Subgroups — C₅₃×#62107 #27269619

Order 13568.

1958 conjugacy classes of subgroups.

The subgroup lattice has 900 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3384	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3367	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3354	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3343	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3330	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3306	C₂	1	4	yes	maximal, complemented
9	C₅₃×#3251	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3239	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#3115	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3100	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#3079	C₂	1	2	yes	maximal, complemented
16	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
17	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
19	C₅₃×#556	C₂²	1	1	yes	characteristic
20	C₅₃×#555	C₂²	1	1	yes	characteristic
21	C₅₃×#554	C₂²	1	1	yes	characteristic
22	C₅₃×#553	—	2	1	no	—
23	C₅₃×#552	—	2	1	no	—
24	C₅₃×#552	C₂²	1	2	yes	—
25	C₅₃×#551	—	2	2	no	—
26	D₄×Q₈×C₅₃	—	2	1	no	—
27	D₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
28	C₅₃×#547	—	2	2	no	—
29	C₅₃×#547	—	2	1	no	—
30	C₅₃×#547	C₂²	1	1	yes	characteristic
31	C₅₃×#547	C₂²	1	4	yes	—
32	C₅₃×#547	C₂²	1	2	yes	complemented
33	C₅₃×#546	C₂²	1	2	yes	—
34	C₅₃×#536	—	2	1	no	—
35	C₅₃×#536	—	2	2	no	—
36	C₅₃×#533	C₂²	1	1	yes	characteristic
37	C₅₃×#532	C₂²	1	1	yes	characteristic
38	C₅₃×#532	C₂²	1	2	yes	complemented
39	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
41	C₅₃×#529	—	2	2	no	—
42	C₅₃×#529	C₂²	1	2	yes	—
43	C₅₃×#525	—	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×#498	C₂²	1	2	yes	complemented
50	C₅₃×#489	C₂²	1	4	yes	—
51	C₅₃×#487	C₂²	1	2	yes	complemented
52	C₅₃×#484	C₂²	1	4	yes	complemented
53	C₅₃×#483	C₂²	1	4	yes	—
54	C₅₃×#482	C₂²	1	4	yes	complemented
55	C₅₃×#481	C₂²	1	4	yes	—
56	C₅₃×#480	C₂²	1	2	yes	—
57	C₅₃×#478	C₂²	1	4	yes	complemented
58	C₅₃×#477	C₂²	1	4	yes	complemented
59	C₅₃×#476	C₂²	1	4	yes	—
60	C₅₃×#476	C₂²	1	2	yes	complemented
61	C₅₃×#475	C₂²	1	2	yes	complemented
62	C₅₃×#475	C₂²	1	2	yes	complemented
63	C₅₃×#475	C₂²	1	2	yes	complemented
64	C₅₃×#474	C₂²	1	1	yes	characteristic
65	C₅₃×#474	C₂²	1	2	yes	—
66	C₅₃×#474	C₂²	1	2	yes	—
67	C₅₃×#474	C₂²	1	2	yes	—
68	C₅₃×#474	C₂²	1	2	yes	—
69	C₅₃×#474	C₂²	1	1	yes	characteristic
70	C₅₃×#469	C₂²	1	4	yes	—
71	C₅₃×#468	C₂²	1	4	yes	complemented
72	C₅₃×#465	C₂²	1	4	yes	—
73	C₅₃×#463	C₂²	1	4	yes	—
74	C₅₃×#461	C₂²	1	2	yes	complemented
75	C₅₃×#460	C₂²	1	2	yes	complemented
76	C₅₃×#459	C₂²	1	1	yes	characteristic
77	C₅₃×#459	C₂²	1	1	yes	characteristic
78	C₅₃×#449	C₂²	1	4	yes	—
79	C₅₃×#447	C₂²	1	2	yes	—
80	C₅₃×#446	C₂²	1	2	yes	complemented
81	C₅₃×#442	C₂²	1	4	yes	—
82	C₅₃×#440	C₂²	1	4	yes	complemented
83	C₄×D₈×C₅₃	C₂²	1	2	yes	—
84	C₄×D₈×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×#107×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	2₋¹⁺⁴×C₅₃	—	4	8	no	—
96	C₂×#41×C₅₃	—	2	4	no	—
97	C₂×#41×C₅₃	—	2	2	no	—
98	C₂×#41×C₅₃	—	2	1	no	—
99	C₂×#41×C₅₃	—	2	2	no	—
100	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
101	C₂×#41×C₅₃	C₂³	1	2	yes	—
102	C₂²×Q₈×C₅₃	—	2	2	no	—
103	C₂²×Q₈×C₅₃	—	2	1	no	—
104	C₂²×Q₈×C₅₃	—	2	2	no	—
105	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₅₃	—	2	1	no	—
107	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
108	C₂×D₈×C₅₃	C₂³	1	2	yes	—
109	C₂×D₈×C₅₃	C₂³	1	2	yes	—
110	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
111	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
112	C₄⋊Q₈×C₅₃	—	2	1	no	—
113	C₄⋊Q₈×C₅₃	—	2	1	no	—
114	C₄⋊Q₈×C₅₃	—	2	2	no	—
115	C₄⋊Q₈×C₅₃	—	2	2	no	—
116	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
117	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
118	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
120	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
121	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
122	#127×C₅₃	C₂³	1	1	yes	characteristic
