Subgroups — C₅₃×#62109 #27269621

Order 13568.

1974 conjugacy classes of subgroups.

The subgroup lattice has 916 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3386	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3368	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3353	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3343	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3324	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3310	C₂	1	4	yes	maximal, complemented
9	C₅₃×#3254	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3235	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#3110	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#3077	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	C₂²×#41×C₅₃	—	2	1	no	—
19	C₅₃×#557	—	2	1	no	—
20	C₅₃×#557	C₂²	1	2	yes	complemented
21	C₅₃×#556	—	2	1	no	—
22	C₅₃×#556	C₂²	1	2	yes	complemented
23	C₅₃×#552	—	2	2	no	—
24	C₅₃×#552	C₂²	1	4	yes	complemented
25	C₅₃×#550	C₂²	1	1	yes	characteristic
26	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#548	—	2	1	no	—
28	C₅₃×#547	C₂²	1	2	yes	—
29	C₅₃×#547	C₂²	1	1	yes	characteristic
30	C₅₃×#546	—	2	2	no	—
31	C₅₃×#546	—	2	1	no	—
32	D₄²×C₅₃	C₂²	1	2	yes	—
33	D₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#541	—	2	2	no	—
35	C₅₃×#536	—	2	1	no	—
36	C₅₃×#533	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#533	C₂²	1	2	yes	complemented
38	C₅₃×#533	C₂²	1	2	yes	complemented
39	C₅₃×#532	C₂²	1	1	yes	characteristic
40	C₅₃×#532	C₂²	1	2	yes	—
41	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#525	—	2	1	no	—
43	C₅₃×#525	—	2	2	no	—
44	C₅₃×#519	—	2	1	no	—
45	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
48	C₅₃×#501	C₂²	1	4	yes	complemented
49	C₅₃×#500	C₂²	1	2	yes	complemented
50	C₅₃×#490	C₂²	1	4	yes	complemented
51	C₅₃×#486	C₂²	1	2	yes	complemented
52	C₅₃×#484	C₂²	1	4	yes	complemented
53	C₅₃×#483	C₂²	1	2	yes	complemented
54	C₅₃×#482	C₂²	1	4	yes	complemented
55	C₅₃×#482	C₂²	1	4	yes	complemented
56	C₅₃×#480	C₂²	1	4	yes	complemented
57	C₅₃×#478	C₂²	1	4	yes	—
58	C₅₃×#478	C₂²	1	4	yes	complemented
59	C₅₃×#477	C₂²	1	4	yes	complemented
60	C₅₃×#476	C₂²	1	1	yes	characteristic
61	C₅₃×#476	C₂²	1	1	yes	characteristic, complemented
62	C₅₃×#476	C₂²	1	2	yes	complemented
63	C₅₃×#475	C₂²	1	2	yes	complemented
64	C₅₃×#474	C₂²	1	2	yes	—
65	C₅₃×#474	C₂²	1	2	yes	—
66	C₅₃×#474	C₂²	1	2	yes	complemented
67	C₅₃×#474	C₂²	1	2	yes	complemented
68	C₅₃×#474	C₂²	1	2	yes	complemented
69	C₅₃×#474	C₂²	1	2	yes	complemented
70	C₅₃×#468	C₂²	1	4	yes	complemented
71	C₅₃×#468	C₂²	1	4	yes	complemented
72	C₅₃×#465	C₂²	1	4	yes	complemented
73	C₅₃×#461	C₂²	1	1	yes	characteristic
74	C₅₃×#461	C₂²	1	2	yes	complemented
75	C₅₃×#461	C₂²	1	1	yes	characteristic, complemented
76	C₅₃×#460	C₂²	1	2	yes	complemented
77	C₅₃×#460	C₂²	1	4	yes	complemented
78	C₅₃×#450	C₂²	1	2	yes	—
79	C₅₃×#450	C₂²	1	4	yes	—
80	C₅₃×#446	C₂²	1	2	yes	complemented
81	C₅₃×#440	C₂²	1	4	yes	—
82	C₅₃×#440	C₂²	1	4	yes	complemented
83	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
84	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
85	C₅₃×#436	C₂²	1	2	yes	complemented
86	C₅₃×#435	C₂²	1	4	yes	complemented
87	C₅₃×#432	C₂²	1	2	yes	complemented
88	C₅₃×#431	C₂²	1	1	yes	characteristic, complemented
89	C₅₃×#428	C₂²	1	4	yes	complemented
90	C₂×#107×C₅₃	C₂²	1	2	yes	complemented
91	C₅₃×#423	C₂²	1	2	yes	complemented
92	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
93	C₅₃×#418	C₂²	1	4	yes	—
94	C₂×#102×C₅₃	C₂²	1	2	yes	—
95	C₂×#41×C₅₃	—	2	1	no	—
96	C₂×#41×C₅₃	—	2	2	no	—
97	C₂×#41×C₅₃	—	2	1	no	—
98	C₂×#41×C₅₃	—	2	2	no	—
99	C₂×#41×C₅₃	—	2	1	no	—
100	C₂×#41×C₅₃	—	4	8	no	—
101	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
102	C₂²×Q₈×C₅₃	—	2	1	no	—
103	C₂²×Q₈×C₅₃	—	2	1	no	—
104	C₂²×D₄×C₅₃	—	2	2	no	—
105	C₂²×D₄×C₅₃	—	2	1	no	—
