Subgroups — C₅₃×#62108 #27269620

Order 13568.

1750 conjugacy classes of subgroups.

The subgroup lattice has 870 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3475	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3385	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3369	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3357	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3344	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3331	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3314	C₂	1	4	yes	maximal, complemented
9	C₅₃×#3252	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3249	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3238	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#3116	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#3079	C₂	1	2	yes	maximal, complemented
16	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
17	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
18	Q₈²×C₅₃	C₂²	1	2	yes	complemented
19	C₅₃×#557	—	2	1	no	—
20	C₅₃×#556	C₂²	1	1	yes	characteristic
21	C₅₃×#555	C₂²	1	1	yes	characteristic
22	C₅₃×#554	C₂²	1	1	yes	characteristic
23	C₅₃×#554	C₂²	1	4	yes	—
24	C₅₃×#554	C₂²	1	2	yes	complemented
25	C₅₃×#553	—	2	1	no	—
26	C₅₃×#552	—	2	1	no	—
27	C₅₃×#552	C₂²	1	2	yes	—
28	C₅₃×#551	—	2	2	no	—
29	D₄×Q₈×C₅₃	—	2	2	no	—
30	D₄×Q₈×C₅₃	—	2	1	no	—
31	C₅₃×#547	C₂²	1	1	yes	characteristic
32	C₅₃×#546	C₂²	1	2	yes	—
33	C₅₃×#544	—	2	1	no	—
34	C₅₃×#544	—	2	2	no	—
35	C₅₃×#533	C₂²	1	1	yes	characteristic
36	C₅₃×#533	C₂²	1	1	yes	characteristic
37	C₅₃×#533	C₂²	1	2	yes	complemented
38	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
40	C₅₃×#529	—	2	2	no	—
41	C₅₃×#529	C₂²	1	2	yes	—
42	C₅₃×#525	—	2	1	no	—
43	C₅₃×#520	—	2	1	no	—
44	C₄×#41×C₅₃	C₂²	1	2	yes	—
45	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#117×C₅₃	—	2	1	no	—
48	C₅₃×#501	C₂²	1	4	yes	complemented
49	C₅₃×#500	C₂²	1	2	yes	complemented
50	C₅₃×#489	C₂²	1	4	yes	—
51	C₅₃×#487	C₂²	1	2	yes	complemented
52	C₅₃×#485	C₂²	1	4	yes	complemented
53	C₅₃×#484	C₂²	1	4	yes	—
54	C₅₃×#483	C₂²	1	2	yes	—
55	C₅₃×#483	C₂²	1	4	yes	complemented
56	C₅₃×#480	C₂²	1	4	yes	—
57	C₅₃×#479	C₂²	1	4	yes	complemented
58	C₅₃×#477	C₂²	1	2	yes	complemented
59	C₅₃×#477	C₂²	1	2	yes	complemented
60	C₅₃×#477	C₂²	1	2	yes	complemented
61	C₅₃×#477	C₂²	1	2	yes	complemented
62	C₅₃×#476	C₂²	1	1	yes	characteristic
63	C₅₃×#476	C₂²	1	2	yes	—
64	C₅₃×#476	C₂²	1	2	yes	—
65	C₅₃×#476	C₂²	1	2	yes	—
66	C₅₃×#476	C₂²	1	2	yes	—
67	C₅₃×#476	C₂²	1	1	yes	characteristic
68	C₅₃×#475	C₂²	1	4	yes	complemented
69	C₅₃×#474	C₂²	1	4	yes	—
70	C₅₃×#470	C₂²	1	4	yes	complemented
71	C₅₃×#468	C₂²	1	4	yes	—
72	C₅₃×#467	C₂²	1	4	yes	—
73	C₅₃×#464	C₂²	1	4	yes	—
74	C₅₃×#462	C₂²	1	2	yes	complemented
75	C₅₃×#462	C₂²	1	2	yes	complemented
76	C₅₃×#461	C₂²	1	1	yes	characteristic
77	C₅₃×#461	C₂²	1	1	yes	characteristic
78	C₅₃×#452	C₂²	1	4	yes	—
79	C₅₃×#450	C₂²	1	2	yes	—
80	C₅₃×#446	C₂²	1	2	yes	complemented
81	C₅₃×#441	C₂²	1	4	yes	complemented
82	C₅₃×#440	C₂²	1	4	yes	—
83	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
84	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
85	C₅₃×#436	C₂²	1	2	yes	complemented
86	C₅₃×#435	C₂²	1	4	yes	—
87	C₅₃×#432	C₂²	1	2	yes	complemented
88	C₅₃×#431	C₂²	1	1	yes	characteristic
89	C₅₃×#428	C₂²	1	4	yes	complemented
90	C₂×#106×C₅₃	C₂²	1	2	yes	—
91	C₅₃×#423	C₂²	1	2	yes	complemented
92	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
93	C₅₃×#417	C₂²	1	4	yes	—
94	C₂×#102×C₅₃	C₂²	1	2	yes	—
95	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
96	C₂×#41×C₅₃	C₂³	1	2	yes	—
97	C₂²×Q₈×C₅₃	—	2	1	no	—
98	C₂²×Q₈×C₅₃	—	2	2	no	—
99	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₅₃	—	2	1	no	—
101	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
102	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
103	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
104	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
105	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
106	C₄⋊Q₈×C₅₃	—	2	2	no	—