123	#126×C₅₃	—	2	2	no	—
124	#126×C₅₃	—	2	2	no	—
125	#126×C₅₃	—	2	2	no	—
126	#125×C₅₃	—	2	2	no	—
127	#125×C₅₃	—	2	1	no	—
128	#125×C₅₃	C₂³	1	2	yes	—
129	#124×C₅₃	—	2	2	no	—
130	#124×C₅₃	—	2	1	no	—
131	#124×C₅₃	—	2	2	no	—
132	#124×C₅₃	C₂³	1	2	yes	—
133	#123×C₅₃	—	2	2	no	—
134	#123×C₅₃	—	2	2	no	—
135	#123×C₅₃	—	2	4	no	—
136	#123×C₅₃	—	2	4	no	—
137	#123×C₅₃	—	2	4	no	—
138	#123×C₅₃	—	2	2	no	—
139	#123×C₅₃	—	2	2	no	—
140	C₂²⋊Q₈×C₅₃	—	2	2	no	—
141	C₂²⋊Q₈×C₅₃	—	2	2	no	—
142	C₂²⋊Q₈×C₅₃	—	2	4	no	—
143	C₂²⋊Q₈×C₅₃	—	2	4	no	—
144	C₂²⋊Q₈×C₅₃	—	2	2	no	—
145	C₂²⋊Q₈×C₅₃	—	2	1	no	—
146	C₂²⋊Q₈×C₅₃	—	2	1	no	—
147	C₂²⋊Q₈×C₅₃	—	2	1	no	—
148	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
149	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
150	#121×C₅₃	—	2	2	no	—
151	#121×C₅₃	—	2	1	no	—
152	#121×C₅₃	—	2	1	no	—
153	#121×C₅₃	—	2	1	no	—
154	#121×C₅₃	—	2	2	no	—
155	#121×C₅₃	C₂³	1	2	yes	—
156	#121×C₅₃	C₂³	1	2	yes	—
157	#121×C₅₃	C₂³	1	4	yes	—
158	#120×C₅₃	—	2	2	no	—
159	C₄×Q₈×C₅₃	—	2	1	no	—
160	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
161	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
162	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
163	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
164	C₄×D₄×C₅₃	—	2	1	no	—
165	C₄×D₄×C₅₃	—	2	1	no	—
166	C₄×D₄×C₅₃	—	2	1	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	—	2	2	no	—
171	C₄×D₄×C₅₃	D₄	1	2	yes	—
172	C₄×D₄×C₅₃	D₄	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	1	yes	characteristic
177	C₄×D₄×C₅₃	C₂³	1	2	yes	—
178	C₄×D₄×C₅₃	C₂³	1	2	yes	—
179	C₄×D₄×C₅₃	C₂³	1	4	yes	—
180	C₄×D₄×C₅₃	C₂³	1	2	yes	—
181	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
182	#117×C₅₃	—	2	1	no	—
183	#117×C₅₃	—	2	1	no	—
184	#117×C₅₃	—	2	2	no	—
185	#117×C₅₃	—	2	2	no	—
186	#117×C₅₃	—	2	2	no	—
187	#117×C₅₃	—	2	1	no	—
188	#117×C₅₃	D₄	1	1	yes	characteristic
189	#117×C₅₃	D₄	1	2	yes	—
190	#117×C₅₃	C₂³	1	2	yes	—
191	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
192	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
193	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
194	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
195	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
197	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
198	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
199	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
200	C₂×C₄²×C₅₃	—	2	2	no	—
201	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
202	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
203	#107×C₅₃	C₂³	1	2	yes	—
204	#107×C₅₃	C₂³	1	4	yes	—
205	#107×C₅₃	C₂³	1	4	yes	—
206	#107×C₅₃	C₂³	1	2	yes	—
207	#106×C₅₃	C₂³	1	4	yes	—
208	#106×C₅₃	C₂³	1	4	yes	—
209	#106×C₅₃	C₂³	1	4	yes	—
210	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
211	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
212	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
213	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
214	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
215	#103×C₅₃	C₂³	1	4	yes	—
216	#103×C₅₃	C₂³	1	4	yes	—
217	#103×C₅₃	C₂³	1	4	yes	—
218	#102×C₅₃	C₂³	1	2	yes	—
219	#102×C₅₃	C₂³	1	4	yes	—
220	#102×C₅₃	C₂³	1	4	yes	—
221	#102×C₅₃	C₂³	1	2	yes	—
222	#102×C₅₃	C₂³	1	2	yes	—
223	#102×C₅₃	C₂³	1	2	yes	—
224	#102×C₅₃	C₂³	1	4	yes	—
225	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
226	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
227	#97×C₅₃	C₂³	1	4	yes	—
228	C₄×C₈×C₅₃	C₂³	1	2	yes	—
229	C₂⁴×C₅₃	—	2	1	no	—
230	#41×C₅₃	—	2	4	no	—
231	#41×C₅₃	—	2	4	no	—
232	#41×C₅₃	—	4	2	no	—
233	#41×C₅₃	—	4	8	no	—
234	#41×C₅₃	—	4	16	no	—
235	#41×C₅₃	—	4	8	no	—
236	#41×C₅₃	—	4	4	no	—
237	C₂×Q₈×C₅₃	—	2	4	no	—
238	C₂×Q₈×C₅₃	—	2	2	no	—
239	C₂×Q₈×C₅₃	—	2	1	no	—
240	C₂×Q₈×C₅₃	—	2	2	no	—
241	C₂×Q₈×C₅₃	—	2	1	no	—
242	C₂×Q₈×C₅₃	—	4	8	no	—
243	C₂×Q₈×C₅₃	—	4	8	no	—
244	C₂×Q₈×C₅₃	—	4	4	no	—
245	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
246	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
247	C₂×D₄×C₅₃	—	2	2	no	—