106	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
108	C₂³×C₄×C₅₃	—	2	2	no	—
109	C₂³×C₄×C₅₃	—	2	1	no	—
110	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
111	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
112	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
113	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
114	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
115	C₄⋊Q₈×C₅₃	—	2	1	no	—
116	C₄⋊Q₈×C₅₃	—	2	1	no	—
117	C₄⋊Q₈×C₅₃	—	2	2	no	—
118	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
119	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
120	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
121	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
122	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
123	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
124	#127×C₅₃	C₂³	1	1	yes	characteristic
125	#127×C₅₃	C₂³	1	2	yes	—
126	#126×C₅₃	—	2	4	no	—
127	#126×C₅₃	—	2	4	no	—
128	#125×C₅₃	—	2	4	no	—
129	#125×C₅₃	—	2	4	no	—
130	#125×C₅₃	—	2	1	no	—
131	#125×C₅₃	—	2	2	no	—
132	#125×C₅₃	C₂³	1	4	yes	—
133	#124×C₅₃	—	2	2	no	—
134	#124×C₅₃	—	2	1	no	—
135	#124×C₅₃	—	2	2	no	—
136	#123×C₅₃	—	2	2	no	—
137	#123×C₅₃	—	2	2	no	—
138	#123×C₅₃	—	2	2	no	—
139	#123×C₅₃	—	2	4	no	—
140	C₂²⋊Q₈×C₅₃	—	2	1	no	—
141	C₂²⋊Q₈×C₅₃	—	2	2	no	—
142	C₂²⋊Q₈×C₅₃	—	2	1	no	—
143	C₂²⋊Q₈×C₅₃	—	2	2	no	—
144	C₂²⋊Q₈×C₅₃	—	2	2	no	—
145	C₂²⋊Q₈×C₅₃	—	2	2	no	—
146	C₂²⋊Q₈×C₅₃	—	2	1	no	—
147	C₂²⋊Q₈×C₅₃	—	2	2	no	—
148	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
149	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
150	#121×C₅₃	—	2	1	no	—
151	#121×C₅₃	—	2	2	no	—
152	#121×C₅₃	—	2	1	no	—
153	#121×C₅₃	—	2	2	no	—
154	#121×C₅₃	—	2	1	no	—
155	#121×C₅₃	—	2	2	no	—
156	#121×C₅₃	C₂³	1	2	yes	—
157	#121×C₅₃	C₂³	1	2	yes	—
158	#121×C₅₃	C₂³	1	2	yes	—
159	#120×C₅₃	—	2	2	no	—
160	#120×C₅₃	—	2	2	no	—
161	#120×C₅₃	—	2	2	no	—
162	C₄×Q₈×C₅₃	—	2	2	no	—
163	C₄×Q₈×C₅₃	—	2	2	no	—
164	C₄×Q₈×C₅₃	—	2	2	no	—
165	C₄×Q₈×C₅₃	—	2	2	no	—
166	C₄×Q₈×C₅₃	—	2	2	no	—
167	C₄×Q₈×C₅₃	—	2	1	no	—
168	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
169	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
170	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
171	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
172	C₄×D₄×C₅₃	—	2	2	no	—
173	C₄×D₄×C₅₃	—	2	1	no	—
174	C₄×D₄×C₅₃	—	2	2	no	—
175	C₄×D₄×C₅₃	—	2	1	no	—
176	C₄×D₄×C₅₃	—	2	2	no	—
177	C₄×D₄×C₅₃	—	2	1	no	—
178	C₄×D₄×C₅₃	D₄	1	2	yes	—
179	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
180	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
181	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
182	C₄×D₄×C₅₃	C₂³	1	2	yes	—
183	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
184	C₄×D₄×C₅₃	C₂³	1	2	yes	—
185	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
186	C₄×D₄×C₅₃	C₂³	1	2	yes	—
187	C₄×D₄×C₅₃	C₂³	1	2	yes	—
188	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
189	#117×C₅₃	—	2	4	no	—
190	#117×C₅₃	—	2	1	no	—
191	#117×C₅₃	—	2	2	no	—
192	#117×C₅₃	D₄	1	1	yes	characteristic
193	#117×C₅₃	D₄	1	2	yes	—
194	#117×C₅₃	C₂³	1	4	yes	—
195	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
196	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
197	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
198	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
199	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
200	C₂×C₂²⋊C₄×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	4	yes	—
204	#107×C₅₃	C₂³	1	4	yes	—
205	#107×C₅₃	C₂³	1	4	yes	—
206	#107×C₅₃	C₂³	1	4	yes	—
207	#106×C₅₃	C₂³	1	2	yes	—
208	#106×C₅₃	C₂³	1	4	yes	—
209	#106×C₅₃	C₂³	1	2	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	2	yes	—
216	#103×C₅₃	C₂³	1	2	yes	—
217	#103×C₅₃	C₂³	1	4	yes	—
218	#102×C₅₃	C₂³	1	4	yes	—
