Subgroups — C₅₃×#8949 #27216461

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×M₁₆×C₅₃	—	2	1	no	—
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×SD₁₆×C₅₃	—	2	2	no	—
13	C₄×SD₁₆×C₅₃	—	2	2	no	—
14	C₅₃×#436	—	2	2	no	—
15	C₅₃×#436	—	2	2	no	—
16	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#103×C₅₃	C₄	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#336	—	2	1	no	—
26	C₅₃×#336	—	2	1	no	—
27	C₅₃×Q₈⋊C₈	—	2	2	no	—
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	—	2	1	no	—
32	C₂×SD₁₆×C₅₃	—	2	2	no	—
33	C₂×SD₁₆×C₅₃	—	2	2	no	—
34	C₂×SD₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	4	no	—
36	C₂×M₁₆×C₅₃	—	4	2	no	—
37	C₂×M₁₆×C₅₃	—	4	2	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	—	2	1	no	—
46	C₄×Q₈×C₅₃	—	2	1	no	—
47	C₄×Q₈×C₅₃	—	2	1	no	—
48	C₄×Q₈×C₅₃	—	2	1	no	—
49	C₄×D₄×C₅₃	—	2	1	no	—
50	C₄×D₄×C₅₃	—	2	1	no	—
51	C₄×D₄×C₅₃	D₄	1	2	yes	—
52	C₄×D₄×C₅₃	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄²×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	#106×C₅₃	—	2	2	no	—
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
63	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
64	#103×C₅₃	—	2	2	no	—
65	#102×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	—	2	4	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	—	2	1	no	—
71	C₄×C₈×C₅₃	—	2	1	no	—
72	C₄×C₈×C₅₃	—	2	1	no	—
73	C₂⁴×C₅₃	—	2	1	no	—
74	C₂×Q₈×C₅₃	—	2	1	no	—
75	C₂×Q₈×C₅₃	—	2	1	no	—
76	C₂×Q₈×C₅₃	—	2	1	no	—
77	C₂×Q₈×C₅₃	—	2	1	no	—
78	C₂×Q₈×C₅₃	—	2	1	no	—
79	C₂×Q₈×C₅₃	M₁₆	1	2	yes	complemented
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	1	no	—
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	4	1	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	—	4	1	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
98	SD₁₆×C₅₃	—	2	8	no	—
99	M₁₆×C₅₃	—	4	8	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	4	1	no	—
113	C₂×C₈×C₅₃	—	4	1	no	—
114	C₂×C₈×C₅₃	—	4	1	no	—
115	C₂×C₈×C₅₃	—	4	1	no	—
116	C₂×C₈×C₅₃	—	4	1	no	—
117	C₂×C₈×C₅₃	—	4	1	no	—
118	C₂×C₈×C₅₃	—	4	1	no	—
119	C₂×C₈×C₅₃	M₁₆	1	2	yes	complemented
120	C₂×C₈×C₅₃	SD₁₆	1	2	yes	complemented
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	4	1	no	—
125	C₄⋊C₄×C₅₃	—	4	1	no	—
126	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	4	1	no	—
130	C₄²×C₅₃	—	4	1	no	—
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅₃	#41	1	1	yes	characteristic
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	—	4	1	no	—
137	C₂³×C₅₃	—	4	1	no	—
138	C₂³×C₅₃	—	4	1	no	—
139	C₂³×C₅₃	—	4	1	no	—
140	C₂³×C₅₃	—	4	1	no	—
141	C₂³×C₅₃	—	4	1	no	—
142	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
143	Q₈×C₅₃	—	2	2	no	—
144	Q₈×C₅₃	—	2	2	no	—
145	Q₈×C₅₃	—	4	1	no	—
146	Q₈×C₅₃	—	4	1	no	—
147	D₄×C₅₃	—	2	2	no	—
148	D₄×C₅₃	—	2	2	no	—
149	D₄×C₅₃	—	4	1	no	—
150	D₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
164	C₂×C₄×C₅₃	—	4	1	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
174	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
175	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
178	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	2	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	4	1	no	—
186	C₈×C₅₃	—	8	1	no	—
187	#8949	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	8	1	no	—
192	C₂²×C₅₃	—	8	1	no	—
193	C₂²×C₅₃	—	8	1	no	—
194	C₂²×C₅₃	—	8	1	no	—
195	C₂²×C₅₃	#386	1	1	yes	characteristic
196	C₂²×C₅₃	#435	1	1	yes	characteristic
197	C₂²×C₅₃	#436	1	1	yes	characteristic
198	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
199	C₂²×C₅₃	#440	1	1	yes	characteristic
200	C₂²×C₅₃	#443	1	1	yes	characteristic
201	C₂²×C₅₃	#444	1	1	yes	characteristic
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	#407	1	1	yes	characteristic
208	C₄×C₅₃	#408	1	1	yes	characteristic
209	C₄×C₅₃	#435	1	1	yes	characteristic
210	C₄×C₅₃	#436	1	1	yes	characteristic
211	C₄×C₅₃	#450	1	1	yes	characteristic
212	C₄×C₅₃	#452	1	1	yes	characteristic
213	C₄×C₅₃	#461	1	1	yes	characteristic
214	C₄×C₅₃	#463	1	1	yes	characteristic
215	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
216	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
217	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
218	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
220	#1277	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1273	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×C₅₃	—	8	2	no	—
223	C₂×C₅₃	#1583	1	1	yes	characteristic
224	C₂×C₅₃	#1588	1	1	yes	characteristic
225	C₂×C₅₃	#1590	1	1	yes	characteristic
226	C₂×C₅₃	#1592	1	1	yes	characteristic
227	C₂×C₅₃	#1870	1	1	yes	characteristic
228	C₂×C₅₃	#1876	1	1	yes	characteristic
229	C₂×C₅₃	#1877	1	1	yes	characteristic
230	C₂²×SD₁₆	C₄×C₅₃	1	1	yes	characteristic