248	C₂×D₄×C₅₃	—	2	2	no	—
249	C₂×D₄×C₅₃	—	2	4	no	—
250	C₂×D₄×C₅₃	—	2	2	no	—
251	C₂×D₄×C₅₃	—	2	2	no	—
252	C₂×D₄×C₅₃	—	2	1	no	—
253	C₂×D₄×C₅₃	—	2	1	no	—
254	C₂×D₄×C₅₃	—	2	1	no	—
255	C₂×D₄×C₅₃	—	4	1	no	—
256	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
257	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
258	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
259	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
260	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
261	C₂²×C₄×C₅₃	—	2	1	no	—
262	C₂²×C₄×C₅₃	—	2	4	no	—
263	C₂²×C₄×C₅₃	—	2	2	no	—
264	C₂²×C₄×C₅₃	—	2	1	no	—
265	C₂²×C₄×C₅₃	—	2	2	no	—
266	C₂²×C₄×C₅₃	—	2	1	no	—
267	C₂²×C₄×C₅₃	—	2	1	no	—
268	C₂²×C₄×C₅₃	—	2	1	no	—
269	C₂²×C₄×C₅₃	—	2	4	no	—
270	C₂²×C₄×C₅₃	—	2	2	no	—
271	C₂²×C₄×C₅₃	—	2	2	no	—
272	C₂²×C₄×C₅₃	—	4	4	no	—
273	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
274	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
275	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
276	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
277	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
279	SD₁₆×C₅₃	—	4	4	no	—
280	D₈×C₅₃	—	2	4	no	—
281	D₈×C₅₃	—	4	2	no	—
282	M₁₆×C₅₃	—	4	4	no	—
283	C₂×C₈×C₅₃	—	4	2	no	—
284	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
285	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
286	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
287	C₄⋊C₄×C₅₃	—	2	2	no	—
288	C₄⋊C₄×C₅₃	—	2	1	no	—
289	C₄⋊C₄×C₅₃	—	2	1	no	—
290	C₄⋊C₄×C₅₃	—	2	1	no	—
291	C₄⋊C₄×C₅₃	—	2	2	no	—
292	C₄⋊C₄×C₅₃	—	2	2	no	—
293	C₄⋊C₄×C₅₃	—	2	4	no	—
294	C₄⋊C₄×C₅₃	—	2	2	no	—
295	C₄⋊C₄×C₅₃	—	2	1	no	—
296	C₄⋊C₄×C₅₃	—	2	4	no	—
297	C₄⋊C₄×C₅₃	—	2	2	no	—
298	C₄⋊C₄×C₅₃	—	2	2	no	—
299	C₄⋊C₄×C₅₃	—	2	2	no	—
300	C₄⋊C₄×C₅₃	—	2	2	no	—
301	C₄⋊C₄×C₅₃	—	2	2	no	—
302	C₄⋊C₄×C₅₃	—	2	1	no	—
303	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
304	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
305	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
306	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
307	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
308	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
309	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
310	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
311	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
312	C₂²⋊C₄×C₅₃	—	2	2	no	—
313	C₂²⋊C₄×C₅₃	—	2	2	no	—
314	C₂²⋊C₄×C₅₃	—	2	2	no	—
315	C₂²⋊C₄×C₅₃	—	2	1	no	—
316	C₂²⋊C₄×C₅₃	—	2	2	no	—
317	C₂²⋊C₄×C₅₃	—	2	2	no	—
318	C₂²⋊C₄×C₅₃	—	2	1	no	—
319	C₂²⋊C₄×C₅₃	—	2	1	no	—
320	C₂²⋊C₄×C₅₃	—	2	1	no	—
321	C₂²⋊C₄×C₅₃	—	2	2	no	—
322	C₂²⋊C₄×C₅₃	—	2	2	no	—
323	C₂²⋊C₄×C₅₃	—	2	4	no	—
324	C₂²⋊C₄×C₅₃	—	2	2	no	—
325	C₂²⋊C₄×C₅₃	—	2	4	no	—
326	C₂²⋊C₄×C₅₃	—	2	2	no	—
327	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
328	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
329	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
330	C₄²×C₅₃	—	2	2	no	—
331	C₄²×C₅₃	—	2	2	no	—
332	C₄²×C₅₃	—	2	2	no	—
333	C₄²×C₅₃	C₂×D₄	1	2	yes	—
334	C₄²×C₅₃	C₂×D₄	1	2	yes	—
335	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
336	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
337	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
338	C₄²×C₅₃	C₂⁴	1	2	yes	—
339	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
340	C₂³×C₅₃	—	2	1	no	—
341	C₂³×C₅₃	—	2	2	no	—
342	C₂³×C₅₃	—	2	1	no	—
343	C₂³×C₅₃	—	4	2	no	—
344	C₂³×C₅₃	—	4	2	no	—
345	C₂³×C₅₃	—	8	1	no	—
346	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
347	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
348	Q₈×C₅₃	—	2	2	no	—
349	Q₈×C₅₃	—	2	2	no	—
350	Q₈×C₅₃	—	2	2	no	—
351	Q₈×C₅₃	—	4	8	no	—
352	Q₈×C₅₃	—	4	2	no	—
353	Q₈×C₅₃	—	4	4	no	—
354	Q₈×C₅₃	—	4	4	no	—
355	Q₈×C₅₃	—	8	1	no	—
356	D₄×C₅₃	—	2	4	no	—
357	D₄×C₅₃	—	2	2	no	—
358	D₄×C₅₃	—	2	1	no	—
359	D₄×C₅₃	—	2	2	no	—
360	D₄×C₅₃	—	2	1	no	—
361	D₄×C₅₃	—	2	1	no	—
362	D₄×C₅₃	—	2	2	no	—
363	D₄×C₅₃	—	4	2	no	—
364	D₄×C₅₃	—	4	2	no	—
365	D₄×C₅₃	—	4	4	no	—
366	D₄×C₅₃	—	4	8	no	—
367	D₄×C₅₃	—	4	4	no	—