219	#102×C₅₃	C₂³	1	4	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	4	yes	—
224	#102×C₅₃	C₂³	1	2	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	2	no	—
230	C₂⁴×C₅₃	—	2	1	no	—
231	#41×C₅₃	—	2	4	no	—
232	#41×C₅₃	—	4	8	no	—
233	#41×C₅₃	—	4	8	no	—
234	#41×C₅₃	—	4	4	no	—
235	#41×C₅₃	—	4	4	no	—
236	#41×C₅₃	—	4	4	no	—
237	C₂×Q₈×C₅₃	—	2	1	no	—
238	C₂×Q₈×C₅₃	—	2	2	no	—
239	C₂×Q₈×C₅₃	—	2	2	no	—
240	C₂×Q₈×C₅₃	—	2	2	no	—
241	C₂×Q₈×C₅₃	—	2	1	no	—
242	C₂×Q₈×C₅₃	—	4	4	no	—
243	C₂×Q₈×C₅₃	—	4	4	no	—
244	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
245	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
246	C₂×D₄×C₅₃	—	2	2	no	—
247	C₂×D₄×C₅₃	—	2	4	no	—
248	C₂×D₄×C₅₃	—	2	2	no	—
249	C₂×D₄×C₅₃	—	2	2	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	2	no	—
254	C₂×D₄×C₅₃	—	2	1	no	—
255	C₂×D₄×C₅₃	—	2	2	no	—
256	C₂×D₄×C₅₃	—	2	1	no	—
257	C₂×D₄×C₅₃	—	4	1	no	—
258	C₂×D₄×C₅₃	—	4	2	no	—
259	C₂×D₄×C₅₃	—	4	8	no	—
260	C₂×D₄×C₅₃	—	4	4	no	—
261	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
262	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
263	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
264	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
265	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₅₃	—	2	2	no	—
267	C₂²×C₄×C₅₃	—	2	1	no	—
268	C₂²×C₄×C₅₃	—	2	2	no	—
269	C₂²×C₄×C₅₃	—	2	1	no	—
270	C₂²×C₄×C₅₃	—	2	2	no	—
271	C₂²×C₄×C₅₃	—	2	1	no	—
272	C₂²×C₄×C₅₃	—	2	2	no	—
273	C₂²×C₄×C₅₃	—	2	1	no	—
274	C₂²×C₄×C₅₃	—	2	2	no	—
275	C₂²×C₄×C₅₃	—	2	1	no	—
276	C₂²×C₄×C₅₃	—	4	8	no	—
277	C₂²×C₄×C₅₃	—	4	4	no	—
278	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
279	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
280	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
281	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
282	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
284	SD₁₆×C₅₃	—	2	4	no	—
285	SD₁₆×C₅₃	—	4	4	no	—
286	SD₁₆×C₅₃	—	4	2	no	—
287	M₁₆×C₅₃	—	4	4	no	—
288	C₂×C₈×C₅₃	—	4	2	no	—
289	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
290	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
291	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
292	C₄⋊C₄×C₅₃	—	2	2	no	—
293	C₄⋊C₄×C₅₃	—	2	2	no	—
294	C₄⋊C₄×C₅₃	—	2	2	no	—
295	C₄⋊C₄×C₅₃	—	2	1	no	—
296	C₄⋊C₄×C₅₃	—	2	2	no	—
297	C₄⋊C₄×C₅₃	—	2	1	no	—
298	C₄⋊C₄×C₅₃	—	2	1	no	—
299	C₄⋊C₄×C₅₃	—	2	1	no	—
300	C₄⋊C₄×C₅₃	—	2	4	no	—
301	C₄⋊C₄×C₅₃	—	2	4	no	—
302	C₄⋊C₄×C₅₃	—	2	2	no	—
303	C₄⋊C₄×C₅₃	—	2	1	no	—
304	C₄⋊C₄×C₅₃	—	2	4	no	—
305	C₄⋊C₄×C₅₃	—	2	2	no	—
306	C₄⋊C₄×C₅₃	—	2	4	no	—
307	C₄⋊C₄×C₅₃	—	2	2	no	—
308	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
309	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
310	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
311	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
312	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
313	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
314	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
315	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
316	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
317	C₂²⋊C₄×C₅₃	—	2	2	no	—
318	C₂²⋊C₄×C₅₃	—	2	4	no	—
319	C₂²⋊C₄×C₅₃	—	2	4	no	—
320	C₂²⋊C₄×C₅₃	—	2	1	no	—
321	C₂²⋊C₄×C₅₃	—	2	2	no	—
322	C₂²⋊C₄×C₅₃	—	2	1	no	—
323	C₂²⋊C₄×C₅₃	—	2	2	no	—
324	C₂²⋊C₄×C₅₃	—	2	1	no	—
325	C₂²⋊C₄×C₅₃	—	2	1	no	—
326	C₂²⋊C₄×C₅₃	—	2	2	no	—
327	C₂²⋊C₄×C₅₃	—	2	2	no	—
328	C₂²⋊C₄×C₅₃	—	2	2	no	—
329	C₂²⋊C₄×C₅₃	—	2	2	no	—
330	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
331	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
332	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
333	C₄²×C₅₃	—	2	4	no	—
334	C₄²×C₅₃	—	2	4	no	—
335	C₄²×C₅₃	—	2	2	no	—
336	C₄²×C₅₃	C₂×D₄	1	2	yes	—
337	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
338	C₄²×C₅₃	C₂×D₄	1	2	yes	—
