Subgroups — C₅₃×#22858 #27230370

Order 13568.

1386 conjugacy classes of subgroups.

The subgroup lattice has 836 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#447	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#102×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2361	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1936	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1926	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1893	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1879	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1879	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1853	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1762	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	—	2	1	no	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₅₃×#447	C₄	1	8	yes	complemented
28	C₅₃×#442	—	2	4	no	—
29	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×#102×C₅₃	—	2	4	no	—
32	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#390	C₂²	1	1	yes	characteristic
44	C₅₃×#390	C₂²	1	1	yes	characteristic
45	C₅₃×#390	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#389	C₂²	1	1	yes	characteristic, complemented
47	C₅₃×#388	—	2	1	no	—
48	C₅₃×#386	—	2	1	no	—
49	C₅₃×#386	—	2	1	no	—
50	C₅₃×#386	—	2	1	no	—
51	C₅₃×#385	—	2	1	no	—
52	C₅₃×#380	C₂²	1	1	yes	characteristic
53	C₅₃×#379	—	2	1	no	—
54	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
56	C₅₃×#340	C₂²	1	1	yes	characteristic, complemented
57	C₅₃×#340	C₂²	1	1	yes	characteristic, complemented
58	C₅₃×#340	C₂²	1	1	yes	characteristic, complemented
59	C₅₃×#340	C₂²	1	1	yes	characteristic, complemented
60	C₅₃×#336	C₂²	1	1	yes	characteristic, complemented
61	C₅₃×#336	C₂²	1	1	yes	characteristic, complemented
62	C₂⁵×C₅₃	—	2	1	no	—
63	C₂²×D₄×C₅₃	—	2	2	no	—
64	C₂²×D₄×C₅₃	—	2	2	no	—
65	C₂²×D₄×C₅₃	—	2	1	no	—
66	C₂²×D₄×C₅₃	—	2	1	no	—
67	C₂²×D₄×C₅₃	—	2	2	no	—
68	C₂²×D₄×C₅₃	—	4	1	no	—
69	C₂²×D₄×C₅₃	—	4	1	no	—
70	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
71	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅₃	—	2	1	no	—
75	C₂³×C₄×C₅₃	—	2	1	no	—
76	C₂³×C₄×C₅₃	—	2	1	no	—
77	C₂³×C₄×C₅₃	—	2	1	no	—
78	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×D₈×C₅₃	—	2	2	no	—
80	C₂×D₈×C₅₃	—	2	2	no	—
81	C₂×D₈×C₅₃	—	4	1	no	—
82	C₂×D₈×C₅₃	—	4	1	no	—
83	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
84	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
85	C₂²×C₈×C₅₃	—	2	2	no	—
86	C₂²×C₈×C₅₃	—	2	2	no	—
87	C₂²×C₈×C₅₃	—	4	1	no	—
88	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	#121×C₅₃	—	2	2	no	—
93	#121×C₅₃	C₂×C₄	1	4	yes	complemented
94	C₄×D₄×C₅₃	—	2	2	no	—
95	C₄×D₄×C₅₃	—	2	2	no	—
96	C₄×D₄×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	4	1	no	—
98	C₄×D₄×C₅₃	—	4	1	no	—
99	C₄×D₄×C₅₃	—	4	1	no	—
100	C₄×D₄×C₅₃	—	4	4	no	—
101	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
122	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
123	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	—	2	1	no	—
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	#106×C₅₃	—	2	2	no	—
128	#106×C₅₃	—	4	1	no	—
129	#102×C₅₃	—	2	2	no	—
130	#102×C₅₃	—	2	2	no	—
131	#102×C₅₃	—	2	2	no	—
132	#102×C₅₃	—	2	2	no	—
133	#102×C₅₃	—	4	1	no	—
134	#102×C₅₃	C₂×C₄	1	4	yes	complemented
135	#102×C₅₃	C₂×C₄	1	4	yes	complemented
136	#102×C₅₃	D₄	1	4	yes	complemented
137	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
138	#97×C₅₃	—	2	2	no	—
