Subgroups — #27290054

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133146	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
6	C₄×#13133124	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133146	—	2	2	no	—
13	C₂×#13133146	—	2	2	no	—
14	C₂×#13133146	C₄	1	4	yes	complemented
15	C₂×C₄×#88322	—	2	2	no	—
16	C₂×C₄×#88322	—	2	2	no	—
17	C₂×C₄×#88322	—	2	2	no	—
18	C₂×#13133124	C₄	1	2	yes	—
19	C₂×#13133124	C₄	1	2	yes	—
20	#16810377	—	2	2	no	—
21	#16810377	—	2	2	no	—
22	#16810377	C₄	1	4	yes	complemented
23	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#336	—	2	2	no	—
27	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	—	2	1	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	—	2	1	no	—
34	C₂²×#88322	—	2	1	no	—
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	—	2	1	no	—
38	C₂²×#88322	C₂×C₄	1	2	yes	—
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	C₂²×#88322	D₄	1	2	yes	—
41	C₂²×#88322	D₄	1	2	yes	—
42	C₂²×#88322	D₄	1	2	yes	—
43	C₂²×#88322	Q₈	1	2	yes	—
44	C₂×C₄×Dic₅₃	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	C₂×C₄×Dic₅₃	—	2	1	no	—
49	C₂×C₄×Dic₅₃	—	2	1	no	—
50	C₂×C₄×Dic₅₃	—	2	1	no	—
51	C₂×C₄×Dic₅₃	—	2	1	no	—
52	C₂×C₄×Dic₅₃	—	2	1	no	—
53	C₂×C₄×Dic₅₃	—	2	1	no	—
54	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	#13133146	—	2	8	no	—
57	C₄×#88322	—	2	8	no	—
58	#13133124	—	2	4	no	—
59	#13133124	C₈	1	8	yes	complemented
60	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	—	2	1	no	—
65	C₂²×C₈×C₅₃	—	2	1	no	—
66	C₂²×C₈×C₅₃	—	2	1	no	—
67	C₂²×C₈×C₅₃	—	2	1	no	—
68	C₂²×C₈×C₅₃	—	2	1	no	—
69	C₂²×C₈×C₅₃	—	2	1	no	—
70	C₂²×C₈×C₅₃	—	2	1	no	—
71	C₂²×C₈×C₅₃	—	2	1	no	—
72	C₂²×C₈×C₅₃	—	2	1	no	—
73	C₂²×C₈×C₅₃	—	2	1	no	—
74	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
75	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₅₃	—	2	1	no	—
77	C₂×C₄²×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	—	2	1	no	—
79	C₂×C₄²×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	D₄	1	2	yes	—
81	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
82	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
89	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
104	C₂²×Dic₅₃	—	2	1	no	—
105	C₂²×Dic₅₃	—	2	1	no	—
106	C₂²×Dic₅₃	—	2	2	no	—
107	C₂²×Dic₅₃	—	2	2	no	—
108	C₂²×Dic₅₃	—	2	1	no	—
109	C₂²×Dic₅₃	—	2	1	no	—
110	C₂²×Dic₅₃	—	2	2	no	—
111	C₂²×Dic₅₃	—	2	2	no	—
112	C₂²×Dic₅₃	—	2	2	no	—
113	C₂²×Dic₅₃	—	2	2	no	—
114	C₂²×Dic₅₃	—	2	2	no	—
115	C₂²×Dic₅₃	—	2	2	no	—
116	C₂²×Dic₅₃	—	2	2	no	—
117	C₂²×Dic₅₃	C₄²	1	2	yes	—
118	C₂²×Dic₅₃	C₄²	1	2	yes	—
119	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
120	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	4	1	no	—
128	C₂×C₈×C₅₃	—	4	1	no	—
129	C₂×C₈×C₅₃	—	4	1	no	—
130	C₂×C₈×C₅₃	—	4	1	no	—
131	C₂×C₈×C₅₃	—	4	1	no	—
132	C₂×C₈×C₅₃	—	4	1	no	—
133	C₂×C₈×C₅₃	—	4	1	no	—
134	C₂×C₈×C₅₃	—	4	1	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄×Dic₅₃	—	2	2	no	—
140	C₄×Dic₅₃	—	2	2	no	—
141	C₄×Dic₅₃	—	2	2	no	—
142	C₄×Dic₅₃	—	2	2	no	—
143	C₄×Dic₅₃	—	4	1	no	—
144	C₄×Dic₅₃	—	4	1	no	—
145	C₄×Dic₅₃	—	4	1	no	—
