Subgroups — C₅₃×#34242 #27241754

Order 13568.

2286 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2373	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2363	C₂	1	2	yes	maximal
4	C₅₃×#2362	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2354	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2352	C₂	1	2	yes	maximal, complemented
9	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2335	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2314	C₂	1	2	yes	maximal, complemented
13	C₄×#125×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₄×#124×C₅₃	C₂	1	2	yes	maximal, complemented
15	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#2274	C₂	1	2	yes	maximal, complemented
17	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₅₃×#529	C₄	1	16	yes	complemented
19	C₂×#126×C₅₃	C₂²	1	2	yes	—
20	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#123×C₅₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#117×C₅₃	C₂²	1	2	yes	—
33	C₂×#117×C₅₃	C₂²	1	2	yes	—
34	C₂×#117×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#391	C₂²	1	2	yes	—
38	C₅₃×#390	C₂²	1	2	yes	—
39	C₅₃×#390	C₂²	1	4	yes	—
40	C₅₃×#389	C₂²	1	2	yes	—
41	C₅₃×#389	C₂²	1	2	yes	—
42	C₅₃×#389	C₂²	1	4	yes	—
43	C₅₃×#388	C₂²	1	4	yes	—
44	C₅₃×#388	C₂²	1	4	yes	—
45	C₅₃×#388	C₂²	1	4	yes	—
46	C₅₃×#388	C₂²	1	4	yes	—
47	C₅₃×#388	C₂²	1	4	yes	—
48	C₅₃×#388	C₂²	1	4	yes	—
49	C₅₃×#387	C₂²	1	4	yes	—
50	C₅₃×#387	C₂²	1	4	yes	—
51	C₅₃×#387	C₂²	1	4	yes	—
52	C₅₃×#387	C₂²	1	4	yes	—
53	C₅₃×#386	C₂²	1	4	yes	—
54	C₅₃×#385	C₂²	1	4	yes	—
55	C₅₃×#384	C₂²	1	1	yes	characteristic
56	C₅₃×#384	C₂²	1	1	yes	characteristic
57	C₅₃×#384	C₂²	1	2	yes	—
58	C₅₃×#382	C₂²	1	2	yes	—
59	C₅₃×#382	C₂²	1	2	yes	—
60	C₅₃×#380	C₂²	1	2	yes	—
61	C₅₃×#380	C₂²	1	2	yes	—
62	C₅₃×#380	C₂²	1	2	yes	—
63	C₅₃×#380	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
67	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
70	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
73	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
76	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
77	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
78	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
80	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
81	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
82	C₅₃×#376	C₂²	1	2	yes	—
83	C₄³×C₅₃	C₂²	1	2	yes	—
84	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
87	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
90	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
91	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
92	#126×C₅₃	C₂×C₄	1	16	yes	complemented
93	#125×C₅₃	C₂×C₄	1	8	yes	complemented
94	#124×C₅₃	C₂×C₄	1	8	yes	—
95	#123×C₅₃	C₂×C₄	1	16	yes	complemented
96	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
97	#121×C₅₃	C₂×C₄	1	8	yes	—
98	C₄×Q₈×C₅₃	—	4	4	no	—
99	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
100	C₄×D₄×C₅₃	—	4	4	no	—
101	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
102	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
103	#117×C₅₃	—	2	4	no	—
104	#117×C₅₃	—	2	4	no	—
105	#117×C₅₃	—	2	8	no	—
106	#117×C₅₃	—	4	4	no	—
107	#117×C₅₃	C₂×C₄	1	16	yes	complemented
108	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
131	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
138	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
139	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
140	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
141	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
142	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
143	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
144	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
145	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
146	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
147	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
148	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
149	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
150	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
154	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
155	#95×C₅₃	C₂³	1	2	yes	—
156	#95×C₅₃	C₂³	1	2	yes	—
157	#95×C₅₃	C₂³	1	2	yes	—
158	#95×C₅₃	C₂³	1	2	yes	—
159	#95×C₅₃	C₂³	1	4	yes	—
160	#95×C₅₃	C₂³	1	4	yes	—
161	#95×C₅₃	C₂³	1	4	yes	—
162	#95×C₅₃	C₂³	1	4	yes	—
163	#95×C₅₃	C₂³	1	4	yes	—
164	#95×C₅₃	C₂³	1	4	yes	—
165	#95×C₅₃	C₂³	1	4	yes	—
166	#95×C₅₃	C₂³	1	4	yes	—
167	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