107	C₄⋊Q₈×C₅₃	—	2	4	no	—
108	C₄⋊Q₈×C₅₃	—	2	1	no	—
109	C₄⋊Q₈×C₅₃	—	2	1	no	—
110	C₄⋊Q₈×C₅₃	—	2	2	no	—
111	C₄⋊Q₈×C₅₃	—	2	2	no	—
112	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
113	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
114	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
115	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
116	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
119	#126×C₅₃	—	2	2	no	—
120	#126×C₅₃	—	2	2	no	—
121	#126×C₅₃	—	2	2	no	—
122	#125×C₅₃	—	2	2	no	—
123	#125×C₅₃	—	2	4	no	—
124	#125×C₅₃	—	2	2	no	—
125	#125×C₅₃	—	2	1	no	—
126	#125×C₅₃	—	2	1	no	—
127	#125×C₅₃	—	2	2	no	—
128	#125×C₅₃	C₂³	1	2	yes	—
129	#124×C₅₃	—	2	2	no	—
130	#124×C₅₃	C₂³	1	2	yes	—
131	#123×C₅₃	—	2	2	no	—
132	#123×C₅₃	—	2	2	no	—
133	#123×C₅₃	—	2	2	no	—
134	#123×C₅₃	—	2	2	no	—
135	C₂²⋊Q₈×C₅₃	—	2	2	no	—
136	C₂²⋊Q₈×C₅₃	—	2	4	no	—
137	C₂²⋊Q₈×C₅₃	—	2	4	no	—
138	C₂²⋊Q₈×C₅₃	—	2	4	no	—
139	C₂²⋊Q₈×C₅₃	—	2	1	no	—
140	C₂²⋊Q₈×C₅₃	—	2	1	no	—
141	C₂²⋊Q₈×C₅₃	—	2	1	no	—
142	C₂²⋊Q₈×C₅₃	—	2	1	no	—
143	C₂²⋊Q₈×C₅₃	—	2	2	no	—
144	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
145	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
146	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
147	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
148	#121×C₅₃	—	2	2	no	—
149	#121×C₅₃	—	2	1	no	—
150	#121×C₅₃	—	2	1	no	—
151	#121×C₅₃	C₂³	1	2	yes	—
152	#120×C₅₃	—	2	2	no	—
153	C₄×Q₈×C₅₃	—	2	1	no	—
154	C₄×Q₈×C₅₃	—	2	1	no	—
155	C₄×Q₈×C₅₃	—	2	2	no	—
156	C₄×Q₈×C₅₃	—	2	2	no	—
157	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
158	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
159	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
160	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
161	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
162	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
163	C₄×D₄×C₅₃	—	2	1	no	—
164	C₄×D₄×C₅₃	—	2	1	no	—
165	C₄×D₄×C₅₃	—	2	4	no	—
166	C₄×D₄×C₅₃	—	2	2	no	—
167	C₄×D₄×C₅₃	—	2	2	no	—
168	C₄×D₄×C₅₃	—	2	2	no	—
169	C₄×D₄×C₅₃	—	2	2	no	—
170	C₄×D₄×C₅₃	D₄	1	2	yes	—
171	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
172	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
173	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
174	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
175	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
176	C₄×D₄×C₅₃	C₂³	1	2	yes	—
177	C₄×D₄×C₅₃	C₂³	1	2	yes	—
178	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
179	#117×C₅₃	—	2	1	no	—
180	#117×C₅₃	—	2	1	no	—
181	#117×C₅₃	—	2	2	no	—
182	#117×C₅₃	—	2	2	no	—
183	#117×C₅₃	—	2	2	no	—
184	#117×C₅₃	—	2	1	no	—
185	#117×C₅₃	—	2	1	no	—
186	#117×C₅₃	—	2	2	no	—
187	#117×C₅₃	D₄	1	1	yes	characteristic
188	#117×C₅₃	D₄	1	2	yes	—
189	#117×C₅₃	C₂³	1	2	yes	—
190	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
191	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
192	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
193	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
194	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
195	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
196	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
197	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
198	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
199	C₂×C₄²×C₅₃	—	2	2	no	—
200	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
201	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
202	#107×C₅₃	C₂³	1	4	yes	—
203	#107×C₅₃	C₂³	1	4	yes	—
204	#107×C₅₃	C₂³	1	4	yes	—
205	#106×C₅₃	C₂³	1	4	yes	—
206	#106×C₅₃	C₂³	1	2	yes	—
207	#106×C₅₃	C₂³	1	4	yes	—
208	#106×C₅₃	C₂³	1	2	yes	—
209	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
210	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
211	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
212	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
213	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
214	#103×C₅₃	C₂³	1	2	yes	—
215	#103×C₅₃	C₂³	1	4	yes	—