368	D₄×C₅₃	—	4	2	no	—
369	D₄×C₅₃	—	8	2	no	—
370	D₄×C₅₃	2₋¹⁺⁴	1	2	yes	complemented
371	C₂×C₄×C₅₃	—	2	4	no	—
372	C₂×C₄×C₅₃	—	2	1	no	—
373	C₂×C₄×C₅₃	—	2	2	no	—
374	C₂×C₄×C₅₃	—	2	1	no	—
375	C₂×C₄×C₅₃	—	2	2	no	—
376	C₂×C₄×C₅₃	—	2	4	no	—
377	C₂×C₄×C₅₃	—	2	2	no	—
378	C₂×C₄×C₅₃	—	2	1	no	—
379	C₂×C₄×C₅₃	—	2	1	no	—
380	C₂×C₄×C₅₃	—	2	2	no	—
381	C₂×C₄×C₅₃	—	2	2	no	—
382	C₂×C₄×C₅₃	—	4	2	no	—
383	C₂×C₄×C₅₃	—	4	1	no	—
384	C₂×C₄×C₅₃	—	4	4	no	—
385	C₂×C₄×C₅₃	—	4	8	no	—
386	C₂×C₄×C₅₃	—	4	8	no	—
387	C₂×C₄×C₅₃	—	4	2	no	—
388	C₂×C₄×C₅₃	—	4	4	no	—
389	C₂×C₄×C₅₃	—	4	2	no	—
390	C₂×C₄×C₅₃	—	4	2	no	—
391	C₂×C₄×C₅₃	—	4	2	no	—
392	C₂×C₄×C₅₃	—	4	2	no	—
393	C₂×C₄×C₅₃	—	4	4	no	—
394	C₂×C₄×C₅₃	—	8	2	no	—
395	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
396	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
397	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
398	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
399	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
400	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
401	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
402	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
403	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
404	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
405	C₈×C₅₃	—	2	2	no	—
406	C₈×C₅₃	—	2	4	no	—
407	C₈×C₅₃	—	2	2	no	—
408	#62107	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
409	C₂²×C₅₃	—	2	1	no	—
410	C₂²×C₅₃	—	2	1	no	—
411	C₂²×C₅₃	—	2	2	no	—
412	C₂²×C₅₃	—	2	2	no	—
413	C₂²×C₅₃	—	2	2	no	—
414	C₂²×C₅₃	—	2	2	no	—
415	C₂²×C₅₃	—	4	1	no	—
416	C₂²×C₅₃	—	4	1	no	—
417	C₂²×C₅₃	—	4	2	no	—
418	C₂²×C₅₃	—	4	1	no	—
419	C₂²×C₅₃	—	4	1	no	—
420	C₂²×C₅₃	—	4	2	no	—
421	C₂²×C₅₃	—	4	2	no	—
422	C₂²×C₅₃	—	4	1	no	—
423	C₂²×C₅₃	—	8	4	no	—
424	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
425	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
426	C₄×C₅₃	—	2	1	no	—
427	C₄×C₅₃	—	2	2	no	—
428	C₄×C₅₃	—	2	1	no	—
429	C₄×C₅₃	—	2	2	no	—
430	C₄×C₅₃	—	2	1	no	—
431	C₄×C₅₃	—	2	2	no	—
432	C₄×C₅₃	—	4	2	no	—
433	C₄×C₅₃	—	4	2	no	—
434	C₄×C₅₃	—	4	1	no	—
435	C₄×C₅₃	—	4	1	no	—
436	C₄×C₅₃	—	4	4	no	—
437	C₄×C₅₃	—	4	2	no	—
438	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
439	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
440	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
441	C₄×C₅₃	#585	1	1	yes	characteristic
442	#3499	C₂×C₅₃	1	1	yes	characteristic, complemented
443	#3475	C₂×C₅₃	1	1	yes	characteristic, complemented
444	#3384	C₂×C₅₃	1	2	yes	complemented
445	#3367	C₂×C₅₃	1	2	yes	complemented
446	#3354	C₂×C₅₃	1	2	yes	complemented
447	#3343	C₂×C₅₃	1	4	yes	complemented
448	#3330	C₂×C₅₃	1	2	yes	complemented
449	#3306	C₂×C₅₃	1	4	yes	complemented
450	#3251	C₂×C₅₃	1	2	yes	complemented
451	#3246	C₂×C₅₃	1	1	yes	characteristic, complemented
452	#3239	C₂×C₅₃	1	2	yes	complemented
453	#3229	C₂×C₅₃	1	1	yes	characteristic, complemented
454	#3115	C₂×C₅₃	1	2	yes	complemented
455	#3100	C₂×C₅₃	1	1	yes	characteristic, complemented
456	#3079	C₂×C₅₃	1	2	yes	complemented
457	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
458	#2989	C₂×C₅₃	1	1	yes	characteristic, complemented
459	C₂×C₅₃	—	2	2	no	—
460	C₂×C₅₃	—	4	2	no	—
461	C₂×C₅₃	—	4	2	no	—
462	C₂×C₅₃	—	8	2	no	—
463	C₂×C₅₃	—	8	1	no	—
464	C₂×C₅₃	#3489	1	1	yes	characteristic
465	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
466	C₂×C₅₃	#3586	1	1	yes	characteristic
467	C₂×2₋¹⁺⁴	—	2	1	no	—
468	#556	C₂²×C₅₃	1	1	yes	characteristic
469	#555	C₂²×C₅₃	1	1	yes	characteristic
470	#554	C₂²×C₅₃	1	1	yes	characteristic
471	#553	—	2	1	no	—
472	#552	—	2	1	no	—
473	#552	C₂²×C₅₃	1	2	yes	—
474	#551	—	2	2	no	—
475	D₄×Q₈	—	2	1	no	—
476	D₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
477	#547	—	2	2	no	—
478	#547	—	2	1	no	—
479	#547	C₂²×C₅₃	1	1	yes	characteristic
480	#547	C₂²×C₅₃	1	4	yes	—
481	#547	C₂²×C₅₃	1	2	yes	complemented
482	#546	C₂²×C₅₃	1	2	yes	—
483	#536	—	2	1	no	—
484	#536	—	2	2	no	—
485	#533	C₂²×C₅₃	1	1	yes	characteristic
486	#532	C₂²×C₅₃	1	1	yes	characteristic