168	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂×Q₈×C₅₃	—	2	2	no	—
170	C₂×Q₈×C₅₃	—	2	2	no	—
171	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
172	C₂×D₄×C₅₃	—	2	2	no	—
173	C₂×D₄×C₅₃	—	2	2	no	—
174	C₂×D₄×C₅₃	—	4	2	no	—
175	C₂×D₄×C₅₃	—	4	2	no	—
176	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
177	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
178	C₂²×C₄×C₅₃	—	2	4	no	—
179	C₂²×C₄×C₅₃	—	2	2	no	—
180	C₂²×C₄×C₅₃	—	2	2	no	—
181	C₂²×C₄×C₅₃	—	2	4	no	—
182	C₂²×C₄×C₅₃	—	2	2	no	—
183	C₂²×C₄×C₅₃	—	2	2	no	—
184	C₂²×C₄×C₅₃	—	2	8	no	—
185	C₂²×C₄×C₅₃	—	4	1	no	—
186	C₂²×C₄×C₅₃	—	4	1	no	—
187	C₂²×C₄×C₅₃	—	4	4	no	—
188	C₂²×C₄×C₅₃	—	4	4	no	—
189	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
190	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
191	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
192	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
193	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
197	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
208	C₄⋊C₄×C₅₃	—	2	8	no	—
209	C₄⋊C₄×C₅₃	—	2	4	no	—
210	C₄⋊C₄×C₅₃	—	2	2	no	—
211	C₄⋊C₄×C₅₃	—	2	2	no	—
212	C₄⋊C₄×C₅₃	—	2	4	no	—
213	C₄⋊C₄×C₅₃	—	2	4	no	—
214	C₄⋊C₄×C₅₃	—	2	4	no	—
215	C₄⋊C₄×C₅₃	—	4	2	no	—
216	C₄⋊C₄×C₅₃	—	4	2	no	—
217	C₄⋊C₄×C₅₃	—	4	4	no	—
218	C₄⋊C₄×C₅₃	—	4	4	no	—
219	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
220	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
221	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
222	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
223	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
224	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
225	C₄⋊C₄×C₅₃	#41	1	4	yes	—
226	C₄⋊C₄×C₅₃	#41	1	4	yes	—
227	C₂²⋊C₄×C₅₃	—	2	4	no	—
228	C₂²⋊C₄×C₅₃	—	2	4	no	—
229	C₂²⋊C₄×C₅₃	—	2	8	no	—
230	C₂²⋊C₄×C₅₃	—	4	4	no	—
231	C₂²⋊C₄×C₅₃	—	4	4	no	—
232	C₂²⋊C₄×C₅₃	—	4	4	no	—
233	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
234	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
235	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
236	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
237	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
238	C₄²×C₅₃	—	2	4	no	—
239	C₄²×C₅₃	—	2	4	no	—
240	C₄²×C₅₃	—	2	8	no	—
241	C₄²×C₅₃	—	2	4	no	—
242	C₄²×C₅₃	—	2	4	no	—
243	C₄²×C₅₃	—	4	2	no	—
244	C₄²×C₅₃	—	4	2	no	—
245	C₄²×C₅₃	—	4	4	no	—
246	C₄²×C₅₃	—	4	4	no	—
247	C₄²×C₅₃	—	4	4	no	—
248	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
249	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
250	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
251	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
252	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
253	C₂³×C₅₃	—	2	2	no	—
254	C₂³×C₅₃	—	2	1	no	—
255	C₂³×C₅₃	—	2	2	no	—
256	C₂³×C₅₃	—	2	1	no	—
257	C₂³×C₅₃	—	2	1	no	—
258	C₂³×C₅₃	—	2	2	no	—
259	C₂³×C₅₃	—	2	1	no	—
260	C₂³×C₅₃	—	2	2	no	—
261	C₂³×C₅₃	—	2	2	no	—
262	C₂³×C₅₃	—	2	1	no	—
263	C₂³×C₅₃	—	2	1	no	—
264	C₂³×C₅₃	—	2	2	no	—
265	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
266	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
267	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
268	Q₈×C₅₃	—	2	8	no	—
269	D₄×C₅₃	—	2	8	no	—
270	D₄×C₅₃	—	4	8	no	—
271	C₂×C₄×C₅₃	—	2	1	no	—
272	C₂×C₄×C₅₃	—	2	1	no	—
273	C₂×C₄×C₅₃	—	2	2	no	—
274	C₂×C₄×C₅₃	—	2	2	no	—
275	C₂×C₄×C₅₃	—	2	1	no	—
276	C₂×C₄×C₅₃	—	2	2	no	—
277	C₂×C₄×C₅₃	—	2	2	no	—
278	C₂×C₄×C₅₃	—	2	1	no	—
279	C₂×C₄×C₅₃	—	2	2	no	—
280	C₂×C₄×C₅₃	—	2	2	no	—
281	C₂×C₄×C₅₃	—	2	2	no	—
282	C₂×C₄×C₅₃	—	2	1	no	—
283	C₂×C₄×C₅₃	—	2	1	no	—
284	C₂×C₄×C₅₃	—	2	1	no	—
285	C₂×C₄×C₅₃	—	2	1	no	—
286	C₂×C₄×C₅₃	—	2	2	no	—
287	C₂×C₄×C₅₃	—	2	2	no	—
288	C₂×C₄×C₅₃	—	2	2	no	—
289	C₂×C₄×C₅₃	—	2	16	no	—
290	C₂×C₄×C₅₃	—	2	8	no	—
291	C₂×C₄×C₅₃	—	2	8	no	—
292	C₂×C₄×C₅₃	—	2	4	no	—
293	C₂×C₄×C₅₃	—	2	2	no	—
294	C₂×C₄×C₅₃	—	2	4	no	—
295	C₂×C₄×C₅₃	—	2	2	no	—
296	C₂×C₄×C₅₃	—	2	4	no	—
297	C₂×C₄×C₅₃	—	2	2	no	—
298	C₂×C₄×C₅₃	—	2	4	no	—
299	C₂×C₄×C₅₃	—	2	2	no	—
300	C₂×C₄×C₅₃	—	2	4	no	—
301	C₂×C₄×C₅₃	—	2	2	no	—
302	C₂×C₄×C₅₃	—	2	2	no	—
303	C₂×C₄×C₅₃	—	2	4	no	—
304	C₂×C₄×C₅₃	—	4	4	no	—
305	C₂×C₄×C₅₃	—	8	8	no	—
306	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
307	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
308	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
309	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
310	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
311	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