216	#103×C₅₃	C₂³	1	4	yes	—
217	#103×C₅₃	C₂³	1	4	yes	—
218	#103×C₅₃	C₂³	1	4	yes	—
219	#103×C₅₃	C₂³	1	2	yes	—
220	#102×C₅₃	C₂³	1	2	yes	—
221	#102×C₅₃	C₂³	1	4	yes	—
222	#102×C₅₃	C₂³	1	4	yes	—
223	#102×C₅₃	C₂³	1	2	yes	—
224	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
225	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
226	#97×C₅₃	C₂³	1	4	yes	—
227	C₄×C₈×C₅₃	C₂³	1	2	yes	—
228	C₂⁴×C₅₃	—	2	1	no	—
229	#41×C₅₃	—	2	4	no	—
230	#41×C₅₃	—	2	4	no	—
231	#41×C₅₃	—	4	2	no	—
232	C₂×Q₈×C₅₃	—	2	1	no	—
233	C₂×Q₈×C₅₃	—	2	2	no	—
234	C₂×Q₈×C₅₃	—	2	1	no	—
235	C₂×Q₈×C₅₃	—	2	4	no	—
236	C₂×Q₈×C₅₃	—	2	2	no	—
237	C₂×Q₈×C₅₃	—	8	2	no	—
238	C₂×Q₈×C₅₃	—	8	4	no	—
239	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
240	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
241	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
242	C₂×D₄×C₅₃	—	2	2	no	—
243	C₂×D₄×C₅₃	—	2	2	no	—
244	C₂×D₄×C₅₃	—	2	2	no	—
245	C₂×D₄×C₅₃	—	2	1	no	—
246	C₂×D₄×C₅₃	—	4	1	no	—
247	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
248	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
249	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂²×C₄×C₅₃	—	2	1	no	—
252	C₂²×C₄×C₅₃	—	2	4	no	—
253	C₂²×C₄×C₅₃	—	2	2	no	—
254	C₂²×C₄×C₅₃	—	2	1	no	—
255	C₂²×C₄×C₅₃	—	2	2	no	—
256	C₂²×C₄×C₅₃	—	2	1	no	—
257	C₂²×C₄×C₅₃	—	2	1	no	—
258	C₂²×C₄×C₅₃	—	2	1	no	—
259	C₂²×C₄×C₅₃	—	2	2	no	—
260	C₂²×C₄×C₅₃	—	4	4	no	—
261	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
262	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
263	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
264	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
265	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
267	Q₁₆×C₅₃	—	4	4	no	—
268	SD₁₆×C₅₃	—	2	4	no	—
269	SD₁₆×C₅₃	—	4	2	no	—
270	M₁₆×C₅₃	—	4	4	no	—
271	C₂×C₈×C₅₃	—	4	2	no	—
272	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
273	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
274	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
275	C₄⋊C₄×C₅₃	—	2	2	no	—
276	C₄⋊C₄×C₅₃	—	2	1	no	—
277	C₄⋊C₄×C₅₃	—	2	1	no	—
278	C₄⋊C₄×C₅₃	—	2	1	no	—
279	C₄⋊C₄×C₅₃	—	2	2	no	—
280	C₄⋊C₄×C₅₃	—	2	2	no	—
281	C₄⋊C₄×C₅₃	—	2	4	no	—
282	C₄⋊C₄×C₅₃	—	2	2	no	—
283	C₄⋊C₄×C₅₃	—	2	1	no	—
284	C₄⋊C₄×C₅₃	—	2	4	no	—
285	C₄⋊C₄×C₅₃	—	2	2	no	—
286	C₄⋊C₄×C₅₃	—	2	1	no	—
287	C₄⋊C₄×C₅₃	—	2	2	no	—
288	C₄⋊C₄×C₅₃	—	2	2	no	—
289	C₄⋊C₄×C₅₃	—	2	2	no	—
290	C₄⋊C₄×C₅₃	—	2	1	no	—
291	C₄⋊C₄×C₅₃	—	2	2	no	—
292	C₄⋊C₄×C₅₃	—	2	4	no	—
293	C₄⋊C₄×C₅₃	—	2	2	no	—
294	C₄⋊C₄×C₅₃	—	2	1	no	—
295	C₄⋊C₄×C₅₃	—	2	4	no	—
296	C₄⋊C₄×C₅₃	—	2	2	no	—
297	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
298	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
299	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
300	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
301	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
302	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
303	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
304	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
305	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
306	C₂²⋊C₄×C₅₃	—	2	2	no	—
307	C₂²⋊C₄×C₅₃	—	2	4	no	—
308	C₂²⋊C₄×C₅₃	—	2	2	no	—
309	C₂²⋊C₄×C₅₃	—	2	2	no	—
310	C₂²⋊C₄×C₅₃	—	2	2	no	—
311	C₂²⋊C₄×C₅₃	—	2	1	no	—
312	C₂²⋊C₄×C₅₃	—	2	2	no	—
313	C₂²⋊C₄×C₅₃	—	2	2	no	—
314	C₂²⋊C₄×C₅₃	—	2	1	no	—
315	C₂²⋊C₄×C₅₃	—	2	1	no	—
316	C₂²⋊C₄×C₅₃	—	2	1	no	—
317	C₂²⋊C₄×C₅₃	—	2	2	no	—
318	C₂²⋊C₄×C₅₃	—	2	2	no	—
319	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
320	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
321	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
322	C₄²×C₅₃	—	2	2	no	—
323	C₄²×C₅₃	—	2	2	no	—
324	C₄²×C₅₃	—	2	2	no	—
325	C₄²×C₅₃	—	2	4	no	—
326	C₄²×C₅₃	—	2	2	no	—
327	C₄²×C₅₃	C₂×D₄	1	2	yes	—
328	C₄²×C₅₃	C₂×D₄	1	2	yes	—
329	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
330	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
331	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
332	C₄²×C₅₃	C₂⁴	1	2	yes	—
