	53	1	no	—
423	C₄²	—	53	1	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
428	C₄²	D₄×D₅₃	1	1	yes	characteristic
429	C₄²	D₄×D₅₃	1	1	yes	characteristic
430	C₂³	—	53	1	no	—
431	C₂³	—	53	1	no	—
432	C₂³	—	106	2	no	—
433	C₂³	—	106	1	no	—
434	C₂³	—	106	1	no	—
435	C₂³	—	212	1	no	—
436	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	53	1	no	—
440	C₂×C₄	—	53	1	no	—
441	C₂×C₄	—	53	4	no	—
442	C₂×C₄	—	53	1	no	—
443	C₂×C₄	—	53	1	no	—
444	C₂×C₄	—	53	1	no	—
445	C₂×C₄	—	53	1	no	—
446	C₂×C₄	—	106	2	no	—
447	C₂×C₄	—	106	2	no	—
448	C₂×C₄	—	106	2	no	—
449	C₂×C₄	—	106	2	no	—
450	C₂×C₄	—	106	2	no	—
451	C₂×C₄	—	106	2	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	—	106	1	no	—
458	C₂×C₄	—	106	2	no	—
459	C₂×C₄	—	106	1	no	—
460	C₂×C₄	—	106	1	no	—
461	C₂×C₄	—	106	1	no	—
462	C₂×C₄	—	106	1	no	—
463	C₂×C₄	—	212	1	no	—
464	C₂×C₄	—	212	1	no	—
465	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
467	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
468	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
470	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
471	C₂×C₄	#13133262	1	1	yes	characteristic
472	C₂×C₄	#13133263	1	1	yes	characteristic
473	C₂×C₄	#13133264	1	1	yes	characteristic
474	C₂×C₄	#13133265	1	1	yes	characteristic
475	C₈	—	2	4	no	—
476	C₈	—	4	2	no	—
477	C₈	—	4	1	no	—
478	C₈	—	4	1	no	—
479	C₈	—	106	4	no	—
480	C₈	—	212	2	no	—
481	C₈	—	212	1	no	—
482	C₈	—	212	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	53	2	no	—
486	C₂²	—	106	1	no	—
487	C₂²	—	106	1	no	—
488	C₂²	—	106	1	no	—
489	C₂²	—	106	1	no	—
490	C₂²	—	106	1	no	—
491	C₂²	—	212	2	no	—
492	C₂²	—	212	2	no	—
493	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
494	C₂²	#16811250	1	1	yes	characteristic
495	C₂²	#16811253	1	1	yes	characteristic
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	53	2	no	—
500	C₄	—	53	2	no	—
501	C₄	—	106	1	no	—
502	C₄	—	106	1	no	—
503	C₄	—	106	1	no	—
504	C₄	—	106	1	no	—
505	C₄	#108×D₅₃	1	2	yes	—
506	C₄	#16811102	1	1	yes	characteristic
507	C₄	#16811105	1	1	yes	characteristic
508	C₄	#16811189	1	1	yes	characteristic
509	C₄	#16811196	1	1	yes	characteristic
510	C₂	—	2	1	no	—
511	C₂	—	2	1	no	—
512	C₂	—	106	2	no	—
513	C₂	—	212	1	no	—
514	C₂	#20277337	1	1	yes	characteristic
515	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
516	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	#21993
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
4. #20275491	C₂	semidirect
5. #20275490	C₂	semidirect
6. #20275488	C₂	semidirect
9. #20274885	C₂	semidirect
10. #20274851	C₂	semidirect
11. C₅₃×#1950	C₂	semidirect
215. C₄×D₅₃	#108	semidirect
258. #1950	D₅₃	semidirect
305. C₅₃	#21993	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
C₂×#16810675	2
#423×D₅₃	2
#20275491	2
#20275490	2
#20275488	2
#20274921	2
#20274918	2
#20274885	2
#20274851	2
C₅₃×#1950	2
#21993	53