139	#97×C₅₃	—	4	1	no	—
140	#95×C₅₃	—	2	1	no	—
141	#95×C₅₃	—	2	1	no	—
142	#95×C₅₃	—	2	1	no	—
143	#95×C₅₃	D₄	1	1	yes	characteristic
144	#95×C₅₃	D₄	1	1	yes	characteristic
145	C₂⁴×C₅₃	—	2	1	no	—
146	C₂⁴×C₅₃	—	2	1	no	—
147	C₂⁴×C₅₃	—	2	1	no	—
148	C₂⁴×C₅₃	—	2	4	no	—
149	C₂⁴×C₅₃	—	4	2	no	—
150	C₂⁴×C₅₃	—	4	2	no	—
151	C₂⁴×C₅₃	—	4	2	no	—
152	C₂⁴×C₅₃	—	4	2	no	—
153	C₂⁴×C₅₃	—	8	1	no	—
154	C₂⁴×C₅₃	—	8	1	no	—
155	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₅₃	—	2	8	no	—
157	C₂×D₄×C₅₃	—	2	4	no	—
158	C₂×D₄×C₅₃	—	2	4	no	—
159	C₂×D₄×C₅₃	—	2	1	no	—
160	C₂×D₄×C₅₃	—	2	1	no	—
161	C₂×D₄×C₅₃	—	2	1	no	—
162	C₂×D₄×C₅₃	—	2	1	no	—
163	C₂×D₄×C₅₃	—	2	1	no	—
164	C₂×D₄×C₅₃	—	2	1	no	—
165	C₂×D₄×C₅₃	—	2	1	no	—
166	C₂×D₄×C₅₃	—	2	1	no	—
167	C₂×D₄×C₅₃	—	2	1	no	—
168	C₂×D₄×C₅₃	—	2	1	no	—
169	C₂×D₄×C₅₃	—	2	1	no	—
170	C₂×D₄×C₅₃	—	2	1	no	—
171	C₂×D₄×C₅₃	—	2	1	no	—
172	C₂×D₄×C₅₃	—	4	4	no	—
173	C₂×D₄×C₅₃	—	4	4	no	—
174	C₂×D₄×C₅₃	—	4	4	no	—
175	C₂×D₄×C₅₃	—	4	2	no	—
176	C₂×D₄×C₅₃	—	4	2	no	—
177	C₂×D₄×C₅₃	—	4	2	no	—
178	C₂×D₄×C₅₃	—	4	2	no	—
179	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
180	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
181	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
182	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
183	C₂×D₄×C₅₃	#41	1	2	yes	—
184	C₂²×C₄×C₅₃	—	2	1	no	—
185	C₂²×C₄×C₅₃	—	2	1	no	—
186	C₂²×C₄×C₅₃	—	2	1	no	—
187	C₂²×C₄×C₅₃	—	2	1	no	—
188	C₂²×C₄×C₅₃	—	2	1	no	—
189	C₂²×C₄×C₅₃	—	2	1	no	—
190	C₂²×C₄×C₅₃	—	2	1	no	—
191	C₂²×C₄×C₅₃	—	2	1	no	—
192	C₂²×C₄×C₅₃	—	2	1	no	—
193	C₂²×C₄×C₅₃	—	4	2	no	—
194	C₂²×C₄×C₅₃	—	4	2	no	—
195	C₂²×C₄×C₅₃	—	4	2	no	—
196	C₂²×C₄×C₅₃	—	4	2	no	—
197	C₂²×C₄×C₅₃	—	4	2	no	—
198	C₂²×C₄×C₅₃	—	4	2	no	—
199	C₂²×C₄×C₅₃	—	4	2	no	—
200	C₂²×C₄×C₅₃	—	4	2	no	—
201	C₂²×C₄×C₅₃	—	8	1	no	—
202	C₂²×C₄×C₅₃	—	8	1	no	—
203	C₂²×C₄×C₅₃	—	8	1	no	—
204	C₂²×C₄×C₅₃	—	8	1	no	—
205	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
206	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₅₃	#41	1	2	yes	—
212	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
213	D₈×C₅₃	—	2	8	no	—
214	D₈×C₅₃	—	4	4	no	—
215	C₂×C₈×C₅₃	—	2	1	no	—
216	C₂×C₈×C₅₃	—	2	1	no	—
217	C₂×C₈×C₅₃	—	2	1	no	—
218	C₂×C₈×C₅₃	—	2	1	no	—
219	C₂×C₈×C₅₃	—	2	1	no	—
220	C₂×C₈×C₅₃	—	2	1	no	—
221	C₂×C₈×C₅₃	—	2	1	no	—
222	C₂×C₈×C₅₃	—	2	1	no	—
223	C₂×C₈×C₅₃	—	2	1	no	—
224	C₂×C₈×C₅₃	—	4	4	no	—
225	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
226	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
227	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
228	C₄⋊C₄×C₅₃	—	2	1	no	—
229	C₄⋊C₄×C₅₃	—	2	1	no	—
230	C₄⋊C₄×C₅₃	—	2	1	no	—
231	C₄⋊C₄×C₅₃	—	2	1	no	—
232	C₄⋊C₄×C₅₃	—	2	1	no	—
233	C₄⋊C₄×C₅₃	—	2	1	no	—
234	C₄⋊C₄×C₅₃	—	2	1	no	—
235	C₄⋊C₄×C₅₃	—	2	1	no	—
236	C₄⋊C₄×C₅₃	—	4	2	no	—
237	C₄⋊C₄×C₅₃	—	4	2	no	—
238	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
239	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
240	C₂²⋊C₄×C₅₃	—	2	2	no	—
241	C₂²⋊C₄×C₅₃	—	2	2	no	—
242	C₂²⋊C₄×C₅₃	—	2	4	no	—
243	C₂²⋊C₄×C₅₃	—	4	2	no	—
244	C₂²⋊C₄×C₅₃	—	4	2	no	—
245	C₂²⋊C₄×C₅₃	—	4	2	no	—
246	C₂²⋊C₄×C₅₃	—	4	2	no	—
247	C₂²⋊C₄×C₅₃	—	4	2	no	—
248	C₂²⋊C₄×C₅₃	—	4	2	no	—
249	C₂²⋊C₄×C₅₃	—	4	2	no	—
250	C₂²⋊C₄×C₅₃	—	8	1	no	—
251	C₂²⋊C₄×C₅₃	—	8	1	no	—
252	C₂²⋊C₄×C₅₃	—	8	1	no	—
253	C₄²×C₅₃	—	2	1	no	—
254	C₄²×C₅₃	—	2	1	no	—
255	C₄²×C₅₃	—	2	1	no	—
256	C₄²×C₅₃	—	2	1	no	—
257	C₄²×C₅₃	—	4	2	no	—
258	C₂³×C₅₃	—	2	1	no	—
259	C₂³×C₅₃	—	2	1	no	—