333	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
334	C₂³×C₅₃	—	2	1	no	—
335	C₂³×C₅₃	—	2	1	no	—
336	C₂³×C₅₃	—	4	2	no	—
337	C₂³×C₅₃	—	4	2	no	—
338	C₂³×C₅₃	—	8	1	no	—
339	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
340	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
341	Q₈×C₅₃	—	2	2	no	—
342	Q₈×C₅₃	—	2	2	no	—
343	Q₈×C₅₃	—	2	2	no	—
344	Q₈×C₅₃	—	2	2	no	—
345	Q₈×C₅₃	—	2	2	no	—
346	Q₈×C₅₃	—	4	2	no	—
347	Q₈×C₅₃	—	4	4	no	—
348	Q₈×C₅₃	—	8	1	no	—
349	Q₈×C₅₃	—	8	4	no	—
350	Q₈×C₅₃	—	8	2	no	—
351	D₄×C₅₃	—	2	4	no	—
352	D₄×C₅₃	—	2	2	no	—
353	D₄×C₅₃	—	2	1	no	—
354	D₄×C₅₃	—	2	1	no	—
355	D₄×C₅₃	—	2	1	no	—
356	D₄×C₅₃	—	4	2	no	—
357	D₄×C₅₃	—	4	4	no	—
358	D₄×C₅₃	—	8	2	no	—
359	D₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
360	C₂×C₄×C₅₃	—	2	4	no	—
361	C₂×C₄×C₅₃	—	2	1	no	—
362	C₂×C₄×C₅₃	—	2	2	no	—
363	C₂×C₄×C₅₃	—	2	1	no	—
364	C₂×C₄×C₅₃	—	2	2	no	—
365	C₂×C₄×C₅₃	—	2	1	no	—
366	C₂×C₄×C₅₃	—	2	1	no	—
367	C₂×C₄×C₅₃	—	2	2	no	—
368	C₂×C₄×C₅₃	—	2	4	no	—
369	C₂×C₄×C₅₃	—	2	2	no	—
370	C₂×C₄×C₅₃	—	2	2	no	—
371	C₂×C₄×C₅₃	—	2	2	no	—
372	C₂×C₄×C₅₃	—	4	2	no	—
373	C₂×C₄×C₅₃	—	4	1	no	—
374	C₂×C₄×C₅₃	—	4	4	no	—
375	C₂×C₄×C₅₃	—	4	2	no	—
376	C₂×C₄×C₅₃	—	4	2	no	—
377	C₂×C₄×C₅₃	—	4	2	no	—
378	C₂×C₄×C₅₃	—	4	2	no	—
379	C₂×C₄×C₅₃	—	4	2	no	—
380	C₂×C₄×C₅₃	—	8	2	no	—
381	C₂×C₄×C₅₃	—	8	2	no	—
382	C₂×C₄×C₅₃	—	8	4	no	—
383	C₂×C₄×C₅₃	#137	1	4	yes	—
384	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
385	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
386	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
387	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
388	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
389	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
390	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
391	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
393	C₈×C₅₃	—	2	2	no	—
394	C₈×C₅₃	—	2	4	no	—
395	C₈×C₅₃	—	2	2	no	—
396	#62108	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
397	C₂²×C₅₃	—	2	1	no	—
398	C₂²×C₅₃	—	2	1	no	—
399	C₂²×C₅₃	—	2	2	no	—
400	C₂²×C₅₃	—	2	2	no	—
401	C₂²×C₅₃	—	2	2	no	—
402	C₂²×C₅₃	—	2	2	no	—
403	C₂²×C₅₃	—	4	1	no	—
404	C₂²×C₅₃	—	4	1	no	—
405	C₂²×C₅₃	—	4	1	no	—
406	C₂²×C₅₃	—	4	1	no	—
407	C₂²×C₅₃	—	4	1	no	—
408	C₂²×C₅₃	—	8	4	no	—
409	C₂²×C₅₃	C₂×#137	1	2	yes	—
410	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
411	C₄×C₅₃	—	2	1	no	—
412	C₄×C₅₃	—	2	2	no	—
413	C₄×C₅₃	—	2	1	no	—
414	C₄×C₅₃	—	2	2	no	—
415	C₄×C₅₃	—	2	1	no	—
416	C₄×C₅₃	—	2	2	no	—
417	C₄×C₅₃	—	4	4	no	—
418	C₄×C₅₃	—	4	2	no	—
419	C₄×C₅₃	—	4	2	no	—
420	C₄×C₅₃	—	4	2	no	—
421	C₄×C₅₃	—	4	2	no	—
422	C₄×C₅₃	—	4	1	no	—
423	C₄×C₅₃	—	4	1	no	—
424	C₄×C₅₃	C₂×#137	1	2	yes	—
425	C₄×C₅₃	C₂×#137	1	2	yes	—
426	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
427	C₄×C₅₃	#585	1	1	yes	characteristic
428	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
429	#3475	C₂×C₅₃	1	1	yes	characteristic
430	#3385	C₂×C₅₃	1	2	yes	complemented
431	#3369	C₂×C₅₃	1	2	yes	complemented
432	#3357	C₂×C₅₃	1	2	yes	complemented
433	#3344	C₂×C₅₃	1	4	yes	complemented
434	#3331	C₂×C₅₃	1	2	yes	complemented
435	#3314	C₂×C₅₃	1	4	yes	complemented
436	#3252	C₂×C₅₃	1	2	yes	complemented
437	#3249	C₂×C₅₃	1	1	yes	characteristic, complemented
438	#3238	C₂×C₅₃	1	2	yes	complemented
439	#3230	C₂×C₅₃	1	1	yes	characteristic, complemented
440	#3116	C₂×C₅₃	1	2	yes	complemented
441	#3101	C₂×C₅₃	1	1	yes	characteristic, complemented
442	#3079	C₂×C₅₃	1	2	yes	complemented
443	C₂×#476	C₂×C₅₃	1	1	yes	characteristic, complemented
444	#2989	C₂×C₅₃	1	1	yes	characteristic, complemented
445	C₂×C₅₃	—	2	2	no	—
446	C₂×C₅₃	—	4	2	no	—
447	C₂×C₅₃	—	4	2	no	—
448	C₂×C₅₃	—	8	1	no	—
449	C₂×C₅₃	#3489	1	1	yes	characteristic
450	C₂×C₅₃	C₂²×#137	1	1	yes	characteristic
451	C₂×C₅₃	#3585	1	1	yes	characteristic
452	Q₈²	C₂²×C₅₃	1	2	yes	complemented
453	#557	—	2	1	no	—
454	#556	C₂²×C₅₃	1	1	yes	characteristic
455	#555	C₂²×C₅₃	1	1	yes	characteristic
456	#554	C₂²×C₅₃	1	1	yes	characteristic
457	#554	C₂²×C₅₃	1	4	yes	—
458	#554	C₂²×C₅₃	1	2	yes	complemented