487	#532	C₂²×C₅₃	1	2	yes	complemented
488	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
489	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
490	#529	—	2	2	no	—
491	#529	C₂²×C₅₃	1	2	yes	—
492	#525	—	2	1	no	—
493	C₄×#41	C₂²×C₅₃	1	2	yes	—
494	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
495	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
496	C₂×#117	—	2	1	no	—
497	#501	C₂²×C₅₃	1	4	yes	complemented
498	#498	C₂²×C₅₃	1	2	yes	complemented
499	#489	C₂²×C₅₃	1	4	yes	—
500	#487	C₂²×C₅₃	1	2	yes	complemented
501	#484	C₂²×C₅₃	1	4	yes	complemented
502	#483	C₂²×C₅₃	1	4	yes	—
503	#482	C₂²×C₅₃	1	4	yes	complemented
504	#481	C₂²×C₅₃	1	4	yes	—
505	#480	C₂²×C₅₃	1	2	yes	—
506	#478	C₂²×C₅₃	1	4	yes	complemented
507	#477	C₂²×C₅₃	1	4	yes	complemented
508	#476	C₂²×C₅₃	1	4	yes	—
509	#476	C₂²×C₅₃	1	2	yes	complemented
510	#475	C₂²×C₅₃	1	2	yes	complemented
511	#475	C₂²×C₅₃	1	2	yes	complemented
512	#475	C₂²×C₅₃	1	2	yes	complemented
513	#474	C₂²×C₅₃	1	2	yes	—
514	#474	C₂²×C₅₃	1	1	yes	characteristic
515	#474	C₂²×C₅₃	1	2	yes	—
516	#474	C₂²×C₅₃	1	2	yes	—
517	#474	C₂²×C₅₃	1	1	yes	characteristic
518	#474	C₂²×C₅₃	1	2	yes	—
519	#469	C₂²×C₅₃	1	4	yes	—
520	#468	C₂²×C₅₃	1	4	yes	complemented
521	#465	C₂²×C₅₃	1	4	yes	—
522	#463	C₂²×C₅₃	1	4	yes	—
523	#461	C₂²×C₅₃	1	2	yes	complemented
524	#460	C₂²×C₅₃	1	2	yes	complemented
525	#459	C₂²×C₅₃	1	1	yes	characteristic
526	#459	C₂²×C₅₃	1	1	yes	characteristic
527	#449	C₂²×C₅₃	1	4	yes	—
528	#447	C₂²×C₅₃	1	2	yes	—
529	#446	C₂²×C₅₃	1	2	yes	complemented
530	#442	C₂²×C₅₃	1	4	yes	—
531	#440	C₂²×C₅₃	1	4	yes	complemented
532	C₄×D₈	C₂²×C₅₃	1	2	yes	—
533	C₄×D₈	C₂²×C₅₃	1	2	yes	—
534	#436	C₂²×C₅₃	1	2	yes	complemented
535	#435	C₂²×C₅₃	1	4	yes	—
536	#432	C₂²×C₅₃	1	2	yes	complemented
537	#431	C₂²×C₅₃	1	1	yes	characteristic
538	#428	C₂²×C₅₃	1	4	yes	complemented
539	C₂×#107	C₂²×C₅₃	1	2	yes	—
540	#423	C₂²×C₅₃	1	2	yes	complemented
541	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
542	#417	C₂²×C₅₃	1	4	yes	—
543	C₂×#102	C₂²×C₅₃	1	2	yes	—
544	C₅₃	#62107	1	1	yes	characteristic, direct product factor, Sylow
545	2₋¹⁺⁴	—	4	8	no	—
546	C₂×#41	—	2	4	no	—
547	C₂×#41	—	2	2	no	—
548	C₂×#41	—	2	1	no	—
549	C₂×#41	—	2	2	no	—
550	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
551	C₂×#41	C₂³×C₅₃	1	2	yes	—
552	C₂²×Q₈	—	2	2	no	—
553	C₂²×Q₈	—	2	1	no	—
554	C₂²×Q₈	—	2	2	no	—
555	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
556	C₂³×C₄	—	2	1	no	—
557	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
558	C₂×D₈	C₂³×C₅₃	1	2	yes	—
559	C₂×D₈	C₂³×C₅₃	1	2	yes	—
560	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
561	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
562	C₄⋊Q₈	—	2	1	no	—
563	C₄⋊Q₈	—	2	1	no	—
564	C₄⋊Q₈	—	2	2	no	—
565	C₄⋊Q₈	—	2	2	no	—
566	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
567	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
568	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
569	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
570	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
571	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
572	#127	C₂³×C₅₃	1	1	yes	characteristic
573	#126	—	2	2	no	—
574	#126	—	2	2	no	—
575	#126	—	2	2	no	—
576	#125	—	2	2	no	—
577	#125	—	2	1	no	—
578	#125	C₂³×C₅₃	1	2	yes	—
579	#124	—	2	2	no	—
580	#124	—	2	1	no	—
581	#124	—	2	2	no	—
582	#124	C₂³×C₅₃	1	2	yes	—
583	#123	—	2	2	no	—
584	#123	—	2	2	no	—
585	#123	—	2	2	no	—
586	#123	—	2	2	no	—
587	#123	—	2	4	no	—
588	#123	—	2	4	no	—
589	#123	—	2	4	no	—
590	C₂²⋊Q₈	—	2	1	no	—
591	C₂²⋊Q₈	—	2	1	no	—
592	C₂²⋊Q₈	—	2	1	no	—
593	C₂²⋊Q₈	—	2	2	no	—
594	C₂²⋊Q₈	—	2	2	no	—
595	C₂²⋊Q₈	—	2	4	no	—
596	C₂²⋊Q₈	—	2	4	no	—
597	C₂²⋊Q₈	—	2	2	no	—
598	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
599	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
600	#121	—	2	1	no	—
601	#121	—	2	1	no	—
602	#121	—	2	1	no	—
603	#121	—	2	2	no	—
604	#121	—	2	2	no	—
605	#121	C₂³×C₅₃	1	2	yes	—
606	#121	C₂³×C₅₃	1	2	yes	—
607	#121	C₂³×C₅₃	1	4	yes	—
608	#120	—	2	2	no	—