339	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
340	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
341	C₄²×C₅₃	C₂⁴	1	2	yes	—
342	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
343	C₂³×C₅₃	—	2	1	no	—
344	C₂³×C₅₃	—	2	2	no	—
345	C₂³×C₅₃	—	2	1	no	—
346	C₂³×C₅₃	—	4	2	no	—
347	C₂³×C₅₃	—	4	4	no	—
348	C₂³×C₅₃	—	4	4	no	—
349	C₂³×C₅₃	—	4	2	no	—
350	C₂³×C₅₃	—	8	1	no	—
351	C₂³×C₅₃	—	8	2	no	—
352	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
353	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
354	Q₈×C₅₃	—	2	2	no	—
355	Q₈×C₅₃	—	2	2	no	—
356	Q₈×C₅₃	—	2	2	no	—
357	Q₈×C₅₃	—	4	2	no	—
358	Q₈×C₅₃	—	4	4	no	—
359	Q₈×C₅₃	—	4	4	no	—
360	Q₈×C₅₃	—	4	4	no	—
361	Q₈×C₅₃	—	4	2	no	—
362	D₄×C₅₃	—	2	4	no	—
363	D₄×C₅₃	—	2	2	no	—
364	D₄×C₅₃	—	2	1	no	—
365	D₄×C₅₃	—	2	2	no	—
366	D₄×C₅₃	—	2	1	no	—
367	D₄×C₅₃	—	2	1	no	—
368	D₄×C₅₃	—	2	2	no	—
369	D₄×C₅₃	—	4	2	no	—
370	D₄×C₅₃	—	4	4	no	—
371	D₄×C₅₃	—	4	4	no	—
372	D₄×C₅₃	—	4	4	no	—
373	D₄×C₅₃	—	4	4	no	—
374	D₄×C₅₃	—	4	4	no	—
375	D₄×C₅₃	—	4	2	no	—
376	D₄×C₅₃	—	8	2	no	—
377	D₄×C₅₃	—	8	1	no	—
378	D₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
379	C₂×C₄×C₅₃	—	2	4	no	—
380	C₂×C₄×C₅₃	—	2	1	no	—
381	C₂×C₄×C₅₃	—	2	2	no	—
382	C₂×C₄×C₅₃	—	2	1	no	—
383	C₂×C₄×C₅₃	—	2	2	no	—
384	C₂×C₄×C₅₃	—	2	1	no	—
385	C₂×C₄×C₅₃	—	2	2	no	—
386	C₂×C₄×C₅₃	—	2	1	no	—
387	C₂×C₄×C₅₃	—	2	4	no	—
388	C₂×C₄×C₅₃	—	2	2	no	—
389	C₂×C₄×C₅₃	—	2	2	no	—
390	C₂×C₄×C₅₃	—	4	1	no	—
391	C₂×C₄×C₅₃	—	4	2	no	—
392	C₂×C₄×C₅₃	—	4	4	no	—
393	C₂×C₄×C₅₃	—	4	4	no	—
394	C₂×C₄×C₅₃	—	4	4	no	—
395	C₂×C₄×C₅₃	—	4	2	no	—
396	C₂×C₄×C₅₃	—	4	4	no	—
397	C₂×C₄×C₅₃	—	4	2	no	—
398	C₂×C₄×C₅₃	—	4	2	no	—
399	C₂×C₄×C₅₃	—	4	2	no	—
400	C₂×C₄×C₅₃	—	4	4	no	—
401	C₂×C₄×C₅₃	—	4	2	no	—
402	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
403	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
404	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
405	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
406	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
407	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
408	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
412	C₈×C₅₃	—	2	2	no	—
413	C₈×C₅₃	—	2	4	no	—
414	C₈×C₅₃	—	2	2	no	—
415	#62109	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
416	C₂²×C₅₃	—	2	1	no	—
417	C₂²×C₅₃	—	2	1	no	—
418	C₂²×C₅₃	—	2	2	no	—
419	C₂²×C₅₃	—	2	2	no	—
420	C₂²×C₅₃	—	2	2	no	—
421	C₂²×C₅₃	—	2	2	no	—
422	C₂²×C₅₃	—	4	1	no	—
423	C₂²×C₅₃	—	4	1	no	—
424	C₂²×C₅₃	—	4	1	no	—
425	C₂²×C₅₃	—	4	2	no	—
426	C₂²×C₅₃	—	4	1	no	—
427	C₂²×C₅₃	—	4	2	no	—
428	C₂²×C₅₃	—	4	2	no	—
429	C₂²×C₅₃	—	4	1	no	—
430	C₂²×C₅₃	—	8	4	no	—
431	C₂²×C₅₃	—	8	8	no	—
432	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
433	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
434	C₄×C₅₃	—	2	1	no	—
435	C₄×C₅₃	—	2	2	no	—
436	C₄×C₅₃	—	2	1	no	—
437	C₄×C₅₃	—	2	1	no	—
438	C₄×C₅₃	—	2	2	no	—
439	C₄×C₅₃	—	2	2	no	—
440	C₄×C₅₃	—	4	2	no	—
441	C₄×C₅₃	—	4	2	no	—
442	C₄×C₅₃	—	4	1	no	—
443	C₄×C₅₃	—	4	2	no	—
444	C₄×C₅₃	—	4	1	no	—
445	C₄×C₅₃	—	4	4	no	—
446	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
447	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
448	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
449	C₄×C₅₃	#585	1	1	yes	characteristic
450	#3509	C₂×C₅₃	1	1	yes	characteristic, complemented
451	D₄×#41	C₂×C₅₃	1	1	yes	characteristic
452	#3386	C₂×C₅₃	1	2	yes	complemented
453	#3368	C₂×C₅₃	1	2	yes	complemented
454	#3353	C₂×C₅₃	1	2	yes	complemented
455	#3343	C₂×C₅₃	1	4	yes	complemented
456	#3324	C₂×C₅₃	1	2	yes	complemented
457	#3310	C₂×C₅₃	1	4	yes	complemented
458	#3254	C₂×C₅₃	1	2	yes	complemented
459	#3246	C₂×C₅₃	1	1	yes	characteristic, complemented
460	#3235	C₂×C₅₃	1	2	yes	complemented
461	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
462	#3110	C₂×C₅₃	1	2	yes	complemented