459	#553	—	2	1	no	—
460	#552	—	2	1	no	—
461	#552	C₂²×C₅₃	1	2	yes	—
462	#551	—	2	2	no	—
463	D₄×Q₈	—	2	2	no	—
464	D₄×Q₈	—	2	1	no	—
465	#547	C₂²×C₅₃	1	1	yes	characteristic
466	#546	C₂²×C₅₃	1	2	yes	—
467	#544	—	2	1	no	—
468	#544	—	2	2	no	—
469	#533	C₂²×C₅₃	1	1	yes	characteristic
470	#533	C₂²×C₅₃	1	1	yes	characteristic
471	#533	C₂²×C₅₃	1	2	yes	complemented
472	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
473	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
474	#529	—	2	2	no	—
475	#529	C₂²×C₅₃	1	2	yes	—
476	#525	—	2	1	no	—
477	#520	—	2	1	no	—
478	C₄×#41	C₂²×C₅₃	1	2	yes	—
479	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
480	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
481	C₂×#117	—	2	1	no	—
482	#501	C₂²×C₅₃	1	4	yes	complemented
483	#500	C₂²×C₅₃	1	2	yes	complemented
484	#489	C₂²×C₅₃	1	4	yes	—
485	#487	C₂²×C₅₃	1	2	yes	complemented
486	#485	C₂²×C₅₃	1	4	yes	complemented
487	#484	C₂²×C₅₃	1	4	yes	—
488	#483	C₂²×C₅₃	1	2	yes	—
489	#483	C₂²×C₅₃	1	4	yes	complemented
490	#480	C₂²×C₅₃	1	4	yes	—
491	#479	C₂²×C₅₃	1	4	yes	complemented
492	#477	C₂²×C₅₃	1	2	yes	complemented
493	#477	C₂²×C₅₃	1	2	yes	complemented
494	#477	C₂²×C₅₃	1	2	yes	complemented
495	#477	C₂²×C₅₃	1	2	yes	complemented
496	#476	C₂²×C₅₃	1	2	yes	—
497	#476	C₂²×C₅₃	1	1	yes	characteristic
498	#476	C₂²×C₅₃	1	2	yes	—
499	#476	C₂²×C₅₃	1	1	yes	characteristic
500	#476	C₂²×C₅₃	1	2	yes	—
501	#476	C₂²×C₅₃	1	2	yes	—
502	#475	C₂²×C₅₃	1	4	yes	complemented
503	#474	C₂²×C₅₃	1	4	yes	—
504	#470	C₂²×C₅₃	1	4	yes	complemented
505	#468	C₂²×C₅₃	1	4	yes	—
506	#467	C₂²×C₅₃	1	4	yes	—
507	#464	C₂²×C₅₃	1	4	yes	—
508	#462	C₂²×C₅₃	1	2	yes	complemented
509	#462	C₂²×C₅₃	1	2	yes	complemented
510	#461	C₂²×C₅₃	1	1	yes	characteristic
511	#461	C₂²×C₅₃	1	1	yes	characteristic
512	#452	C₂²×C₅₃	1	4	yes	—
513	#450	C₂²×C₅₃	1	2	yes	—
514	#446	C₂²×C₅₃	1	2	yes	complemented
515	#441	C₂²×C₅₃	1	4	yes	complemented
516	#440	C₂²×C₅₃	1	4	yes	—
517	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
518	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
519	#436	C₂²×C₅₃	1	2	yes	complemented
520	#435	C₂²×C₅₃	1	4	yes	—
521	#432	C₂²×C₅₃	1	2	yes	complemented
522	#431	C₂²×C₅₃	1	1	yes	characteristic
523	#428	C₂²×C₅₃	1	4	yes	complemented
524	C₂×#106	C₂²×C₅₃	1	2	yes	—
525	#423	C₂²×C₅₃	1	2	yes	complemented
526	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
527	#417	C₂²×C₅₃	1	4	yes	—
528	C₂×#102	C₂²×C₅₃	1	2	yes	—
529	C₅₃	#62108	1	1	yes	characteristic, direct product factor, Sylow
530	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
531	C₂×#41	C₂³×C₅₃	1	2	yes	—
532	C₂²×Q₈	—	2	1	no	—
533	C₂²×Q₈	—	2	2	no	—
534	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
535	C₂³×C₄	—	2	1	no	—
536	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
537	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
538	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
539	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
540	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
541	C₄⋊Q₈	—	2	2	no	—
542	C₄⋊Q₈	—	2	4	no	—
543	C₄⋊Q₈	—	2	1	no	—
544	C₄⋊Q₈	—	2	1	no	—
545	C₄⋊Q₈	—	2	2	no	—
546	C₄⋊Q₈	—	2	2	no	—
547	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
548	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
549	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
550	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
551	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
552	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
553	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
554	#126	—	2	2	no	—
555	#126	—	2	2	no	—
556	#126	—	2	2	no	—
557	#125	—	2	2	no	—
558	#125	—	2	4	no	—
559	#125	—	2	2	no	—
560	#125	—	2	1	no	—
561	#125	—	2	1	no	—
562	#125	—	2	2	no	—
563	#125	C₂³×C₅₃	1	2	yes	—
564	#124	—	2	2	no	—
565	#124	C₂³×C₅₃	1	2	yes	—
566	#123	—	2	2	no	—
567	#123	—	2	2	no	—
568	#123	—	2	2	no	—
569	#123	—	2	2	no	—
570	C₂²⋊Q₈	—	2	1	no	—
571	C₂²⋊Q₈	—	2	1	no	—
572	C₂²⋊Q₈	—	2	1	no	—
573	C₂²⋊Q₈	—	2	1	no	—
574	C₂²⋊Q₈	—	2	2	no	—