609	C₄×Q₈	—	2	1	no	—
610	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
611	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
612	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
613	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
614	C₄×D₄	—	2	2	no	—
615	C₄×D₄	—	2	2	no	—
616	C₄×D₄	—	2	2	no	—
617	C₄×D₄	—	2	1	no	—
618	C₄×D₄	—	2	1	no	—
619	C₄×D₄	—	2	1	no	—
620	C₄×D₄	—	2	2	no	—
621	C₄×D₄	D₄×C₅₃	1	2	yes	—
622	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
623	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
624	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
625	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
626	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
627	C₄×D₄	C₂³×C₅₃	1	2	yes	—
628	C₄×D₄	C₂³×C₅₃	1	2	yes	—
629	C₄×D₄	C₂³×C₅₃	1	4	yes	—
630	C₄×D₄	C₂³×C₅₃	1	2	yes	—
631	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
632	#117	—	2	1	no	—
633	#117	—	2	1	no	—
634	#117	—	2	1	no	—
635	#117	—	2	2	no	—
636	#117	—	2	2	no	—
637	#117	—	2	2	no	—
638	#117	D₄×C₅₃	1	1	yes	characteristic
639	#117	D₄×C₅₃	1	2	yes	—
640	#117	C₂³×C₅₃	1	2	yes	—
641	C₂×C₄⋊C₄	—	2	1	no	—
642	C₂×C₄⋊C₄	—	2	2	no	—
643	C₂×C₄⋊C₄	—	2	1	no	—
644	C₂×C₄⋊C₄	—	2	1	no	—
645	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
646	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
647	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
648	C₂×C₂²⋊C₄	—	2	1	no	—
649	C₂×C₂²⋊C₄	—	2	2	no	—
650	C₂×C₄²	—	2	2	no	—
651	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
652	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
653	#107	C₂³×C₅₃	1	4	yes	—
654	#107	C₂³×C₅₃	1	4	yes	—
655	#107	C₂³×C₅₃	1	2	yes	—
656	#107	C₂³×C₅₃	1	2	yes	—
657	#106	C₂³×C₅₃	1	4	yes	—
658	#106	C₂³×C₅₃	1	4	yes	—
659	#106	C₂³×C₅₃	1	4	yes	—
660	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
661	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
662	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
663	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
664	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
665	#103	C₂³×C₅₃	1	4	yes	—
666	#103	C₂³×C₅₃	1	4	yes	—
667	#103	C₂³×C₅₃	1	4	yes	—
668	#102	C₂³×C₅₃	1	4	yes	—
669	#102	C₂³×C₅₃	1	4	yes	—
670	#102	C₂³×C₅₃	1	2	yes	—
671	#102	C₂³×C₅₃	1	2	yes	—
672	#102	C₂³×C₅₃	1	2	yes	—
673	#102	C₂³×C₅₃	1	2	yes	—
674	#102	C₂³×C₅₃	1	4	yes	—
675	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
676	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
677	#97	C₂³×C₅₃	1	4	yes	—
678	C₄×C₈	C₂³×C₅₃	1	2	yes	—
679	C₂⁴	—	2	1	no	—
680	#41	—	2	4	no	—
681	#41	—	2	4	no	—
682	#41	—	4	2	no	—
683	#41	—	4	8	no	—
684	#41	—	4	16	no	—
685	#41	—	4	8	no	—
686	#41	—	4	4	no	—
687	C₂×Q₈	—	2	4	no	—
688	C₂×Q₈	—	2	2	no	—
689	C₂×Q₈	—	2	1	no	—
690	C₂×Q₈	—	2	2	no	—
691	C₂×Q₈	—	2	1	no	—
692	C₂×Q₈	—	4	8	no	—
693	C₂×Q₈	—	4	8	no	—
694	C₂×Q₈	—	4	4	no	—
695	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
696	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
697	C₂×D₄	—	2	2	no	—
698	C₂×D₄	—	2	2	no	—
699	C₂×D₄	—	2	4	no	—
700	C₂×D₄	—	2	2	no	—
701	C₂×D₄	—	2	2	no	—
702	C₂×D₄	—	2	1	no	—
703	C₂×D₄	—	2	1	no	—
704	C₂×D₄	—	2	1	no	—
705	C₂×D₄	—	4	1	no	—
706	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
707	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
708	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
709	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
710	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
711	C₂²×C₄	—	2	1	no	—
712	C₂²×C₄	—	2	4	no	—
713	C₂²×C₄	—	2	2	no	—
714	C₂²×C₄	—	2	1	no	—
715	C₂²×C₄	—	2	2	no	—
716	C₂²×C₄	—	2	1	no	—
717	C₂²×C₄	—	2	1	no	—
718	C₂²×C₄	—	2	1	no	—
719	C₂²×C₄	—	2	4	no	—
720	C₂²×C₄	—	2	2	no	—
721	C₂²×C₄	—	2	2	no	—
722	C₂²×C₄	—	4	4	no	—
723	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
724	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
725	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
726	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
727	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