463	#3101	C₂×C₅₃	1	1	yes	characteristic, complemented
464	#3077	C₂×C₅₃	1	2	yes	complemented
465	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
466	#2989	C₂×C₅₃	1	1	yes	characteristic, complemented
467	C₂×C₅₃	—	2	2	no	—
468	C₂×C₅₃	—	4	2	no	—
469	C₂×C₅₃	—	4	2	no	—
470	C₂×C₅₃	—	8	1	no	—
471	C₂×C₅₃	—	8	2	no	—
472	C₂×C₅₃	#3489	1	1	yes	characteristic
473	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
474	C₂×C₅₃	#3586	1	1	yes	characteristic
475	C₂²×#41	—	2	1	no	—
476	#557	—	2	1	no	—
477	#557	C₂²×C₅₃	1	2	yes	complemented
478	#556	—	2	1	no	—
479	#556	C₂²×C₅₃	1	2	yes	complemented
480	#552	—	2	2	no	—
481	#552	C₂²×C₅₃	1	4	yes	complemented
482	#550	C₂²×C₅₃	1	1	yes	characteristic
483	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
484	#548	—	2	1	no	—
485	#547	C₂²×C₅₃	1	2	yes	—
486	#547	C₂²×C₅₃	1	1	yes	characteristic
487	#546	—	2	2	no	—
488	#546	—	2	1	no	—
489	D₄²	C₂²×C₅₃	1	2	yes	—
490	D₄²	C₂²×C₅₃	1	1	yes	characteristic
491	#541	—	2	2	no	—
492	#536	—	2	1	no	—
493	#533	C₂²×C₅₃	1	1	yes	characteristic, complemented
494	#533	C₂²×C₅₃	1	2	yes	complemented
495	#533	C₂²×C₅₃	1	2	yes	complemented
496	#532	C₂²×C₅₃	1	1	yes	characteristic
497	#532	C₂²×C₅₃	1	2	yes	—
498	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
499	#525	—	2	1	no	—
500	#525	—	2	2	no	—
501	#519	—	2	1	no	—
502	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
503	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
504	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
505	#501	C₂²×C₅₃	1	4	yes	complemented
506	#500	C₂²×C₅₃	1	2	yes	complemented
507	#490	C₂²×C₅₃	1	4	yes	complemented
508	#486	C₂²×C₅₃	1	2	yes	complemented
509	#484	C₂²×C₅₃	1	4	yes	complemented
510	#483	C₂²×C₅₃	1	2	yes	complemented
511	#482	C₂²×C₅₃	1	4	yes	complemented
512	#482	C₂²×C₅₃	1	4	yes	complemented
513	#480	C₂²×C₅₃	1	4	yes	complemented
514	#478	C₂²×C₅₃	1	4	yes	—
515	#478	C₂²×C₅₃	1	4	yes	complemented
516	#477	C₂²×C₅₃	1	4	yes	complemented
517	#476	C₂²×C₅₃	1	1	yes	characteristic
518	#476	C₂²×C₅₃	1	2	yes	complemented
519	#476	C₂²×C₅₃	1	1	yes	characteristic, complemented
520	#475	C₂²×C₅₃	1	2	yes	complemented
521	#474	C₂²×C₅₃	1	2	yes	—
522	#474	C₂²×C₅₃	1	2	yes	—
523	#474	C₂²×C₅₃	1	2	yes	complemented
524	#474	C₂²×C₅₃	1	2	yes	complemented
525	#474	C₂²×C₅₃	1	2	yes	complemented
526	#474	C₂²×C₅₃	1	2	yes	complemented
527	#468	C₂²×C₅₃	1	4	yes	complemented
528	#468	C₂²×C₅₃	1	4	yes	complemented
529	#465	C₂²×C₅₃	1	4	yes	complemented
530	#461	C₂²×C₅₃	1	1	yes	characteristic
531	#461	C₂²×C₅₃	1	1	yes	characteristic, complemented
532	#461	C₂²×C₅₃	1	2	yes	complemented
533	#460	C₂²×C₅₃	1	2	yes	complemented
534	#460	C₂²×C₅₃	1	4	yes	complemented
535	#450	C₂²×C₅₃	1	2	yes	—
536	#450	C₂²×C₅₃	1	4	yes	—
537	#446	C₂²×C₅₃	1	2	yes	complemented
538	#440	C₂²×C₅₃	1	4	yes	—
539	#440	C₂²×C₅₃	1	4	yes	complemented
540	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
541	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
542	#436	C₂²×C₅₃	1	2	yes	complemented
543	#435	C₂²×C₅₃	1	4	yes	complemented
544	#432	C₂²×C₅₃	1	2	yes	complemented
545	#431	C₂²×C₅₃	1	1	yes	characteristic, complemented
546	#428	C₂²×C₅₃	1	4	yes	complemented
547	C₂×#107	C₂²×C₅₃	1	2	yes	complemented
548	#423	C₂²×C₅₃	1	2	yes	complemented
549	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
550	#418	C₂²×C₅₃	1	4	yes	—
551	C₂×#102	C₂²×C₅₃	1	2	yes	—
552	C₅₃	#62109	1	1	yes	characteristic, direct product factor, Sylow
553	C₂×#41	—	2	1	no	—
554	C₂×#41	—	2	2	no	—
555	C₂×#41	—	2	1	no	—
556	C₂×#41	—	2	2	no	—
557	C₂×#41	—	2	1	no	—
558	C₂×#41	—	4	8	no	—
559	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
560	C₂²×Q₈	—	2	1	no	—
561	C₂²×Q₈	—	2	1	no	—
562	C₂²×D₄	—	2	2	no	—
563	C₂²×D₄	—	2	1	no	—
564	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
565	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