312	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
313	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
314	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
315	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
316	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
317	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
318	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
319	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
320	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
321	#34242	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
322	C₂²×C₅₃	—	2	4	no	—
323	C₂²×C₅₃	—	2	2	no	—
324	C₂²×C₅₃	—	2	2	no	—
325	C₂²×C₅₃	—	2	4	no	—
326	C₂²×C₅₃	—	2	2	no	—
327	C₂²×C₅₃	—	2	4	no	—
328	C₂²×C₅₃	—	4	2	no	—
329	C₂²×C₅₃	—	4	1	no	—
330	C₂²×C₅₃	—	4	1	no	—
331	C₂²×C₅₃	—	4	2	no	—
332	C₂²×C₅₃	—	4	1	no	—
333	C₂²×C₅₃	—	4	2	no	—
334	C₂²×C₅₃	—	4	2	no	—
335	C₂²×C₅₃	—	4	1	no	—
336	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
337	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
338	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
339	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
340	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
341	C₂²×C₅₃	#585	1	1	yes	characteristic
342	C₂²×C₅₃	#585	1	1	yes	characteristic
343	C₄×C₅₃	—	2	2	no	—
344	C₄×C₅₃	—	2	4	no	—
345	C₄×C₅₃	—	2	2	no	—
346	C₄×C₅₃	—	2	4	no	—
347	C₄×C₅₃	—	2	2	no	—
348	C₄×C₅₃	—	2	2	no	—
349	C₄×C₅₃	—	2	4	no	—
350	C₄×C₅₃	—	2	4	no	—
351	C₄×C₅₃	—	2	4	no	—
352	C₄×C₅₃	—	4	4	no	—
353	C₄×C₅₃	—	4	2	no	—
354	C₄×C₅₃	—	4	4	no	—
355	C₄×C₅₃	—	4	2	no	—
356	C₄×C₅₃	C₄×#41	1	4	yes	—
357	C₄×C₅₃	#520	1	4	yes	—
358	C₂×#529	C₂×C₅₃	1	1	yes	characteristic
359	#2373	C₂×C₅₃	1	4	yes	complemented
360	#2363	C₂×C₅₃	1	2	yes	—
361	#2362	C₂×C₅₃	1	2	yes	complemented
362	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
363	#2354	C₂×C₅₃	1	2	yes	complemented
364	#2354	C₂×C₅₃	1	1	yes	characteristic, complemented
365	#2352	C₂×C₅₃	1	2	yes	complemented
366	Q₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
367	D₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
368	#2335	C₂×C₅₃	1	4	yes	complemented
369	#2314	C₂×C₅₃	1	2	yes	complemented
370	C₄×#125	C₂×C₅₃	1	2	yes	complemented
371	C₄×#124	C₂×C₅₃	1	2	yes	complemented
372	#2285	C₂×C₅₃	1	1	yes	characteristic, complemented
373	#2274	C₂×C₅₃	1	2	yes	complemented
374	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
375	C₂×C₅₃	—	4	2	no	—
376	C₂×C₅₃	—	4	4	no	—
377	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
378	C₂×C₅₃	C₂×#520	1	1	yes	characteristic
379	C₂×C₅₃	#3428	1	1	yes	characteristic
380	C₂×C₅₃	#3514	1	1	yes	characteristic
381	C₂×C₅₃	#3515	1	1	yes	characteristic
382	C₂×C₅₃	#3517	1	1	yes	characteristic
383	C₂×C₅₃	#3518	1	1	yes	characteristic
384	#529	C₄×C₅₃	1	16	yes	complemented
385	C₂×#126	C₂²×C₅₃	1	2	yes	—
386	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
387	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
388	C₂×#123	C₂²×C₅₃	1	2	yes	—
389	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
390	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
391	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
392	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
393	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
394	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
395	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
396	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
397	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
398	C₂×#117	C₂²×C₅₃	1	2	yes	—
399	C₂×#117	C₂²×C₅₃	1	2	yes	—
400	C₂×#117	C₂²×C₅₃	1	2	yes	—
401	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
402	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
403	#391	C₂²×C₅₃	1	2	yes	—
404	#390	C₂²×C₅₃	1	4	yes	—
405	#390	C₂²×C₅₃	1	2	yes	—
406	#389	C₂²×C₅₃	1	2	yes	—
407	#389	C₂²×C₅₃	1	2	yes	—
408	#389	C₂²×C₅₃	1	4	yes	—
409	#388	C₂²×C₅₃	1	4	yes	—
410	#388	C₂²×C₅₃	1	4	yes	—
411	#388	C₂²×C₅₃	1	4	yes	—
412	#388	C₂²×C₅₃	1	4	yes	—
413	#388	C₂²×C₅₃	1	4	yes	—
414	#388	C₂²×C₅₃	1	4	yes	—
415	#387	C₂²×C₅₃	1	4	yes	—
416	#387	C₂²×C₅₃	1	4	yes	—
417	#387	C₂²×C₅₃	1	4	yes	—
418	#387	C₂²×C₅₃	1	4	yes	—
419	#386	C₂²×C₅₃	1	4	yes	—
420	#385	C₂²×C₅₃	1	4	yes	—
421	#384	C₂²×C₅₃	1	2	yes	—
422	#384	C₂²×C₅₃	1	1	yes	characteristic
423	#384	C₂²×C₅₃	1	1	yes	characteristic
424	#382	C₂²×C₅₃	1	2	yes	—
425	#382	C₂²×C₅₃	1	2	yes	—
426	#380	C₂²×C₅₃	1	2	yes	—
427	#380	C₂²×C₅₃	1	2	yes	—
428	#380	C₂²×C₅₃	1	2	yes	—
429	#380	C₂²×C₅₃	1	2	yes	—