146	C₄×Dic₅₃	—	4	1	no	—
147	C₂×#88322	—	2	2	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	2	no	—
150	C₂×#88322	—	2	2	no	—
151	C₂×#88322	—	2	2	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	2	no	—
154	C₂×#88322	—	2	2	no	—
155	C₂×#88322	—	2	2	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	C₂²⋊C₄	1	4	yes	—
160	C₂×#88322	C₄⋊C₄	1	4	yes	—
161	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
162	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂³×C₅₃	#117	1	1	yes	characteristic
164	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
167	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
168	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
169	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
170	C₂³×C₅₃	#131	1	2	yes	—
171	C₂³×C₅₃	#131	1	2	yes	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	4	no	—
177	C₂×C₄×C₅₃	—	2	4	no	—
178	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	complemented
179	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
182	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
191	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
192	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
193	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
199	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
200	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
201	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
202	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
203	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
204	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
205	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
207	C₂×Dic₅₃	—	2	2	no	—
208	C₂×Dic₅₃	—	2	2	no	—
209	C₂×Dic₅₃	—	2	4	no	—
210	C₂×Dic₅₃	—	2	4	no	—
211	C₂×Dic₅₃	—	2	2	no	—
212	C₂×Dic₅₃	—	2	2	no	—
213	C₂×Dic₅₃	—	2	2	no	—
214	C₂×Dic₅₃	—	2	4	no	—
215	C₂×Dic₅₃	—	2	4	no	—
216	C₂×Dic₅₃	—	2	4	no	—
217	C₂×Dic₅₃	—	4	2	no	—
218	C₂×Dic₅₃	—	4	2	no	—
219	C₂×Dic₅₃	—	4	2	no	—
220	C₂×Dic₅₃	—	4	2	no	—
221	C₂×Dic₅₃	—	4	2	no	—
222	C₂×Dic₅₃	—	4	2	no	—
223	C₂×Dic₅₃	C₄×C₈	1	4	yes	complemented
224	C₈×C₅₃	—	4	2	no	—
225	C₈×C₅₃	—	4	2	no	—
226	#88322	—	2	4	no	—
227	#88322	—	2	4	no	—
228	#10876	—	53	1	no	maximal, Sylow
229	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
231	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
232	C₂²×C₅₃	#385	1	1	yes	characteristic
233	C₂²×C₅₃	#387	1	1	yes	characteristic
234	C₂²×C₅₃	#388	1	1	yes	characteristic
235	C₂²×C₅₃	#389	1	1	yes	characteristic
236	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
237	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
238	C₂²×C₅₃	#405	1	1	yes	characteristic
239	C₂²×C₅₃	#405	1	1	yes	characteristic
240	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
241	C₂²×C₅₃	#424	1	2	yes	—
242	C₂²×C₅₃	#431	1	2	yes	—
243	C₂²×C₅₃	#433	1	2	yes	—
244	C₂²×C₅₃	C₈×D₄	1	2	yes	—
245	C₂²×C₅₃	C₈×D₄	1	2	yes	—
246	C₂²×C₅₃	C₈×D₄	1	2	yes	—
247	C₂²×C₅₃	#435	1	2	yes	—
248	C₂²×C₅₃	#435	1	2	yes	—
249	C₂²×C₅₃	#436	1	2	yes	—