566	C₂³×C₄	—	2	2	no	—
567	C₂³×C₄	—	2	1	no	—
568	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
569	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
570	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
571	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
572	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
573	C₄⋊Q₈	—	2	2	no	—
574	C₄⋊Q₈	—	2	1	no	—
575	C₄⋊Q₈	—	2	1	no	—
576	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
577	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
578	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
579	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
580	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
581	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
582	#127	C₂³×C₅₃	1	1	yes	characteristic
583	#127	C₂³×C₅₃	1	2	yes	—
584	#126	—	2	4	no	—
585	#126	—	2	4	no	—
586	#125	—	2	4	no	—
587	#125	—	2	4	no	—
588	#125	—	2	1	no	—
589	#125	—	2	2	no	—
590	#125	C₂³×C₅₃	1	4	yes	—
591	#124	—	2	2	no	—
592	#124	—	2	2	no	—
593	#124	—	2	1	no	—
594	#123	—	2	4	no	—
595	#123	—	2	2	no	—
596	#123	—	2	2	no	—
597	#123	—	2	2	no	—
598	C₂²⋊Q₈	—	2	2	no	—
599	C₂²⋊Q₈	—	2	2	no	—
600	C₂²⋊Q₈	—	2	1	no	—
601	C₂²⋊Q₈	—	2	2	no	—
602	C₂²⋊Q₈	—	2	1	no	—
603	C₂²⋊Q₈	—	2	2	no	—
604	C₂²⋊Q₈	—	2	1	no	—
605	C₂²⋊Q₈	—	2	2	no	—
606	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
607	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
608	#121	—	2	1	no	—
609	#121	—	2	2	no	—
610	#121	—	2	1	no	—
611	#121	—	2	2	no	—
612	#121	—	2	1	no	—
613	#121	—	2	2	no	—
614	#121	C₂³×C₅₃	1	2	yes	—
615	#121	C₂³×C₅₃	1	2	yes	—
616	#121	C₂³×C₅₃	1	2	yes	—
617	#120	—	2	2	no	—
618	#120	—	2	2	no	—
619	#120	—	2	2	no	—
620	C₄×Q₈	—	2	2	no	—
621	C₄×Q₈	—	2	2	no	—
622	C₄×Q₈	—	2	2	no	—
623	C₄×Q₈	—	2	2	no	—
624	C₄×Q₈	—	2	2	no	—
625	C₄×Q₈	—	2	1	no	—
626	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
627	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
628	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
629	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
630	C₄×D₄	—	2	2	no	—
631	C₄×D₄	—	2	1	no	—
632	C₄×D₄	—	2	2	no	—
633	C₄×D₄	—	2	1	no	—
634	C₄×D₄	—	2	2	no	—
635	C₄×D₄	—	2	1	no	—
636	C₄×D₄	D₄×C₅₃	1	2	yes	—
637	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
638	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
639	C₄×D₄	C₂³×C₅₃	1	2	yes	—
640	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
641	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
642	C₄×D₄	C₂³×C₅₃	1	2	yes	—
643	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
644	C₄×D₄	C₂³×C₅₃	1	2	yes	—
645	C₄×D₄	C₂³×C₅₃	1	2	yes	—
646	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
647	#117	—	2	1	no	—
648	#117	—	2	2	no	—
649	#117	—	2	4	no	—
650	#117	D₄×C₅₃	1	1	yes	characteristic
651	#117	D₄×C₅₃	1	2	yes	—
652	#117	C₂³×C₅₃	1	4	yes	—
653	C₂×C₄⋊C₄	—	2	1	no	—
654	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
655	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
656	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
657	C₂×C₂²⋊C₄	—	2	1	no	—
658	C₂×C₂²⋊C₄	—	2	2	no	—
659	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
660	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
661	#107	C₂³×C₅₃	1	4	yes	—
662	#107	C₂³×C₅₃	1	4	yes	—
663	#107	C₂³×C₅₃	1	4	yes	—
664	#107	C₂³×C₅₃	1	4	yes	—
665	#106	C₂³×C₅₃	1	2	yes	—
666	#106	C₂³×C₅₃	1	4	yes	—
667	#106	C₂³×C₅₃	1	2	yes	—
668	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
669	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
670	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
671	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
672	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
673	#103	C₂³×C₅₃	1	2	yes	—
674	#103	C₂³×C₅₃	1	2	yes	—
675	#103	C₂³×C₅₃	1	4	yes	—
676	#102	C₂³×C₅₃	1	4	yes	—