575	C₂²⋊Q₈	—	2	2	no	—
576	C₂²⋊Q₈	—	2	4	no	—
577	C₂²⋊Q₈	—	2	4	no	—
578	C₂²⋊Q₈	—	2	4	no	—
579	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
580	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
581	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
582	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
583	#121	—	2	1	no	—
584	#121	—	2	1	no	—
585	#121	—	2	2	no	—
586	#121	C₂³×C₅₃	1	2	yes	—
587	#120	—	2	2	no	—
588	C₄×Q₈	—	2	2	no	—
589	C₄×Q₈	—	2	1	no	—
590	C₄×Q₈	—	2	1	no	—
591	C₄×Q₈	—	2	2	no	—
592	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
593	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
594	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
595	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
596	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
597	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
598	C₄×D₄	—	2	4	no	—
599	C₄×D₄	—	2	2	no	—
600	C₄×D₄	—	2	2	no	—
601	C₄×D₄	—	2	2	no	—
602	C₄×D₄	—	2	2	no	—
603	C₄×D₄	—	2	1	no	—
604	C₄×D₄	—	2	1	no	—
605	C₄×D₄	D₄×C₅₃	1	2	yes	—
606	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
607	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
608	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
609	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
610	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
611	C₄×D₄	C₂³×C₅₃	1	2	yes	—
612	C₄×D₄	C₂³×C₅₃	1	2	yes	—
613	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
614	#117	—	2	1	no	—
615	#117	—	2	1	no	—
616	#117	—	2	2	no	—
617	#117	—	2	1	no	—
618	#117	—	2	1	no	—
619	#117	—	2	2	no	—
620	#117	—	2	2	no	—
621	#117	—	2	2	no	—
622	#117	D₄×C₅₃	1	1	yes	characteristic
623	#117	D₄×C₅₃	1	2	yes	—
624	#117	C₂³×C₅₃	1	2	yes	—
625	C₂×C₄⋊C₄	—	2	1	no	—
626	C₂×C₄⋊C₄	—	2	2	no	—
627	C₂×C₄⋊C₄	—	2	1	no	—
628	C₂×C₄⋊C₄	—	2	1	no	—
629	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
630	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
631	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
632	C₂×C₂²⋊C₄	—	2	1	no	—
633	C₂×C₂²⋊C₄	—	2	2	no	—
634	C₂×C₄²	—	2	2	no	—
635	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
636	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
637	#107	C₂³×C₅₃	1	4	yes	—
638	#107	C₂³×C₅₃	1	4	yes	—
639	#107	C₂³×C₅₃	1	4	yes	—
640	#106	C₂³×C₅₃	1	4	yes	—
641	#106	C₂³×C₅₃	1	4	yes	—
642	#106	C₂³×C₅₃	1	2	yes	—
643	#106	C₂³×C₅₃	1	2	yes	—
644	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
645	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
646	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
647	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
648	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
649	#103	C₂³×C₅₃	1	4	yes	—
650	#103	C₂³×C₅₃	1	4	yes	—
651	#103	C₂³×C₅₃	1	2	yes	—
652	#103	C₂³×C₅₃	1	2	yes	—
653	#103	C₂³×C₅₃	1	4	yes	—
654	#103	C₂³×C₅₃	1	4	yes	—
655	#102	C₂³×C₅₃	1	4	yes	—
656	#102	C₂³×C₅₃	1	4	yes	—
657	#102	C₂³×C₅₃	1	2	yes	—
658	#102	C₂³×C₅₃	1	2	yes	—
659	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
660	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
661	#97	C₂³×C₅₃	1	4	yes	—
662	C₄×C₈	C₂³×C₅₃	1	2	yes	—
663	C₂⁴	—	2	1	no	—
664	#41	—	2	4	no	—
665	#41	—	2	4	no	—
666	#41	—	4	2	no	—
667	C₂×Q₈	—	2	1	no	—
668	C₂×Q₈	—	2	2	no	—
669	C₂×Q₈	—	2	1	no	—
670	C₂×Q₈	—	2	4	no	—
671	C₂×Q₈	—	2	2	no	—
672	C₂×Q₈	—	8	2	no	—
673	C₂×Q₈	—	8	4	no	—
674	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
675	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
676	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
677	C₂×D₄	—	2	2	no	—
678	C₂×D₄	—	2	2	no	—
679	C₂×D₄	—	2	2	no	—
680	C₂×D₄	—	2	1	no	—
681	C₂×D₄	—	4	1	no	—
682	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
683	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
684	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
685	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
686	C₂²×C₄	—	2	1	no	—
687	C₂²×C₄	—	2	4	no	—
688	C₂²×C₄	—	2	2	no	—
689	C₂²×C₄	—	2	1	no	—
690	C₂²×C₄	—	2	2	no	—
691	C₂²×C₄	—	2	1	no	—