250	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
251	C₂²×C₅₃	#446	1	2	yes	—
252	C₄×C₅₃	—	2	8	no	—
253	C₄×C₅₃	#385	1	1	yes	characteristic
254	C₄×C₅₃	#386	1	1	yes	characteristic
255	C₄×C₅₃	#387	1	1	yes	characteristic
256	C₄×C₅₃	#387	1	1	yes	characteristic
257	C₄×C₅₃	#387	1	1	yes	characteristic
258	C₄×C₅₃	#388	1	1	yes	characteristic
259	C₄×C₅₃	#389	1	1	yes	characteristic
260	C₄×C₅₃	#391	1	1	yes	characteristic
261	Dic₅₃	—	2	4	no	—
262	Dic₅₃	—	4	4	no	—
263	Dic₅₃	—	4	2	no	—
264	C₂²×C₄⋊C₈	—	53	1	no	—
265	C₂²×C₄×C₈	—	53	1	no	—
266	C₂×#336	—	53	1	no	—
267	C₂×#336	—	53	1	no	—
268	C₂×#336	—	53	1	no	—
269	C₂×#336	D₅₃	1	1	yes	characteristic
270	C₄×#95	—	53	1	no	—
271	C₂×C₅₃	#1436	1	1	yes	characteristic
272	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₅₃	#1753	1	1	yes	characteristic
274	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
275	C₂×C₅₃	#1768	1	1	yes	characteristic
276	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₅₃	#1771	1	1	yes	characteristic
278	C₂×C₅₃	#1850	1	2	yes	—
279	C₂×C₅₃	#1916	1	2	yes	—
280	C₂×C₅₃	#1923	1	2	yes	—
281	C₂×C₅₃	#1939	1	2	yes	—
282	C₂³×C₈	—	53	1	no	—
283	C₂³×C₈	—	53	1	no	—
284	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
285	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
286	C₂²×C₄²	—	53	1	no	—
287	C₂²×C₄²	—	53	1	no	—
288	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₈	—	53	4	no	—
290	C₂×C₄⋊C₈	—	106	2	no	—
291	C₂×C₄⋊C₈	—	106	2	no	—
292	C₂×C₄×C₈	—	106	2	no	—
293	C₂×C₄×C₈	—	106	2	no	—
294	C₂×C₄×C₈	—	106	2	no	—
295	C₂×#95	—	53	2	no	—
296	C₂×#95	—	53	2	no	—
297	#336	—	2	2	no	—
298	#336	—	53	4	no	—
299	#336	—	106	2	no	—
300	#336	—	106	2	no	—
301	C₅₃	#10876	1	1	yes	characteristic, complemented, Sylow
302	C₂³×C₄	—	53	1	no	—
303	C₂³×C₄	—	53	1	no	—
304	C₂³×C₄	—	53	2	no	—
305	C₂³×C₄	C₄×D₅₃	1	2	yes	—
306	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	—	2	1	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	1	no	—
319	C₂²×C₈	—	53	2	no	—
320	C₂²×C₈	—	53	2	no	—
321	C₂²×C₈	—	53	2	no	—
322	C₂²×C₈	—	53	2	no	—
323	C₂²×C₈	—	53	2	no	—
324	C₂²×C₈	—	53	2	no	—
325	C₂²×C₈	—	106	1	no	—
326	C₂²×C₈	—	106	1	no	—
327	C₂²×C₈	—	106	1	no	—
328	C₂²×C₈	—	106	1	no	—
329	C₂²×C₈	—	106	1	no	—
330	C₂²×C₈	—	106	1	no	—
331	C₂²×C₈	—	106	1	no	—
332	C₂²×C₈	—	106	1	no	—
333	C₂²×C₈	C₄×D₅₃	1	2	yes	—
334	C₂²×C₈	C₄×D₅₃	1	2	yes	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	53	2	no	—
340	C₂×C₄²	—	53	2	no	—
341	C₂×C₄²	—	106	1	no	—
342	C₂×C₄²	—	106	1	no	—
343	C₂×C₄²	—	106	1	no	—
344	C₂×C₄²	—	106	1	no	—
345	C₂×C₄²	—	106	1	no	—
346	C₂×C₄²	—	106	1	no	—
347	C₂×C₄²	—	106	1	no	—
348	C₂×C₄²	—	106	1	no	—
349	C₂×C₄²	—	106	1	no	—
350	C₂×C₄²	—	106	1	no	—
351	C₂×C₄²	Dic₁₀₆	1	2	yes	—
352	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
353	C₄⋊C₈	—	106	8	no	—
354	C₄×C₈	—	106	8	no	—
355	#95	—	53	8	no	—
356	#95	—	106	4	no	—
357	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	53	2	no	—
363	C₂²×C₄	—	53	4	no	—
364	C₂²×C₄	—	53	2	no	—
365	C₂²×C₄	—	53	4	no	—
366	C₂²×C₄	—	106	1	no	—
367	C₂²×C₄	—	106	1	no	—
368	C₂²×C₄	—	106	2	no	—
369	C₂²×C₄	—	106	2	no	—