231	C₂²×M₁₆	—	2	1	no	—
232	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
234	C₄×SD₁₆	—	2	2	no	—
235	C₄×SD₁₆	—	2	2	no	—
236	#436	—	2	2	no	—
237	#436	—	2	2	no	—
238	C₂×#106	C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×#103	C₄×C₅₃	1	1	yes	characteristic
242	C₂×#102	C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₂²⋊C₈	—	2	1	no	—
244	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
247	#336	—	2	1	no	—
248	#336	—	2	1	no	—
249	Q₈⋊C₈	—	2	2	no	—
250	D₄⋊C₈	—	2	2	no	—
251	C₅₃	#8949	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
253	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
254	C₂³×C₄	—	2	1	no	—
255	C₂×SD₁₆	—	2	2	no	—
256	C₂×SD₁₆	—	2	2	no	—
257	C₂×SD₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	—	4	2	no	—
260	C₂×M₁₆	—	4	2	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	—	2	1	no	—
270	C₄×Q₈	—	2	1	no	—
271	C₄×Q₈	—	2	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	D₄×C₅₃	1	2	yes	—
275	C₄×D₄	D₄×C₅₃	1	2	yes	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₄²	—	2	1	no	—
281	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
282	#106	—	2	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
287	#103	—	2	2	no	—
288	#102	—	2	2	no	—
289	C₂²⋊C₈	—	2	4	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₂⁴	—	2	1	no	—
297	C₂×Q₈	—	2	1	no	—
298	C₂×Q₈	—	2	1	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	M₁₆×C₅₃	1	2	yes	complemented
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	—	2	1	no	—
308	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
320	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
321	SD₁₆	—	2	8	no	—
322	M₁₆	—	4	8	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	—	4	1	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	M₁₆×C₅₃	1	2	yes	complemented
343	C₂×C₈	SD₁₆×C₅₃	1	2	yes	complemented
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₄²	—	4	1	no	—
353	C₄²	—	4	1	no	—
354	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₄²	#41×C₅₃	1	1	yes	characteristic
358	C₂³	—	2	2	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
366	Q₈	—	2	2	no	—
367	Q₈	—	2	2	no	—
368	Q₈	—	4	1	no	—
369	Q₈	—	4	1	no	—
370	D₄	—	2	2	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	1	no	—
373	D₄	—	4	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
392	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
393	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
394	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
397	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
398	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
400	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
401	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
402	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	1	no	—
407	C₈	—	4	2	no	—
408	C₈	—	4	1	no	—
409	C₈	—	8	1	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	1	no	—
417	C₂²	C₅₃×#386	1	1	yes	characteristic
418	C₂²	C₅₃×#435	1	1	yes	characteristic
419	C₂²	C₅₃×#436	1	1	yes	characteristic
420	C₂²	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
421	C₂²	C₅₃×#440	1	1	yes	characteristic
422	C₂²	C₅₃×#443	1	1	yes	characteristic
423	C₂²	C₅₃×#444	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	1	no	—
429	C₄	C₅₃×#407	1	1	yes	characteristic
430	C₄	C₅₃×#408	1	1	yes	characteristic
431	C₄	C₅₃×#435	1	1	yes	characteristic
432	C₄	C₅₃×#436	1	1	yes	characteristic
433	C₄	C₅₃×#450	1	1	yes	characteristic
434	C₄	C₅₃×#452	1	1	yes	characteristic
435	C₄	C₅₃×#461	1	1	yes	characteristic
436	C₄	C₅₃×#463	1	1	yes	characteristic
437	C₂	—	8	2	no	—
438	C₂	C₅₃×#1583	1	1	yes	characteristic
439	C₂	C₅₃×#1588	1	1	yes	characteristic
440	C₂	C₅₃×#1590	1	1	yes	characteristic
441	C₂	C₅₃×#1592	1	1	yes	characteristic
442	C₂	C₅₃×#1870	1	1	yes	characteristic
443	C₂	C₅₃×#1876	1	1	yes	characteristic
444	C₂	C₅₃×#1877	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1984	C₂	semidirect
4. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
6. C₅₃×#1277	C₂	semidirect
7. C₅₃×#1273	C₂	semidirect
79. C₂×Q₈×C₅₃	M₁₆	semidirect
119. C₂×C₈×C₅₃	M₁₆	semidirect
120. C₂×C₈×C₅₃	SD₁₆	semidirect
187. #8949	C₅₃	direct (N × H)
217. #1984	C₂×C₅₃	semidirect
218. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
220. #1277	C₂×C₅₃	semidirect
221. #1273	C₂×C₅₃	semidirect
251. C₅₃	#8949	direct (N × H)
302. C₂×Q₈	M₁₆×C₅₃	semidirect
342. C₂×C₈	M₁₆×C₅₃	semidirect
343. C₂×C₈	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₅₃	2
C₂×C₅₃×#436	2
C₅₃×#1984	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1277	2
C₅₃×#1273	2
#8949	53

Subgroups — C53×#8949 #27216461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8949 #27216461