692	C₂²×C₄	—	2	1	no	—
693	C₂²×C₄	—	2	1	no	—
694	C₂²×C₄	—	2	2	no	—
695	C₂²×C₄	—	4	4	no	—
696	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
697	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
698	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
699	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
700	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
701	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
702	Q₁₆	—	4	4	no	—
703	SD₁₆	—	2	4	no	—
704	SD₁₆	—	4	2	no	—
705	M₁₆	—	4	4	no	—
706	C₂×C₈	—	4	2	no	—
707	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
708	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
709	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
710	C₄⋊C₄	—	2	2	no	—
711	C₄⋊C₄	—	2	1	no	—
712	C₄⋊C₄	—	2	1	no	—
713	C₄⋊C₄	—	2	1	no	—
714	C₄⋊C₄	—	2	2	no	—
715	C₄⋊C₄	—	2	2	no	—
716	C₄⋊C₄	—	2	4	no	—
717	C₄⋊C₄	—	2	2	no	—
718	C₄⋊C₄	—	2	1	no	—
719	C₄⋊C₄	—	2	4	no	—
720	C₄⋊C₄	—	2	2	no	—
721	C₄⋊C₄	—	2	1	no	—
722	C₄⋊C₄	—	2	2	no	—
723	C₄⋊C₄	—	2	2	no	—
724	C₄⋊C₄	—	2	2	no	—
725	C₄⋊C₄	—	2	1	no	—
726	C₄⋊C₄	—	2	2	no	—
727	C₄⋊C₄	—	2	4	no	—
728	C₄⋊C₄	—	2	2	no	—
729	C₄⋊C₄	—	2	1	no	—
730	C₄⋊C₄	—	2	4	no	—
731	C₄⋊C₄	—	2	2	no	—
732	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
733	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
734	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
735	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
736	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
737	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
738	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
739	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
740	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
741	C₂²⋊C₄	—	2	2	no	—
742	C₂²⋊C₄	—	2	4	no	—
743	C₂²⋊C₄	—	2	2	no	—
744	C₂²⋊C₄	—	2	2	no	—
745	C₂²⋊C₄	—	2	2	no	—
746	C₂²⋊C₄	—	2	1	no	—
747	C₂²⋊C₄	—	2	2	no	—
748	C₂²⋊C₄	—	2	2	no	—
749	C₂²⋊C₄	—	2	1	no	—
750	C₂²⋊C₄	—	2	1	no	—
751	C₂²⋊C₄	—	2	1	no	—
752	C₂²⋊C₄	—	2	2	no	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
755	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
756	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
757	C₄²	—	2	2	no	—
758	C₄²	—	2	2	no	—
759	C₄²	—	2	2	no	—
760	C₄²	—	2	4	no	—
761	C₄²	—	2	2	no	—
762	C₄²	C₂×D₄×C₅₃	1	2	yes	—
763	C₄²	C₂×D₄×C₅₃	1	2	yes	—
764	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
765	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
766	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
767	C₄²	C₂⁴×C₅₃	1	2	yes	—
768	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
769	C₂³	—	2	1	no	—
770	C₂³	—	2	1	no	—
771	C₂³	—	4	2	no	—
772	C₂³	—	4	2	no	—
773	C₂³	—	8	1	no	—
774	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
775	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
776	Q₈	—	2	2	no	—
777	Q₈	—	2	2	no	—
778	Q₈	—	2	2	no	—
779	Q₈	—	2	2	no	—
780	Q₈	—	2	2	no	—
781	Q₈	—	4	2	no	—
782	Q₈	—	4	4	no	—
783	Q₈	—	8	1	no	—
784	Q₈	—	8	4	no	—
785	Q₈	—	8	2	no	—
786	D₄	—	2	4	no	—
787	D₄	—	2	2	no	—
788	D₄	—	2	1	no	—
789	D₄	—	2	1	no	—
790	D₄	—	2	1	no	—
791	D₄	—	4	2	no	—
792	D₄	—	4	4	no	—
793	D₄	—	8	2	no	—
794	D₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
795	C₂×C₄	—	2	4	no	—
796	C₂×C₄	—	2	1	no	—
797	C₂×C₄	—	2	2	no	—
798	C₂×C₄	—	2	1	no	—
799	C₂×C₄	—	2	2	no	—
800	C₂×C₄	—	2	1	no	—
801	C₂×C₄	—	2	2	no	—
802	C₂×C₄	—	2	2	no	—
803	C₂×C₄	—	2	2	no	—
804	C₂×C₄	—	2	4	no	—
805	C₂×C₄	—	2	2	no	—
806	C₂×C₄	—	2	1	no	—
807	C₂×C₄	—	4	2	no	—
808	C₂×C₄	—	4	1	no	—
809	C₂×C₄	—	4	4	no	—
810	C₂×C₄	—	4	2	no	—
811	C₂×C₄	—	4	2	no	—
812	C₂×C₄	—	4	2	no	—
813	C₂×C₄	—	4	2	no	—
814	C₂×C₄	—	4	2	no	—
815	C₂×C₄	—	8	2	no	—
816	C₂×C₄	—	8	2	no	—
817	C₂×C₄	—	8	4	no	—
818	C₂×C₄	#137×C₅₃	1	4	yes	—
819	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
820	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
821	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
822	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
823	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