728	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
729	SD₁₆	—	4	4	no	—
730	D₈	—	2	4	no	—
731	D₈	—	4	2	no	—
732	M₁₆	—	4	4	no	—
733	C₂×C₈	—	4	2	no	—
734	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
735	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
736	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
737	C₄⋊C₄	—	2	2	no	—
738	C₄⋊C₄	—	2	1	no	—
739	C₄⋊C₄	—	2	1	no	—
740	C₄⋊C₄	—	2	1	no	—
741	C₄⋊C₄	—	2	2	no	—
742	C₄⋊C₄	—	2	2	no	—
743	C₄⋊C₄	—	2	4	no	—
744	C₄⋊C₄	—	2	2	no	—
745	C₄⋊C₄	—	2	1	no	—
746	C₄⋊C₄	—	2	4	no	—
747	C₄⋊C₄	—	2	2	no	—
748	C₄⋊C₄	—	2	2	no	—
749	C₄⋊C₄	—	2	2	no	—
750	C₄⋊C₄	—	2	2	no	—
751	C₄⋊C₄	—	2	2	no	—
752	C₄⋊C₄	—	2	1	no	—
753	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
754	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
755	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
756	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
757	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
758	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
759	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
760	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
761	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
762	C₂²⋊C₄	—	2	2	no	—
763	C₂²⋊C₄	—	2	2	no	—
764	C₂²⋊C₄	—	2	2	no	—
765	C₂²⋊C₄	—	2	1	no	—
766	C₂²⋊C₄	—	2	2	no	—
767	C₂²⋊C₄	—	2	2	no	—
768	C₂²⋊C₄	—	2	1	no	—
769	C₂²⋊C₄	—	2	1	no	—
770	C₂²⋊C₄	—	2	1	no	—
771	C₂²⋊C₄	—	2	2	no	—
772	C₂²⋊C₄	—	2	2	no	—
773	C₂²⋊C₄	—	2	4	no	—
774	C₂²⋊C₄	—	2	2	no	—
775	C₂²⋊C₄	—	2	4	no	—
776	C₂²⋊C₄	—	2	2	no	—
777	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
778	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
779	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
780	C₄²	—	2	2	no	—
781	C₄²	—	2	2	no	—
782	C₄²	—	2	2	no	—
783	C₄²	C₂×D₄×C₅₃	1	2	yes	—
784	C₄²	C₂×D₄×C₅₃	1	2	yes	—
785	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
786	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
787	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
788	C₄²	C₂⁴×C₅₃	1	2	yes	—
789	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
790	C₂³	—	2	1	no	—
791	C₂³	—	2	1	no	—
792	C₂³	—	2	2	no	—
793	C₂³	—	4	2	no	—
794	C₂³	—	4	2	no	—
795	C₂³	—	8	1	no	—
796	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
797	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
798	Q₈	—	2	2	no	—
799	Q₈	—	2	2	no	—
800	Q₈	—	2	2	no	—
801	Q₈	—	4	8	no	—
802	Q₈	—	4	2	no	—
803	Q₈	—	4	4	no	—
804	Q₈	—	4	4	no	—
805	Q₈	—	8	1	no	—
806	D₄	—	2	4	no	—
807	D₄	—	2	2	no	—
808	D₄	—	2	2	no	—
809	D₄	—	2	1	no	—
810	D₄	—	2	1	no	—
811	D₄	—	2	1	no	—
812	D₄	—	2	2	no	—
813	D₄	—	4	2	no	—
814	D₄	—	4	2	no	—
815	D₄	—	4	4	no	—
816	D₄	—	4	8	no	—
817	D₄	—	4	4	no	—
818	D₄	—	4	2	no	—
819	D₄	—	8	2	no	—
820	D₄	2₋¹⁺⁴×C₅₃	1	2	yes	complemented
821	C₂×C₄	—	2	4	no	—
822	C₂×C₄	—	2	1	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	1	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	1	no	—
827	C₂×C₄	—	2	2	no	—
828	C₂×C₄	—	2	2	no	—
829	C₂×C₄	—	2	1	no	—
830	C₂×C₄	—	2	4	no	—
831	C₂×C₄	—	2	2	no	—
832	C₂×C₄	—	4	2	no	—
833	C₂×C₄	—	4	1	no	—
834	C₂×C₄	—	4	4	no	—
835	C₂×C₄	—	4	8	no	—
836	C₂×C₄	—	4	8	no	—
837	C₂×C₄	—	4	2	no	—
838	C₂×C₄	—	4	4	no	—
839	C₂×C₄	—	4	2	no	—
840	C₂×C₄	—	4	2	no	—
841	C₂×C₄	—	4	2	no	—
842	C₂×C₄	—	4	2	no	—
843	C₂×C₄	—	4	4	no	—
844	C₂×C₄	—	8	2	no	—
845	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
846	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
847	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
848	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
849	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
850	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
851	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
852	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
853	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
854	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
855	C₈	—	2	2	no	—
856	C₈	—	2	4	no	—
857	C₈	—	2	2	no	—
858	C₂²	—	2	1	no	—
859	C₂²	—	2	1	no	—