430	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
431	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
432	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
433	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
434	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
435	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
436	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
437	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
438	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
439	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
440	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
441	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
442	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
443	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
444	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
445	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
446	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
447	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
448	#376	C₂²×C₅₃	1	2	yes	—
449	C₄³	C₂²×C₅₃	1	2	yes	—
450	C₅₃	#34242	1	1	yes	characteristic, direct product factor, Sylow
451	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
452	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
453	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
454	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
455	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
456	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
457	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
458	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
459	#126	C₂×C₄×C₅₃	1	16	yes	complemented
460	#125	C₂×C₄×C₅₃	1	8	yes	complemented
461	#124	C₂×C₄×C₅₃	1	8	yes	—
462	#123	C₂×C₄×C₅₃	1	16	yes	complemented
463	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
464	#121	C₂×C₄×C₅₃	1	8	yes	—
465	C₄×Q₈	—	4	4	no	—
466	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
467	C₄×D₄	—	4	4	no	—
468	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
469	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
470	#117	—	2	4	no	—
471	#117	—	2	4	no	—
472	#117	—	2	8	no	—
473	#117	—	4	4	no	—
474	#117	C₂×C₄×C₅₃	1	16	yes	complemented
475	C₂×C₄⋊C₄	—	2	8	no	—
476	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
477	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
478	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
479	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
480	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
481	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
482	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
483	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
484	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
485	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
486	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
487	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
488	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
489	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
490	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
491	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
492	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
494	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
495	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
496	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
497	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
498	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
499	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
500	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
501	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
502	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
503	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
504	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
505	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
506	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
507	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
508	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
509	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
510	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
511	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
512	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
513	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
514	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