677	#102	C₂³×C₅₃	1	4	yes	—
678	#102	C₂³×C₅₃	1	2	yes	—
679	#102	C₂³×C₅₃	1	4	yes	—
680	#102	C₂³×C₅₃	1	4	yes	—
681	#102	C₂³×C₅₃	1	2	yes	—
682	#102	C₂³×C₅₃	1	4	yes	—
683	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
684	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
685	#97	C₂³×C₅₃	1	4	yes	—
686	C₄×C₈	C₂³×C₅₃	1	2	yes	—
687	C₂⁴	—	2	2	no	—
688	C₂⁴	—	2	1	no	—
689	#41	—	2	4	no	—
690	#41	—	4	8	no	—
691	#41	—	4	8	no	—
692	#41	—	4	4	no	—
693	#41	—	4	4	no	—
694	#41	—	4	4	no	—
695	C₂×Q₈	—	2	1	no	—
696	C₂×Q₈	—	2	2	no	—
697	C₂×Q₈	—	2	2	no	—
698	C₂×Q₈	—	2	2	no	—
699	C₂×Q₈	—	2	1	no	—
700	C₂×Q₈	—	4	4	no	—
701	C₂×Q₈	—	4	4	no	—
702	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
703	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
704	C₂×D₄	—	2	2	no	—
705	C₂×D₄	—	2	4	no	—
706	C₂×D₄	—	2	2	no	—
707	C₂×D₄	—	2	2	no	—
708	C₂×D₄	—	2	2	no	—
709	C₂×D₄	—	2	2	no	—
710	C₂×D₄	—	2	1	no	—
711	C₂×D₄	—	2	2	no	—
712	C₂×D₄	—	2	1	no	—
713	C₂×D₄	—	2	2	no	—
714	C₂×D₄	—	2	1	no	—
715	C₂×D₄	—	4	1	no	—
716	C₂×D₄	—	4	2	no	—
717	C₂×D₄	—	4	8	no	—
718	C₂×D₄	—	4	4	no	—
719	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
720	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
721	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
722	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
723	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
724	C₂²×C₄	—	2	2	no	—
725	C₂²×C₄	—	2	1	no	—
726	C₂²×C₄	—	2	2	no	—
727	C₂²×C₄	—	2	1	no	—
728	C₂²×C₄	—	2	2	no	—
729	C₂²×C₄	—	2	1	no	—
730	C₂²×C₄	—	2	2	no	—
731	C₂²×C₄	—	2	1	no	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	1	no	—
734	C₂²×C₄	—	4	8	no	—
735	C₂²×C₄	—	4	4	no	—
736	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
737	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
738	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
739	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
740	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
741	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
742	SD₁₆	—	2	4	no	—
743	SD₁₆	—	4	4	no	—
744	SD₁₆	—	4	2	no	—
745	M₁₆	—	4	4	no	—
746	C₂×C₈	—	4	2	no	—
747	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
748	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
749	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
750	C₄⋊C₄	—	2	2	no	—
751	C₄⋊C₄	—	2	2	no	—
752	C₄⋊C₄	—	2	2	no	—
753	C₄⋊C₄	—	2	1	no	—
754	C₄⋊C₄	—	2	2	no	—
755	C₄⋊C₄	—	2	1	no	—
756	C₄⋊C₄	—	2	1	no	—
757	C₄⋊C₄	—	2	1	no	—
758	C₄⋊C₄	—	2	4	no	—
759	C₄⋊C₄	—	2	4	no	—
760	C₄⋊C₄	—	2	2	no	—
761	C₄⋊C₄	—	2	1	no	—
762	C₄⋊C₄	—	2	4	no	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	4	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
767	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
768	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
769	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
770	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
771	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
772	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
773	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
774	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
775	C₂²⋊C₄	—	2	2	no	—
776	C₂²⋊C₄	—	2	4	no	—
777	C₂²⋊C₄	—	2	4	no	—
778	C₂²⋊C₄	—	2	1	no	—
779	C₂²⋊C₄	—	2	2	no	—
780	C₂²⋊C₄	—	2	1	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	1	no	—
783	C₂²⋊C₄	—	2	1	no	—
784	C₂²⋊C₄	—	2	2	no	—
785	C₂²⋊C₄	—	2	2	no	—
786	C₂²⋊C₄	—	2	2	no	—
787	C₂²⋊C₄	—	2	2	no	—
788	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
789	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
790	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
791	C₄²	—	2	4	no	—
792	C₄²	—	2	4	no	—
793	C₄²	—	2	2	no	—
794	C₄²	C₂×D₄×C₅₃	1	2	yes	—
795	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