824	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
825	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
826	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
827	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
828	C₈	—	2	2	no	—
829	C₈	—	2	4	no	—
830	C₈	—	2	2	no	—
831	C₂²	—	2	1	no	—
832	C₂²	—	2	1	no	—
833	C₂²	—	2	2	no	—
834	C₂²	—	2	2	no	—
835	C₂²	—	2	2	no	—
836	C₂²	—	2	2	no	—
837	C₂²	—	4	1	no	—
838	C₂²	—	4	1	no	—
839	C₂²	—	4	1	no	—
840	C₂²	—	4	1	no	—
841	C₂²	—	4	1	no	—
842	C₂²	—	8	4	no	—
843	C₂²	C₂×#137×C₅₃	1	2	yes	—
844	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
845	C₄	—	2	2	no	—
846	C₄	—	2	1	no	—
847	C₄	—	2	2	no	—
848	C₄	—	2	1	no	—
849	C₄	—	2	2	no	—
850	C₄	—	2	1	no	—
851	C₄	—	4	2	no	—
852	C₄	—	4	2	no	—
853	C₄	—	4	1	no	—
854	C₄	—	4	1	no	—
855	C₄	—	4	4	no	—
856	C₄	—	4	2	no	—
857	C₄	—	4	2	no	—
858	C₄	C₂×#137×C₅₃	1	2	yes	—
859	C₄	C₂×#137×C₅₃	1	2	yes	—
860	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
861	C₄	C₅₃×#585	1	1	yes	characteristic
862	C₂	—	2	2	no	—
863	C₂	—	4	2	no	—
864	C₂	—	4	2	no	—
865	C₂	—	8	1	no	—
866	C₂	C₅₃×#3489	1	1	yes	characteristic
867	C₂	C₂²×#137×C₅₃	1	1	yes	characteristic
868	C₂	C₅₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3503	C₂	semidirect
3. C₅₃×#3385	C₂	semidirect
4. C₅₃×#3369	C₂	semidirect
5. C₅₃×#3357	C₂	semidirect
6. C₅₃×#3344	C₂	semidirect
7. C₅₃×#3331	C₂	semidirect
8. C₅₃×#3314	C₂	semidirect
9. C₅₃×#3252	C₂	semidirect
10. C₅₃×#3249	C₂	semidirect
11. C₅₃×#3238	C₂	semidirect
12. C₅₃×#3230	C₂	semidirect
13. C₅₃×#3116	C₂	semidirect
14. C₅₃×#3101	C₂	semidirect
15. C₅₃×#3079	C₂	semidirect
16. C₂×C₅₃×#476	C₂	semidirect
17. C₅₃×#2989	C₂	semidirect
18. Q₈²×C₅₃	C₂²	semidirect
24. C₅₃×#554	C₂²	semidirect
37. C₅₃×#533	C₂²	semidirect
39. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
48. C₅₃×#501	C₂²	semidirect
49. C₅₃×#500	C₂²	semidirect
51. C₅₃×#487	C₂²	semidirect
52. C₅₃×#485	C₂²	semidirect
55. C₅₃×#483	C₂²	semidirect
57. C₅₃×#479	C₂²	semidirect
58. C₅₃×#477	C₂²	semidirect
59. C₅₃×#477	C₂²	semidirect
60. C₅₃×#477	C₂²	semidirect
61. C₅₃×#477	C₂²	semidirect
68. C₅₃×#475	C₂²	semidirect
70. C₅₃×#470	C₂²	semidirect
74. C₅₃×#462	C₂²	semidirect
75. C₅₃×#462	C₂²	semidirect
80. C₅₃×#446	C₂²	semidirect
81. C₅₃×#441	C₂²	semidirect
85. C₅₃×#436	C₂²	semidirect
87. C₅₃×#432	C₂²	semidirect
89. C₅₃×#428	C₂²	semidirect
91. C₅₃×#423	C₂²	semidirect
396. #62108	C₅₃	direct (N × H)
428. #3503	C₂×C₅₃	semidirect
430. #3385	C₂×C₅₃	semidirect
431. #3369	C₂×C₅₃	semidirect
432. #3357	C₂×C₅₃	semidirect
433. #3344	C₂×C₅₃	semidirect
434. #3331	C₂×C₅₃	semidirect
435. #3314	C₂×C₅₃	semidirect
436. #3252	C₂×C₅₃	semidirect
437. #3249	C₂×C₅₃	semidirect
438. #3238	C₂×C₅₃	semidirect
439. #3230	C₂×C₅₃	semidirect
440. #3116	C₂×C₅₃	semidirect
441. #3101	C₂×C₅₃	semidirect
442. #3079	C₂×C₅₃	semidirect
443. C₂×#476	C₂×C₅₃	semidirect
444. #2989	C₂×C₅₃	semidirect
452. Q₈²	C₂²×C₅₃	semidirect
458. #554	C₂²×C₅₃	semidirect
471. #533	C₂²×C₅₃	semidirect
473. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
482. #501	C₂²×C₅₃	semidirect
483. #500	C₂²×C₅₃	semidirect
485. #487	C₂²×C₅₃	semidirect
486. #485	C₂²×C₅₃	semidirect
489. #483	C₂²×C₅₃	semidirect
491. #479	C₂²×C₅₃	semidirect
492. #477	C₂²×C₅₃	semidirect
493. #477	C₂²×C₅₃	semidirect
494. #477	C₂²×C₅₃	semidirect
495. #477	C₂²×C₅₃	semidirect
502. #475	C₂²×C₅₃	semidirect
504. #470	C₂²×C₅₃	semidirect
508. #462	C₂²×C₅₃	semidirect
509. #462	C₂²×C₅₃	semidirect
514. #446	C₂²×C₅₃	semidirect
515. #441	C₂²×C₅₃	semidirect
519. #436	C₂²×C₅₃	semidirect
521. #432	C₂²×C₅₃	semidirect
523. #428	C₂²×C₅₃	semidirect
525. #423	C₂²×C₅₃	semidirect
529. C₅₃	#62108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3503	2
C₅₃×#3475	2
C₅₃×#3385	2
C₅₃×#3369	2
C₅₃×#3357	2
C₅₃×#3344	2
C₅₃×#3331	2
C₅₃×#3314	2
C₅₃×#3252	2
C₅₃×#3249	2
C₅₃×#3238	2
C₅₃×#3230	2
C₅₃×#3116	2
C₅₃×#3101	2
C₅₃×#3079	2
C₂×C₅₃×#476	2
C₅₃×#2989	2
#62108	53

Subgroups — C53×#62108 #27269620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62108 #27269620