860	C₂²	—	2	2	no	—
861	C₂²	—	2	2	no	—
862	C₂²	—	2	2	no	—
863	C₂²	—	2	2	no	—
864	C₂²	—	4	1	no	—
865	C₂²	—	4	1	no	—
866	C₂²	—	4	1	no	—
867	C₂²	—	4	2	no	—
868	C₂²	—	4	2	no	—
869	C₂²	—	4	1	no	—
870	C₂²	—	4	2	no	—
871	C₂²	—	4	1	no	—
872	C₂²	—	8	4	no	—
873	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
874	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
875	C₄	—	2	2	no	—
876	C₄	—	2	1	no	—
877	C₄	—	2	2	no	—
878	C₄	—	2	1	no	—
879	C₄	—	2	2	no	—
880	C₄	—	2	1	no	—
881	C₄	—	4	2	no	—
882	C₄	—	4	4	no	—
883	C₄	—	4	2	no	—
884	C₄	—	4	1	no	—
885	C₄	—	4	1	no	—
886	C₄	—	4	2	no	—
887	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
888	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
889	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
890	C₄	C₅₃×#585	1	1	yes	characteristic
891	C₂	—	2	2	no	—
892	C₂	—	4	2	no	—
893	C₂	—	4	2	no	—
894	C₂	—	8	1	no	—
895	C₂	—	8	2	no	—
896	C₂	C₅₃×#3489	1	1	yes	characteristic
897	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
898	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
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×#3499	C₂	semidirect
2. C₅₃×#3475	C₂	semidirect
3. C₅₃×#3384	C₂	semidirect
4. C₅₃×#3367	C₂	semidirect
5. C₅₃×#3354	C₂	semidirect
6. C₅₃×#3343	C₂	semidirect
7. C₅₃×#3330	C₂	semidirect
8. C₅₃×#3306	C₂	semidirect
9. C₅₃×#3251	C₂	semidirect
10. C₅₃×#3246	C₂	semidirect
11. C₅₃×#3239	C₂	semidirect
12. C₅₃×#3229	C₂	semidirect
13. C₅₃×#3115	C₂	semidirect
14. C₅₃×#3100	C₂	semidirect
15. C₅₃×#3079	C₂	semidirect
16. C₂×C₅₃×#474	C₂	semidirect
17. C₅₃×#2989	C₂	semidirect
27. D₄×Q₈×C₅₃	C₂²	semidirect
32. C₅₃×#547	C₂²	semidirect
38. C₅₃×#532	C₂²	semidirect
40. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
48. C₅₃×#501	C₂²	semidirect
49. C₅₃×#498	C₂²	semidirect
51. C₅₃×#487	C₂²	semidirect
52. C₅₃×#484	C₂²	semidirect
54. C₅₃×#482	C₂²	semidirect
57. C₅₃×#478	C₂²	semidirect
58. C₅₃×#477	C₂²	semidirect
60. C₅₃×#476	C₂²	semidirect
61. C₅₃×#475	C₂²	semidirect
62. C₅₃×#475	C₂²	semidirect
63. C₅₃×#475	C₂²	semidirect
71. C₅₃×#468	C₂²	semidirect
74. C₅₃×#461	C₂²	semidirect
75. C₅₃×#460	C₂²	semidirect
80. C₅₃×#446	C₂²	semidirect
82. C₅₃×#440	C₂²	semidirect
85. C₅₃×#436	C₂²	semidirect
87. C₅₃×#432	C₂²	semidirect
89. C₅₃×#428	C₂²	semidirect
91. C₅₃×#423	C₂²	semidirect
370. D₄×C₅₃	2₋¹⁺⁴	semidirect
408. #62107	C₅₃	direct (N × H)
442. #3499	C₂×C₅₃	semidirect
443. #3475	C₂×C₅₃	semidirect
444. #3384	C₂×C₅₃	semidirect
445. #3367	C₂×C₅₃	semidirect
446. #3354	C₂×C₅₃	semidirect
447. #3343	C₂×C₅₃	semidirect
448. #3330	C₂×C₅₃	semidirect
449. #3306	C₂×C₅₃	semidirect
450. #3251	C₂×C₅₃	semidirect
451. #3246	C₂×C₅₃	semidirect
452. #3239	C₂×C₅₃	semidirect
453. #3229	C₂×C₅₃	semidirect
454. #3115	C₂×C₅₃	semidirect
455. #3100	C₂×C₅₃	semidirect
456. #3079	C₂×C₅₃	semidirect
457. C₂×#474	C₂×C₅₃	semidirect
458. #2989	C₂×C₅₃	semidirect
476. D₄×Q₈	C₂²×C₅₃	semidirect
481. #547	C₂²×C₅₃	semidirect
487. #532	C₂²×C₅₃	semidirect
489. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
497. #501	C₂²×C₅₃	semidirect
498. #498	C₂²×C₅₃	semidirect
500. #487	C₂²×C₅₃	semidirect
501. #484	C₂²×C₅₃	semidirect
503. #482	C₂²×C₅₃	semidirect
506. #478	C₂²×C₅₃	semidirect
507. #477	C₂²×C₅₃	semidirect
509. #476	C₂²×C₅₃	semidirect
510. #475	C₂²×C₅₃	semidirect
511. #475	C₂²×C₅₃	semidirect
512. #475	C₂²×C₅₃	semidirect
520. #468	C₂²×C₅₃	semidirect
523. #461	C₂²×C₅₃	semidirect
524. #460	C₂²×C₅₃	semidirect
529. #446	C₂²×C₅₃	semidirect
531. #440	C₂²×C₅₃	semidirect
534. #436	C₂²×C₅₃	semidirect
536. #432	C₂²×C₅₃	semidirect
538. #428	C₂²×C₅₃	semidirect
540. #423	C₂²×C₅₃	semidirect
544. C₅₃	#62107	direct (N × H)
820. D₄	2₋¹⁺⁴×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3499	2
C₅₃×#3475	2
C₅₃×#3384	2
C₅₃×#3367	2
C₅₃×#3354	2
C₅₃×#3343	2
C₅₃×#3330	2
C₅₃×#3306	2
C₅₃×#3251	2
C₅₃×#3246	2
C₅₃×#3239	2
C₅₃×#3229	2
C₅₃×#3115	2
C₅₃×#3100	2
C₅₃×#3079	2
C₂×C₅₃×#474	2
C₅₃×#2989	2
#62107	53

Subgroups — C53×#62107 #27269619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62107 #27269619