370	C₂²×C₄	—	106	2	no	—
371	C₂²×C₄	—	106	1	no	—
372	C₂²×C₄	—	106	2	no	—
373	C₂²×C₄	—	106	1	no	—
374	C₂²×C₄	—	106	2	no	—
375	C₂²×C₄	—	106	2	no	—
376	C₂²×C₄	—	106	2	no	—
377	C₂²×C₄	—	106	2	no	—
378	C₂²×C₄	—	106	2	no	—
379	C₂²×C₄	C₈×D₅₃	1	4	yes	—
380	C₂²×C₄	#11719912	1	4	yes	—
381	C₂²×C₄	#11719919	1	4	yes	—
382	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
383	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₂²×C₄	#11719945	1	1	yes	characteristic
386	C₂²×C₄	#11719945	1	1	yes	characteristic
387	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
388	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
390	C₂²×C₄	#11719947	1	1	yes	characteristic
391	C₂²×C₄	#11719947	1	1	yes	characteristic
392	C₂²×C₄	#11719947	1	1	yes	characteristic
393	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
394	C₂²×C₄	#11719949	1	1	yes	characteristic
395	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	4	1	no	—
403	C₂×C₈	—	4	1	no	—
404	C₂×C₈	—	4	1	no	—
405	C₂×C₈	—	4	1	no	—
406	C₂×C₈	—	4	1	no	—
407	C₂×C₈	—	4	1	no	—
408	C₂×C₈	—	4	1	no	—
409	C₂×C₈	—	4	1	no	—
410	C₂×C₈	—	53	4	no	—
411	C₂×C₈	—	53	4	no	—
412	C₂×C₈	—	106	2	no	—
413	C₂×C₈	—	106	2	no	—
414	C₂×C₈	—	106	2	no	—
415	C₂×C₈	—	106	2	no	—
416	C₂×C₈	—	106	2	no	—
417	C₂×C₈	—	106	2	no	—
418	C₂×C₈	—	106	2	no	—
419	C₂×C₈	—	106	2	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	—	106	2	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	106	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	106	2	no	—
429	C₄²	—	106	2	no	—
430	C₄²	—	106	2	no	—
431	C₄²	—	106	2	no	—
432	C₄²	—	212	1	no	—
433	C₄²	—	212	1	no	—
434	C₄²	—	212	1	no	—
435	C₄²	—	212	1	no	—
436	C₂³	C₄²×D₅₃	1	1	yes	characteristic
437	C₂³	C₂×#11719919	1	1	yes	characteristic
438	C₂³	#13133174	1	1	yes	characteristic
439	C₂³	#13133177	1	1	yes	characteristic
440	C₂³	#13133177	1	1	yes	characteristic
441	C₂³	#13133179	1	1	yes	characteristic
442	C₂³	#13133182	1	1	yes	characteristic
443	C₂³	C₂×C₈×D₅₃	1	2	yes	—
444	C₂³	C₂×#11719912	1	2	yes	—
445	C₂³	#13133225	1	2	yes	—
446	C₂³	#13133225	1	2	yes	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	53	4	no	—
454	C₂×C₄	—	106	2	no	—
455	C₂×C₄	—	106	4	no	—
456	C₂×C₄	—	106	4	no	—
457	C₂×C₄	—	106	2	no	—
458	C₂×C₄	—	106	2	no	—
459	C₂×C₄	—	106	2	no	—
460	C₂×C₄	—	106	4	no	—
461	C₂×C₄	—	106	4	no	—
462	C₂×C₄	—	106	4	no	—
463	C₂×C₄	—	106	2	no	—
464	C₂×C₄	—	212	2	no	—
465	C₂×C₄	—	212	2	no	—
466	C₂×C₄	—	212	2	no	—
467	C₂×C₄	—	212	2	no	—
468	C₂×C₄	—	212	2	no	—
469	C₂×C₄	—	212	2	no	—
470	C₂×C₄	#13133146	1	8	yes	complemented
471	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
472	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
473	C₂×C₄	#13133173	1	1	yes	characteristic
474	C₂×C₄	#13133175	1	1	yes	characteristic
475	C₂×C₄	#13133177	1	1	yes	characteristic
476	C₂×C₄	#13133177	1	1	yes	characteristic
477	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
478	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
479	C₂×C₄	#13133179	1	1	yes	characteristic
480	C₂×C₄	#13133182	1	1	yes	characteristic