515	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
516	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
517	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
518	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
519	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
520	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
521	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
522	#95	C₂³×C₅₃	1	2	yes	—
523	#95	C₂³×C₅₃	1	2	yes	—
524	#95	C₂³×C₅₃	1	2	yes	—
525	#95	C₂³×C₅₃	1	2	yes	—
526	#95	C₂³×C₅₃	1	4	yes	—
527	#95	C₂³×C₅₃	1	4	yes	—
528	#95	C₂³×C₅₃	1	4	yes	—
529	#95	C₂³×C₅₃	1	4	yes	—
530	#95	C₂³×C₅₃	1	4	yes	—
531	#95	C₂³×C₅₃	1	4	yes	—
532	#95	C₂³×C₅₃	1	4	yes	—
533	#95	C₂³×C₅₃	1	4	yes	—
534	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
535	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
536	C₂×Q₈	—	2	2	no	—
537	C₂×Q₈	—	2	2	no	—
538	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	2	no	—
541	C₂×D₄	—	4	2	no	—
542	C₂×D₄	—	4	2	no	—
543	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
544	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
545	C₂²×C₄	—	2	4	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	8	no	—
552	C₂²×C₄	—	4	1	no	—
553	C₂²×C₄	—	4	1	no	—
554	C₂²×C₄	—	4	4	no	—
555	C₂²×C₄	—	4	4	no	—
556	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
557	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
558	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
559	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
560	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
561	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
562	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
563	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
564	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
565	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
566	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
567	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
568	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
569	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
570	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
571	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
572	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
573	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
574	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
575	C₄⋊C₄	—	2	8	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	4	4	no	—
583	C₄⋊C₄	—	4	2	no	—
584	C₄⋊C₄	—	4	2	no	—
585	C₄⋊C₄	—	4	4	no	—
586	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
587	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
588	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
589	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
590	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
591	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
592	C₄⋊C₄	#41×C₅₃	1	4	yes	—
593	C₄⋊C₄	#41×C₅₃	1	4	yes	—
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	8	no	—
596	C₂²⋊C₄	—	2	4	no	—
597	C₂²⋊C₄	—	4	4	no	—
598	C₂²⋊C₄	—	4	4	no	—
599	C₂²⋊C₄	—	4	4	no	—
600	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
601	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
602	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
603	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
604	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
605	C₄²	—	2	4	no	—
606	C₄²	—	2	8	no	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	4	no	—
610	C₄²	—	4	4	no	—
611	C₄²	—	4	4	no	—
612	C₄²	—	4	2	no	—
613	C₄²	—	4	4	no	—
614	C₄²	—	4	2	no	—
615	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
616	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
617	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
618	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
619	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	2	no	—
623	C₂³	—	2	2	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	2	no	—
626	C₂³	—	2	1	no	—
627	C₂³	—	2	2	no	—
628	C₂³	—	2	2	no	—
629	C₂³	—	2	2	no	—
630	C₂³	—	2	1	no	—
631	C₂³	—	2	1	no	—
632	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
633	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
634	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
635	Q₈	—	2	8	no	—
636	D₄	—	2	8	no	—
637	D₄	—	4	8	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	16	no	—
657	C₂×C₄	—	2	8	no	—