260	C₂³×C₅₃	—	2	1	no	—
261	C₂³×C₅₃	—	2	1	no	—
262	C₂³×C₅₃	—	2	2	no	—
263	C₂³×C₅₃	—	2	2	no	—
264	C₂³×C₅₃	—	2	2	no	—
265	C₂³×C₅₃	—	2	2	no	—
266	C₂³×C₅₃	—	2	2	no	—
267	C₂³×C₅₃	—	4	8	no	—
268	C₂³×C₅₃	—	4	1	no	—
269	C₂³×C₅₃	—	4	1	no	—
270	C₂³×C₅₃	—	4	1	no	—
271	C₂³×C₅₃	—	4	1	no	—
272	C₂³×C₅₃	—	4	1	no	—
273	C₂³×C₅₃	—	4	1	no	—
274	C₂³×C₅₃	—	4	1	no	—
275	C₂³×C₅₃	—	4	1	no	—
276	C₂³×C₅₃	—	4	1	no	—
277	C₂³×C₅₃	—	4	1	no	—
278	C₂³×C₅₃	—	4	1	no	—
279	C₂³×C₅₃	—	4	1	no	—
280	C₂³×C₅₃	—	4	1	no	—
281	C₂³×C₅₃	—	4	1	no	—
282	C₂³×C₅₃	—	4	1	no	—
283	C₂³×C₅₃	—	4	1	no	—
284	C₂³×C₅₃	—	4	8	no	—
285	C₂³×C₅₃	—	8	4	no	—
286	C₂³×C₅₃	—	8	4	no	—
287	C₂³×C₅₃	—	8	4	no	—
288	C₂³×C₅₃	—	8	4	no	—
289	C₂³×C₅₃	—	8	4	no	—
290	C₂³×C₅₃	C₄×D₄	1	2	yes	—
291	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
292	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
293	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
294	D₄×C₅₃	—	2	2	no	—
295	D₄×C₅₃	—	2	2	no	—
296	D₄×C₅₃	—	2	2	no	—
297	D₄×C₅₃	—	2	2	no	—
298	D₄×C₅₃	—	2	2	no	—
299	D₄×C₅₃	—	2	2	no	—
300	D₄×C₅₃	—	2	2	no	—
301	D₄×C₅₃	—	4	8	no	—
302	D₄×C₅₃	—	4	8	no	—
303	D₄×C₅₃	—	4	1	no	—
304	D₄×C₅₃	—	4	1	no	—
305	D₄×C₅₃	—	4	1	no	—
306	D₄×C₅₃	—	4	1	no	—
307	D₄×C₅₃	C₄×D₄	1	4	yes	complemented
308	C₂×C₄×C₅₃	—	2	1	no	—
309	C₂×C₄×C₅₃	—	2	1	no	—
310	C₂×C₄×C₅₃	—	2	2	no	—
311	C₂×C₄×C₅₃	—	2	2	no	—
312	C₂×C₄×C₅₃	—	2	2	no	—
313	C₂×C₄×C₅₃	—	2	2	no	—
314	C₂×C₄×C₅₃	—	2	2	no	—
315	C₂×C₄×C₅₃	—	2	2	no	—
316	C₂×C₄×C₅₃	—	2	1	no	—
317	C₂×C₄×C₅₃	—	2	1	no	—
318	C₂×C₄×C₅₃	—	2	1	no	—
319	C₂×C₄×C₅₃	—	2	1	no	—
320	C₂×C₄×C₅₃	—	2	1	no	—
321	C₂×C₄×C₅₃	—	2	1	no	—
322	C₂×C₄×C₅₃	—	2	1	no	—
323	C₂×C₄×C₅₃	—	2	1	no	—
324	C₂×C₄×C₅₃	—	2	1	no	—
325	C₂×C₄×C₅₃	—	2	1	no	—
326	C₂×C₄×C₅₃	—	2	1	no	—
327	C₂×C₄×C₅₃	—	2	1	no	—
328	C₂×C₄×C₅₃	—	4	1	no	—
329	C₂×C₄×C₅₃	—	4	1	no	—
330	C₂×C₄×C₅₃	—	4	1	no	—
331	C₂×C₄×C₅₃	—	4	1	no	—
332	C₂×C₄×C₅₃	—	4	1	no	—
333	C₂×C₄×C₅₃	—	4	1	no	—
334	C₂×C₄×C₅₃	—	4	1	no	—
335	C₂×C₄×C₅₃	—	4	1	no	—
336	C₂×C₄×C₅₃	—	4	1	no	—
337	C₂×C₄×C₅₃	—	4	1	no	—
338	C₂×C₄×C₅₃	—	4	1	no	—
339	C₂×C₄×C₅₃	—	4	1	no	—
340	C₂×C₄×C₅₃	—	4	1	no	—
341	C₂×C₄×C₅₃	—	8	4	no	—
342	C₂×C₄×C₅₃	—	8	4	no	—
343	C₂×C₄×C₅₃	—	8	4	no	—
344	C₂×C₄×C₅₃	—	8	4	no	—
345	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
346	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
347	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
348	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
349	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
350	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
351	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
352	C₈×C₅₃	—	2	2	no	—
353	C₈×C₅₃	—	2	2	no	—
354	C₈×C₅₃	—	2	2	no	—
355	C₈×C₅₃	—	4	1	no	—
356	#22858	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
357	C₂²×C₅₃	—	2	2	no	—
358	C₂²×C₅₃	—	2	2	no	—
359	C₂²×C₅₃	—	2	2	no	—
360	C₂²×C₅₃	—	2	2	no	—
361	C₂²×C₅₃	—	2	2	no	—
362	C₂²×C₅₃	—	2	2	no	—
363	C₂²×C₅₃	—	2	4	no	—
364	C₂²×C₅₃	—	4	2	no	—
365	C₂²×C₅₃	—	4	2	no	—
366	C₂²×C₅₃	—	4	2	no	—
367	C₂²×C₅₃	—	4	2	no	—
368	C₂²×C₅₃	—	4	2	no	—
369	C₂²×C₅₃	—	4	2	no	—
370	C₂²×C₅₃	—	4	2	no	—
371	C₂²×C₅₃	—	4	2	no	—
372	C₂²×C₅₃	—	4	2	no	—
373	C₂²×C₅₃	—	4	2	no	—
374	C₂²×C₅₃	—	4	2	no	—
375	C₂²×C₅₃	—	4	2	no	—
376	C₂²×C₅₃	—	8	16	no	—
377	C₂²×C₅₃	—	8	1	no	—
378	C₂²×C₅₃	—	8	1	no	—
379	C₂²×C₅₃	—	8	1	no	—
380	C₂²×C₅₃	—	8	1	no	—
381	C₂²×C₅₃	—	8	1	no	—
382	C₂²×C₅₃	—	8	1	no	—
383	C₂²×C₅₃	—	8	1	no	—
384	C₂²×C₅₃	—	8	1	no	—
385	C₂²×C₅₃	#442	1	4	yes	complemented