481	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
482	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
483	C₂×C₄	#13133185	1	1	yes	characteristic
484	C₂×C₄	#13133188	1	1	yes	characteristic
485	C₂×C₄	#13133190	1	1	yes	characteristic
486	C₂×C₄	#13133193	1	1	yes	characteristic
487	C₂×C₄	#13133194	1	1	yes	characteristic
488	C₂×C₄	#13133194	1	1	yes	characteristic
489	C₂×C₄	#13133195	1	1	yes	characteristic
490	C₂×C₄	#13133198	1	1	yes	characteristic
491	C₂×C₄	#13133199	1	1	yes	characteristic
492	C₂×C₄	#13133202	1	1	yes	characteristic
493	C₂×C₄	#13133203	1	1	yes	characteristic
494	C₂×C₄	#13133207	1	1	yes	characteristic
495	C₂×C₄	#13133208	1	1	yes	characteristic
496	C₂×C₄	#13133210	1	1	yes	characteristic
497	C₂×C₄	#13133212	1	1	yes	characteristic
498	C₂×C₄	#13133216	1	1	yes	characteristic
499	C₈	—	4	2	no	—
500	C₈	—	4	2	no	—
501	C₈	—	106	4	no	—
502	C₈	—	106	4	no	—
503	C₂²	#16810470	1	1	yes	characteristic
504	C₂²	#16810474	1	1	yes	characteristic
505	C₂²	C₄×#11719919	1	1	yes	characteristic
506	C₂²	#16810495	1	1	yes	characteristic
507	C₂²	#16810505	1	1	yes	characteristic
508	C₂²	#16810512	1	1	yes	characteristic
509	C₂²	#16810519	1	1	yes	characteristic
510	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
511	C₂²	C₄×#11719912	1	1	yes	characteristic
512	C₂²	#16810580	1	1	yes	characteristic
513	C₂²	#16810580	1	1	yes	characteristic
514	C₂²	C₂×#13133146	1	2	yes	—
515	C₂²	#16810651	1	2	yes	—
516	C₂²	#16810674	1	2	yes	—
517	C₂²	#16810685	1	2	yes	—
518	C₂²	#16810686	1	2	yes	—
519	C₂²	#16810686	1	2	yes	—
520	C₂²	#16810688	1	2	yes	—
521	C₂²	#16810691	1	2	yes	—
522	C₂²	#16810691	1	2	yes	—
523	C₂²	#16810700	1	2	yes	—
524	C₂²	#16810747	1	2	yes	—
525	C₂²	#16810750	1	2	yes	—
526	C₄	—	2	8	no	—
527	C₄	—	106	4	no	—
528	C₄	—	212	2	no	—
529	C₄	—	212	4	no	—
530	C₄	#16810494	1	1	yes	characteristic
531	C₄	#16810499	1	1	yes	characteristic
532	C₄	#16810504	1	1	yes	characteristic
533	C₄	#16810506	1	1	yes	characteristic
534	C₄	#16810510	1	1	yes	characteristic
535	C₄	#16810511	1	1	yes	characteristic
536	C₄	#16810520	1	1	yes	characteristic
537	C₄	#16810527	1	1	yes	characteristic
538	C₂	#20272946	1	1	yes	characteristic
539	C₂	#20274485	1	1	yes	characteristic
540	C₂	#20274495	1	1	yes	characteristic
541	C₂	C₄×#13133146	1	1	yes	characteristic
542	C₂	#20274575	1	1	yes	characteristic
543	C₂	#20274576	1	1	yes	characteristic
544	C₂	#20274586	1	1	yes	characteristic
545	C₂	#20274892	1	2	yes	—
546	C₂	#20275270	1	2	yes	—
547	C₂	#20275319	1	2	yes	—
548	C₂	#20275431	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#336	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#336	D₅₃
hypocenter	C₅₃	#10876
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
14. C₂×#13133146	C₄	semidirect
22. #16810377	C₄	semidirect
59. #13133124	C₈	semidirect
178. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
223. C₂×Dic₅₃	C₄×C₈	semidirect
301. C₅₃	#10876	semidirect
470. C₂×C₄	#13133146	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133146	2
C₂²×C₄×#88322	2
C₂×#16810377	2
C₄×#13133124	2
C₂×C₅₃×#336	2
#10876	53