658	C₂×C₄	—	2	8	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	4	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	4	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	4	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	4	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	4	4	no	—
672	C₂×C₄	—	8	8	no	—
673	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
674	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
675	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
676	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
677	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
678	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
679	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
680	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
681	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
682	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
683	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
684	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
685	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
686	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
687	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
688	C₂²	—	2	2	no	—
689	C₂²	—	2	4	no	—
690	C₂²	—	2	2	no	—
691	C₂²	—	2	4	no	—
692	C₂²	—	2	2	no	—
693	C₂²	—	2	4	no	—
694	C₂²	—	4	2	no	—
695	C₂²	—	4	1	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	2	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	2	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	2	no	—
702	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
703	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
704	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
705	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
706	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
707	C₂²	C₅₃×#585	1	1	yes	characteristic
708	C₂²	C₅₃×#585	1	1	yes	characteristic
709	C₄	—	2	4	no	—
710	C₄	—	2	4	no	—
711	C₄	—	2	2	no	—
712	C₄	—	2	4	no	—
713	C₄	—	2	2	no	—
714	C₄	—	2	4	no	—
715	C₄	—	2	2	no	—
716	C₄	—	2	2	no	—
717	C₄	—	2	4	no	—
718	C₄	—	4	4	no	—
719	C₄	—	4	2	no	—
720	C₄	—	4	4	no	—
721	C₄	—	4	2	no	—
722	C₄	C₄×#41×C₅₃	1	4	yes	—
723	C₄	C₅₃×#520	1	4	yes	—
724	C₂	—	4	2	no	—
725	C₂	—	4	4	no	—
726	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
727	C₂	C₂×C₅₃×#520	1	1	yes	characteristic
728	C₂	C₅₃×#3428	1	1	yes	characteristic
729	C₂	C₅₃×#3514	1	1	yes	characteristic
730	C₂	C₅₃×#3515	1	1	yes	characteristic
731	C₂	C₅₃×#3517	1	1	yes	characteristic
732	C₂	C₅₃×#3518	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×#121×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2373	C₂	semidirect
4. C₅₃×#2362	C₂	semidirect
5. C₅₃×#2355	C₂	semidirect
6. C₅₃×#2354	C₂	semidirect
7. C₅₃×#2354	C₂	semidirect
8. C₅₃×#2352	C₂	semidirect
9. Q₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. D₄×C₄⋊C₄×C₅₃	C₂	semidirect
11. C₅₃×#2335	C₂	semidirect
12. C₅₃×#2314	C₂	semidirect
13. C₄×#125×C₅₃	C₂	semidirect
14. C₄×#124×C₅₃	C₂	semidirect
15. C₅₃×#2285	C₂	semidirect
16. C₅₃×#2274	C₂	semidirect
17. C₄×#117×C₅₃	C₂	semidirect
18. C₅₃×#529	C₄	semidirect
92. #126×C₅₃	C₂×C₄	semidirect
93. #125×C₅₃	C₂×C₄	semidirect
95. #123×C₅₃	C₂×C₄	semidirect
96. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
99. C₄×Q₈×C₅₃	C₂×C₄	semidirect
102. C₄×D₄×C₅₃	C₂×C₄	semidirect
107. #117×C₅₃	C₂×C₄	semidirect
139. C₂×C₄²×C₅₃	C₂×C₄	semidirect
321. #34242	C₅₃	direct (N × H)
359. #2373	C₂×C₅₃	semidirect
361. #2362	C₂×C₅₃	semidirect
362. #2355	C₂×C₅₃	semidirect
363. #2354	C₂×C₅₃	semidirect
364. #2354	C₂×C₅₃	semidirect
365. #2352	C₂×C₅₃	semidirect
366. Q₈×C₄⋊C₄	C₂×C₅₃	semidirect
367. D₄×C₄⋊C₄	C₂×C₅₃	semidirect
368. #2335	C₂×C₅₃	semidirect
369. #2314	C₂×C₅₃	semidirect
370. C₄×#125	C₂×C₅₃	semidirect
371. C₄×#124	C₂×C₅₃	semidirect
372. #2285	C₂×C₅₃	semidirect
373. #2274	C₂×C₅₃	semidirect
374. C₄×#117	C₂×C₅₃	semidirect
384. #529	C₄×C₅₃	semidirect
450. C₅₃	#34242	direct (N × H)
459. #126	C₂×C₄×C₅₃	semidirect
460. #125	C₂×C₄×C₅₃	semidirect
462. #123	C₂×C₄×C₅₃	semidirect
463. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
466. C₄×Q₈	C₂×C₄×C₅₃	semidirect
469. C₄×D₄	C₂×C₄×C₅₃	semidirect
474. #117	C₂×C₄×C₅₃	semidirect
506. C₂×C₄²	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#529	2
C₅₃×#2373	2
C₅₃×#2363	2
C₅₃×#2362	2
C₅₃×#2355	2
C₅₃×#2354	2
C₅₃×#2352	2
Q₈×C₄⋊C₄×C₅₃	2
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#2335	2
C₅₃×#2314	2
C₄×#125×C₅₃	2
C₄×#124×C₅₃	2
C₅₃×#2285	2
C₅₃×#2274	2
C₄×#117×C₅₃	2
#34242	53

Subgroups — C53×#34242 #27241754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34242 #27241754