386	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
387	C₂²×C₅₃	D₄²	1	1	yes	characteristic
388	C₂²×C₅₃	#546	1	1	yes	characteristic
389	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
391	C₂²×C₅₃	#576	1	1	yes	characteristic
392	C₂²×C₅₃	#577	1	1	yes	characteristic
393	C₄×C₅₃	—	2	2	no	—
394	C₄×C₅₃	—	4	2	no	—
395	C₄×C₅₃	—	4	1	no	—
396	C₄×C₅₃	—	4	2	no	—
397	C₄×C₅₃	—	4	1	no	—
398	C₄×C₅₃	—	4	1	no	—
399	C₄×C₅₃	—	4	1	no	—
400	C₄×C₅₃	—	8	1	no	—
401	C₄×C₅₃	—	8	1	no	—
402	C₄×C₅₃	—	8	1	no	—
403	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
404	C₄×C₅₃	#518	1	1	yes	characteristic
405	C₄×C₅₃	#546	1	1	yes	characteristic
406	C₄×C₅₃	#551	1	1	yes	characteristic
407	C₂×#447	C₂×C₅₃	1	1	yes	characteristic
408	C₂×#442	C₂×C₅₃	1	1	yes	characteristic, complemented
409	C₂²×#102	C₂×C₅₃	1	1	yes	characteristic, complemented
410	#2361	C₂×C₅₃	1	1	yes	characteristic, complemented
411	D₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
412	#1972	C₂×C₅₃	1	1	yes	characteristic, complemented
413	#1936	C₂×C₅₃	1	1	yes	characteristic, complemented
414	#1926	C₂×C₅₃	1	1	yes	characteristic, complemented
415	#1893	C₂×C₅₃	1	1	yes	characteristic, complemented
416	#1879	C₂×C₅₃	1	1	yes	characteristic, complemented
417	#1879	C₂×C₅₃	1	1	yes	characteristic, complemented
418	#1864	C₂×C₅₃	1	1	yes	characteristic, complemented
419	#1853	C₂×C₅₃	1	1	yes	characteristic, complemented
420	#1762	C₂×C₅₃	1	1	yes	characteristic, complemented
421	#1752	C₂×C₅₃	1	1	yes	characteristic, complemented
422	C₂×C₅₃	—	2	4	no	—
423	C₂×C₅₃	—	4	4	no	—
424	C₂×C₅₃	—	8	2	no	—
425	C₂×C₅₃	—	8	2	no	—
426	C₂×C₅₃	#2358	1	1	yes	characteristic
427	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
428	C₂×C₅₃	#2946	1	1	yes	characteristic
429	C₂×C₅₃	#3264	1	1	yes	characteristic
430	C₂×C₅₃	#3265	1	1	yes	characteristic
431	C₂×C₅₃	#3306	1	1	yes	characteristic
432	C₂×C₅₃	#3318	1	1	yes	characteristic
433	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
434	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
435	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
436	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
437	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
438	C₂×C₄×D₄	—	2	1	no	—
439	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
440	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
441	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
442	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
443	C₂²×C₂²⋊C₄	—	2	1	no	—
444	#447	C₄×C₅₃	1	8	yes	complemented
445	#442	—	2	4	no	—
446	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic, complemented
447	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic, complemented
448	C₂×#102	—	2	4	no	—
449	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
450	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
451	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
452	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
453	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
454	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
455	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic, complemented
456	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic, complemented
457	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
458	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic, complemented
459	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic, complemented
460	#390	C₂²×C₅₃	1	1	yes	characteristic
461	#390	C₂²×C₅₃	1	1	yes	characteristic
462	#390	C₂²×C₅₃	1	1	yes	characteristic, complemented
463	#389	C₂²×C₅₃	1	1	yes	characteristic, complemented
464	#388	—	2	1	no	—
465	#386	—	2	1	no	—
466	#386	—	2	1	no	—
467	#386	—	2	1	no	—
468	#385	—	2	1	no	—
469	#380	C₂²×C₅₃	1	1	yes	characteristic
470	#379	—	2	1	no	—
471	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
472	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
473	#340	C₂²×C₅₃	1	1	yes	characteristic, complemented
474	#340	C₂²×C₅₃	1	1	yes	characteristic, complemented
475	#340	C₂²×C₅₃	1	1	yes	characteristic, complemented
476	#340	C₂²×C₅₃	1	1	yes	characteristic, complemented
477	#336	C₂²×C₅₃	1	1	yes	characteristic, complemented
478	#336	C₂²×C₅₃	1	1	yes	characteristic, complemented
479	C₅₃	#22858	1	1	yes	characteristic, direct product factor, Sylow
480	C₂⁵	—	2	1	no	—
481	C₂²×D₄	—	2	2	no	—
482	C₂²×D₄	—	2	2	no	—
483	C₂²×D₄	—	2	1	no	—
484	C₂²×D₄	—	2	1	no	—
485	C₂²×D₄	—	2	2	no	—
486	C₂²×D₄	—	4	1	no	—
487	C₂²×D₄	—	4	1	no	—
488	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
489	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
490	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
491	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
492	C₂³×C₄	—	2	1	no	—
493	C₂³×C₄	—	2	1	no	—
494	C₂³×C₄	—	2	1	no	—
495	C₂³×C₄	—	2	1	no	—
496	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
497	C₂×D₈	—	2	2	no	—
498	C₂×D₈	—	2	2	no	—
499	C₂×D₈	—	4	1	no	—
500	C₂×D₈	—	4	1	no	—
501	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
502	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
503	C₂²×C₈	—	2	2	no	—
504	C₂²×C₈	—	2	2	no	—
505	C₂²×C₈	—	4	1	no	—
506	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
507	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
508	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
509	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
510	#121	—	2	2	no	—
511	#121	C₂×C₄×C₅₃	1	4	yes	complemented
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	—	2	2	no	—
514	C₄×D₄	—	2	2	no	—
515	C₄×D₄	—	4	1	no	—
516	C₄×D₄	—	4	1	no	—
517	C₄×D₄	—	4	1	no	—
518	C₄×D₄	—	4	4	no	—
519	C₂×C₄⋊C₄	—	2	2	no	—
520	C₂×C₄⋊C₄	—	2	1	no	—
521	C₂×C₄⋊C₄	—	2	1	no	—
522	C₂×C₄⋊C₄	—	4	1	no	—
523	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
524	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
525	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
526	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
527	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
528	C₂×C₂²⋊C₄	—	2	1	no	—
529	C₂×C₂²⋊C₄	—	2	1	no	—
530	C₂×C₂²⋊C₄	—	2	1	no	—
531	C₂×C₂²⋊C₄	—	2	1	no	—
532	C₂×C₂²⋊C₄	—	2	1	no	—
533	C₂×C₂²⋊C₄	—	2	1	no	—
534	C₂×C₂²⋊C₄	—	2	1	no	—
535	C₂×C₂²⋊C₄	—	2	1	no	—
536	C₂×C₂²⋊C₄	—	2	1	no	—
537	C₂×C₂²⋊C₄	—	2	1	no	—
538	C₂×C₂²⋊C₄	—	2	1	no	—
539	C₂×C₂²⋊C₄	—	4	4	no	—
540	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
541	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
542	C₂×C₄²	—	2	1	no	—
543	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
544	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
545	#106	—	2	2	no	—
546	#106	—	4	1	no	—
547	#102	—	2	2	no	—
548	#102	—	2	2	no	—
549	#102	—	2	2	no	—
550	#102	—	2	2	no	—
551	#102	—	4	1	no	—
552	#102	C₂×C₄×C₅₃	1	4	yes	complemented
553	#102	C₂×C₄×C₅₃	1	4	yes	complemented
554	#102	D₄×C₅₃	1	4	yes	complemented
555	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
556	#97	—	2	2	no	—
557	#97	—	4	1	no	—
558	#95	—	2	1	no	—
559	#95	—	2	1	no	—
560	#95	—	2	1	no	—
561	#95	D₄×C₅₃	1	1	yes	characteristic
562	#95	D₄×C₅₃	1	1	yes	characteristic
563	C₂⁴	—	2	1	no	—
564	C₂⁴	—	2	1	no	—
565	C₂⁴	—	2	1	no	—
566	C₂⁴	—	2	4	no	—
567	C₂⁴	—	4	2	no	—
568	C₂⁴	—	4	2	no	—
569	C₂⁴	—	4	2	no	—
570	C₂⁴	—	4	2	no	—
571	C₂⁴	—	8	1	no	—
572	C₂⁴	—	8	1	no	—
573	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
574	C₂×D₄	—	2	8	no	—
575	C₂×D₄	—	2	4	no	—
576	C₂×D₄	—	2	4	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	2	1	no	—
583	C₂×D₄	—	2	1	no	—
584	C₂×D₄	—	2	1	no	—
585	C₂×D₄	—	2	1	no	—
586	C₂×D₄	—	2	1	no	—
587	C₂×D₄	—	2	1	no	—
588	C₂×D₄	—	2	1	no	—
589	C₂×D₄	—	2	1	no	—
590	C₂×D₄	—	4	4	no	—
591	C₂×D₄	—	4	4	no	—
592	C₂×D₄	—	4	4	no	—
593	C₂×D₄	—	4	2	no	—
594	C₂×D₄	—	4	2	no	—
595	C₂×D₄	—	4	2	no	—
596	C₂×D₄	—	4	2	no	—
597	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
598	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
599	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
600	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
601	C₂×D₄	#41×C₅₃	1	2	yes	—
602	C₂²×C₄	—	2	1	no	—
603	C₂²×C₄	—	2	1	no	—
604	C₂²×C₄	—	2	1	no	—
605	C₂²×C₄	—	2	1	no	—
606	C₂²×C₄	—	2	1	no	—
607	C₂²×C₄	—	2	1	no	—
608	C₂²×C₄	—	2	1	no	—
609	C₂²×C₄	—	2	1	no	—
610	C₂²×C₄	—	2	1	no	—
611	C₂²×C₄	—	4	2	no	—
612	C₂²×C₄	—	4	2	no	—
613	C₂²×C₄	—	4	2	no	—
614	C₂²×C₄	—	4	2	no	—
615	C₂²×C₄	—	4	2	no	—
616	C₂²×C₄	—	4	2	no	—
617	C₂²×C₄	—	4	2	no	—
618	C₂²×C₄	—	4	2	no	—
619	C₂²×C₄	—	8	1	no	—
620	C₂²×C₄	—	8	1	no	—
621	C₂²×C₄	—	8	1	no	—
622	C₂²×C₄	—	8	1	no	—
623	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
624	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
625	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
626	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
627	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
628	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
629	C₂²×C₄	#41×C₅₃	1	2	yes	—
630	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
631	D₈	—	2	8	no	—
632	D₈	—	4	4	no	—
633	C₂×C₈	—	2	1	no	—
634	C₂×C₈	—	2	1	no	—
635	C₂×C₈	—	2	1	no	—
636	C₂×C₈	—	2	1	no	—
637	C₂×C₈	—	2	1	no	—
638	C₂×C₈	—	2	1	no	—
639	C₂×C₈	—	2	1	no	—
640	C₂×C₈	—	2	1	no	—
641	C₂×C₈	—	2	1	no	—
642	C₂×C₈	—	4	4	no	—
643	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
644	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
645	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
646	C₄⋊C₄	—	2	1	no	—
647	C₄⋊C₄	—	2	1	no	—
648	C₄⋊C₄	—	2	1	no	—
649	C₄⋊C₄	—	2	1	no	—
650	C₄⋊C₄	—	2	1	no	—
651	C₄⋊C₄	—	2	1	no	—
652	C₄⋊C₄	—	2	1	no	—
653	C₄⋊C₄	—	2	1	no	—
654	C₄⋊C₄	—	4	2	no	—
655	C₄⋊C₄	—	4	2	no	—
656	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
657	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
658	C₂²⋊C₄	—	2	2	no	—
659	C₂²⋊C₄	—	2	2	no	—
660	C₂²⋊C₄	—	2	4	no	—
661	C₂²⋊C₄	—	4	2	no	—
662	C₂²⋊C₄	—	4	2	no	—
663	C₂²⋊C₄	—	4	2	no	—
664	C₂²⋊C₄	—	4	2	no	—
665	C₂²⋊C₄	—	4	2	no	—
666	C₂²⋊C₄	—	4	2	no	—
667	C₂²⋊C₄	—	4	2	no	—
668	C₂²⋊C₄	—	8	1	no	—
669	C₂²⋊C₄	—	8	1	no	—
670	C₂²⋊C₄	—	8	1	no	—
671	C₄²	—	2	1	no	—
672	C₄²	—	2	1	no	—
673	C₄²	—	2	1	no	—
674	C₄²	—	2	1	no	—
675	C₄²	—	4	2	no	—
676	C₂³	—	2	1	no	—
677	C₂³	—	2	1	no	—
678	C₂³	—	2	1	no	—
679	C₂³	—	2	1	no	—
680	C₂³	—	2	2	no	—
681	C₂³	—	2	2	no	—
682	C₂³	—	2	2	no	—
683	C₂³	—	2	2	no	—
684	C₂³	—	2	2	no	—
685	C₂³	—	4	8	no	—
686	C₂³	—	4	1	no	—
687	C₂³	—	4	1	no	—
688	C₂³	—	4	1	no	—
689	C₂³	—	4	1	no	—
690	C₂³	—	4	1	no	—
691	C₂³	—	4	1	no	—
692	C₂³	—	4	1	no	—
693	C₂³	—	4	1	no	—
694	C₂³	—	4	1	no	—
695	C₂³	—	4	1	no	—
696	C₂³	—	4	1	no	—
697	C₂³	—	4	1	no	—
698	C₂³	—	4	1	no	—
699	C₂³	—	4	1	no	—
700	C₂³	—	4	1	no	—
701	C₂³	—	4	1	no	—
702	C₂³	—	4	8	no	—
703	C₂³	—	8	4	no	—
704	C₂³	—	8	4	no	—
705	C₂³	—	8	4	no	—
706	C₂³	—	8	4	no	—
707	C₂³	—	8	4	no	—
708	C₂³	C₄×D₄×C₅₃	1	2	yes	—
709	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
710	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
711	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
712	D₄	—	2	2	no	—
713	D₄	—	2	2	no	—
714	D₄	—	2	2	no	—
715	D₄	—	2	2	no	—
716	D₄	—	2	2	no	—
717	D₄	—	2	2	no	—
718	D₄	—	2	2	no	—
719	D₄	—	4	8	no	—
720	D₄	—	4	8	no	—
721	D₄	—	4	1	no	—
722	D₄	—	4	1	no	—
723	D₄	—	4	1	no	—
724	D₄	—	4	1	no	—
725	D₄	C₄×D₄×C₅₃	1	4	yes	complemented
726	C₂×C₄	—	2	1	no	—
727	C₂×C₄	—	2	1	no	—
728	C₂×C₄	—	2	2	no	—
729	C₂×C₄	—	2	2	no	—
730	C₂×C₄	—	2	2	no	—
731	C₂×C₄	—	2	2	no	—
732	C₂×C₄	—	2	2	no	—
733	C₂×C₄	—	2	1	no	—
734	C₂×C₄	—	2	1	no	—
735	C₂×C₄	—	2	2	no	—
736	C₂×C₄	—	2	1	no	—
737	C₂×C₄	—	2	1	no	—
738	C₂×C₄	—	2	1	no	—
739	C₂×C₄	—	2	1	no	—
740	C₂×C₄	—	2	1	no	—
741	C₂×C₄	—	2	1	no	—
742	C₂×C₄	—	2	1	no	—
743	C₂×C₄	—	2	1	no	—
744	C₂×C₄	—	2	1	no	—
745	C₂×C₄	—	2	1	no	—
746	C₂×C₄	—	4	1	no	—
747	C₂×C₄	—	4	1	no	—
748	C₂×C₄	—	4	1	no	—
749	C₂×C₄	—	4	1	no	—
750	C₂×C₄	—	4	1	no	—
751	C₂×C₄	—	4	1	no	—
752	C₂×C₄	—	4	1	no	—
753	C₂×C₄	—	4	1	no	—
754	C₂×C₄	—	4	1	no	—
755	C₂×C₄	—	4	1	no	—
756	C₂×C₄	—	4	1	no	—
757	C₂×C₄	—	4	1	no	—
758	C₂×C₄	—	4	1	no	—
759	C₂×C₄	—	8	4	no	—
760	C₂×C₄	—	8	4	no	—
761	C₂×C₄	—	8	4	no	—
762	C₂×C₄	—	8	4	no	—
763	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
764	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
765	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
766	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
767	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
768	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
769	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
770	C₈	—	2	2	no	—
771	C₈	—	2	2	no	—
772	C₈	—	2	2	no	—
773	C₈	—	4	1	no	—
774	C₂²	—	2	2	no	—
775	C₂²	—	2	2	no	—
776	C₂²	—	2	2	no	—
777	C₂²	—	2	2	no	—
778	C₂²	—	2	2	no	—
779	C₂²	—	2	2	no	—
780	C₂²	—	2	4	no	—
781	C₂²	—	4	2	no	—
782	C₂²	—	4	2	no	—
783	C₂²	—	4	2	no	—
784	C₂²	—	4	2	no	—
785	C₂²	—	4	2	no	—
786	C₂²	—	4	2	no	—
787	C₂²	—	4	2	no	—
788	C₂²	—	4	2	no	—
789	C₂²	—	4	2	no	—
790	C₂²	—	4	2	no	—
791	C₂²	—	4	2	no	—
792	C₂²	—	4	2	no	—
793	C₂²	—	8	16	no	—
794	C₂²	—	8	1	no	—
795	C₂²	—	8	1	no	—
796	C₂²	—	8	1	no	—
797	C₂²	—	8	1	no	—
798	C₂²	—	8	1	no	—
799	C₂²	—	8	1	no	—
800	C₂²	—	8	1	no	—
801	C₂²	—	8	1	no	—
802	C₂²	C₅₃×#442	1	4	yes	complemented
803	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
804	C₂²	D₄²×C₅₃	1	1	yes	characteristic
805	C₂²	C₅₃×#546	1	1	yes	characteristic
806	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
807	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
808	C₂²	C₅₃×#576	1	1	yes	characteristic
809	C₂²	C₅₃×#577	1	1	yes	characteristic
810	C₄	—	2	2	no	—
811	C₄	—	4	2	no	—
812	C₄	—	4	1	no	—
813	C₄	—	4	2	no	—
814	C₄	—	4	1	no	—
815	C₄	—	4	1	no	—
816	C₄	—	4	1	no	—
817	C₄	—	8	1	no	—
818	C₄	—	8	1	no	—
819	C₄	—	8	1	no	—
820	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
821	C₄	C₅₃×#518	1	1	yes	characteristic
822	C₄	C₅₃×#546	1	1	yes	characteristic
823	C₄	C₅₃×#551	1	1	yes	characteristic
824	C₂	—	2	4	no	—
825	C₂	—	4	4	no	—
826	C₂	—	8	2	no	—
827	C₂	—	8	2	no	—
828	C₂	C₅₃×#2358	1	1	yes	characteristic
829	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
830	C₂	C₅₃×#2946	1	1	yes	characteristic
831	C₂	C₅₃×#3264	1	1	yes	characteristic
832	C₂	C₅₃×#3265	1	1	yes	characteristic
833	C₂	C₅₃×#3306	1	1	yes	characteristic
834	C₂	C₅₃×#3318	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₂²⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#442	C₂	semidirect
3. C₂²×#102×C₅₃	C₂	semidirect
4. C₅₃×#2361	C₂	semidirect
5. D₄×C₂²⋊C₄×C₅₃	C₂	semidirect
6. C₅₃×#1972	C₂	semidirect
7. C₅₃×#1936	C₂	semidirect
8. C₅₃×#1926	C₂	semidirect
9. C₅₃×#1893	C₂	semidirect
10. C₅₃×#1879	C₂	semidirect
11. C₅₃×#1879	C₂	semidirect
12. C₅₃×#1864	C₂	semidirect
13. C₅₃×#1853	C₂	semidirect
14. C₅₃×#1762	C₂	semidirect
15. C₅₃×#1752	C₂	semidirect
24. C₂×C₄×D₄×C₅₃	C₂²	semidirect
27. C₅₃×#447	C₄	semidirect
29. C₂×#106×C₅₃	C₂²	semidirect
30. C₂×#106×C₅₃	C₂²	semidirect
38. C₂×#102×C₅₃	C₂²	semidirect
39. C₂×#102×C₅₃	C₂²	semidirect
41. C₂×#97×C₅₃	C₂²	semidirect
42. C₂×#97×C₅₃	C₂²	semidirect
45. C₅₃×#390	C₂²	semidirect
46. C₅₃×#389	C₂²	semidirect
54. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
56. C₅₃×#340	C₂²	semidirect
57. C₅₃×#340	C₂²	semidirect
58. C₅₃×#340	C₂²	semidirect
59. C₅₃×#340	C₂²	semidirect
60. C₅₃×#336	C₂²	semidirect
61. C₅₃×#336	C₂²	semidirect
70. C₂²×D₄×C₅₃	C₂×C₄	semidirect
83. C₂×D₈×C₅₃	C₂×C₄	semidirect
84. C₂×D₈×C₅₃	C₂×C₄	semidirect
93. #121×C₅₃	C₂×C₄	semidirect
134. #102×C₅₃	C₂×C₄	semidirect
135. #102×C₅₃	C₂×C₄	semidirect
136. #102×C₅₃	D₄	semidirect
137. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
307. D₄×C₅₃	C₄×D₄	semidirect
356. #22858	C₅₃	direct (N × H)
385. C₂²×C₅₃	#442	semidirect
408. C₂×#442	C₂×C₅₃	semidirect
409. C₂²×#102	C₂×C₅₃	semidirect
410. #2361	C₂×C₅₃	semidirect
411. D₄×C₂²⋊C₄	C₂×C₅₃	semidirect
412. #1972	C₂×C₅₃	semidirect
413. #1936	C₂×C₅₃	semidirect
414. #1926	C₂×C₅₃	semidirect
415. #1893	C₂×C₅₃	semidirect
416. #1879	C₂×C₅₃	semidirect
417. #1879	C₂×C₅₃	semidirect
418. #1864	C₂×C₅₃	semidirect
419. #1853	C₂×C₅₃	semidirect
420. #1762	C₂×C₅₃	semidirect
421. #1752	C₂×C₅₃	semidirect
441. C₂×C₄×D₄	C₂²×C₅₃	semidirect
444. #447	C₄×C₅₃	semidirect
446. C₂×#106	C₂²×C₅₃	semidirect
447. C₂×#106	C₂²×C₅₃	semidirect
455. C₂×#102	C₂²×C₅₃	semidirect
456. C₂×#102	C₂²×C₅₃	semidirect
458. C₂×#97	C₂²×C₅₃	semidirect
459. C₂×#97	C₂²×C₅₃	semidirect
462. #390	C₂²×C₅₃	semidirect
463. #389	C₂²×C₅₃	semidirect
471. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
473. #340	C₂²×C₅₃	semidirect
474. #340	C₂²×C₅₃	semidirect
475. #340	C₂²×C₅₃	semidirect
476. #340	C₂²×C₅₃	semidirect
477. #336	C₂²×C₅₃	semidirect
478. #336	C₂²×C₅₃	semidirect
479. C₅₃	#22858	direct (N × H)
488. C₂²×D₄	C₂×C₄×C₅₃	semidirect
501. C₂×D₈	C₂×C₄×C₅₃	semidirect
502. C₂×D₈	C₂×C₄×C₅₃	semidirect
511. #121	C₂×C₄×C₅₃	semidirect
552. #102	C₂×C₄×C₅₃	semidirect
553. #102	C₂×C₄×C₅₃	semidirect
554. #102	D₄×C₅₃	semidirect
555. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
725. D₄	C₄×D₄×C₅₃	semidirect
802. C₂²	C₅₃×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#447	2
C₂×C₅₃×#442	2
C₂²×#102×C₅₃	2
C₅₃×#2361	2
D₄×C₂²⋊C₄×C₅₃	2
C₅₃×#1972	2
C₅₃×#1936	2
C₅₃×#1926	2
C₅₃×#1893	2
C₅₃×#1879	2
C₅₃×#1864	2
C₅₃×#1853	2
C₅₃×#1762	2
C₅₃×#1752	2
#22858	53

Subgroups — C53×#22858 #27230370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22858 #27230370