796	C₄²	C₂×D₄×C₅₃	1	2	yes	—
797	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
798	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
799	C₄²	C₂⁴×C₅₃	1	2	yes	—
800	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
801	C₂³	—	2	1	no	—
802	C₂³	—	2	2	no	—
803	C₂³	—	2	1	no	—
804	C₂³	—	4	2	no	—
805	C₂³	—	4	4	no	—
806	C₂³	—	4	4	no	—
807	C₂³	—	4	2	no	—
808	C₂³	—	8	1	no	—
809	C₂³	—	8	2	no	—
810	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
811	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
812	Q₈	—	2	2	no	—
813	Q₈	—	2	2	no	—
814	Q₈	—	2	2	no	—
815	Q₈	—	4	2	no	—
816	Q₈	—	4	4	no	—
817	Q₈	—	4	4	no	—
818	Q₈	—	4	4	no	—
819	Q₈	—	4	2	no	—
820	D₄	—	2	4	no	—
821	D₄	—	2	2	no	—
822	D₄	—	2	1	no	—
823	D₄	—	2	2	no	—
824	D₄	—	2	1	no	—
825	D₄	—	2	2	no	—
826	D₄	—	2	1	no	—
827	D₄	—	4	2	no	—
828	D₄	—	4	4	no	—
829	D₄	—	4	4	no	—
830	D₄	—	4	4	no	—
831	D₄	—	4	4	no	—
832	D₄	—	4	4	no	—
833	D₄	—	4	2	no	—
834	D₄	—	8	2	no	—
835	D₄	—	8	1	no	—
836	D₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
837	C₂×C₄	—	2	4	no	—
838	C₂×C₄	—	2	1	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	1	no	—
841	C₂×C₄	—	2	2	no	—
842	C₂×C₄	—	2	2	no	—
843	C₂×C₄	—	2	2	no	—
844	C₂×C₄	—	2	4	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	1	no	—
848	C₂×C₄	—	4	1	no	—
849	C₂×C₄	—	4	2	no	—
850	C₂×C₄	—	4	4	no	—
851	C₂×C₄	—	4	4	no	—
852	C₂×C₄	—	4	4	no	—
853	C₂×C₄	—	4	2	no	—
854	C₂×C₄	—	4	4	no	—
855	C₂×C₄	—	4	2	no	—
856	C₂×C₄	—	4	2	no	—
857	C₂×C₄	—	4	2	no	—
858	C₂×C₄	—	4	4	no	—
859	C₂×C₄	—	4	2	no	—
860	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
861	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
862	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
863	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
864	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
865	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
866	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
867	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
868	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
869	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
870	C₈	—	2	2	no	—
871	C₈	—	2	4	no	—
872	C₈	—	2	2	no	—
873	C₂²	—	2	1	no	—
874	C₂²	—	2	1	no	—
875	C₂²	—	2	2	no	—
876	C₂²	—	2	2	no	—
877	C₂²	—	2	2	no	—
878	C₂²	—	2	2	no	—
879	C₂²	—	4	1	no	—
880	C₂²	—	4	1	no	—
881	C₂²	—	4	1	no	—
882	C₂²	—	4	2	no	—
883	C₂²	—	4	2	no	—
884	C₂²	—	4	1	no	—
885	C₂²	—	4	1	no	—
886	C₂²	—	4	2	no	—
887	C₂²	—	8	4	no	—
888	C₂²	—	8	8	no	—
889	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
890	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
891	C₄	—	2	1	no	—
892	C₄	—	2	2	no	—
893	C₄	—	2	2	no	—
894	C₄	—	2	1	no	—
895	C₄	—	2	2	no	—
896	C₄	—	2	1	no	—
897	C₄	—	4	2	no	—
898	C₄	—	4	1	no	—
899	C₄	—	4	2	no	—
900	C₄	—	4	1	no	—
901	C₄	—	4	4	no	—
902	C₄	—	4	2	no	—
903	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
904	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
905	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
906	C₄	C₅₃×#585	1	1	yes	characteristic
907	C₂	—	2	2	no	—
908	C₂	—	4	2	no	—
909	C₂	—	4	2	no	—
910	C₂	—	8	1	no	—
911	C₂	—	8	2	no	—
912	C₂	C₅₃×#3489	1	1	yes	characteristic
913	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
914	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	D₄×#41×C₅₃	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×#3509	C₂	semidirect
3. C₅₃×#3386	C₂	semidirect
4. C₅₃×#3368	C₂	semidirect
5. C₅₃×#3353	C₂	semidirect
6. C₅₃×#3343	C₂	semidirect
7. C₅₃×#3324	C₂	semidirect
8. C₅₃×#3310	C₂	semidirect
9. C₅₃×#3254	C₂	semidirect
10. C₅₃×#3246	C₂	semidirect
11. C₅₃×#3235	C₂	semidirect
12. C₅₃×#3226	C₂	semidirect
13. C₅₃×#3110	C₂	semidirect
14. C₅₃×#3101	C₂	semidirect
15. C₅₃×#3077	C₂	semidirect
16. C₂×C₅₃×#474	C₂	semidirect
17. C₅₃×#2989	C₂	semidirect
20. C₅₃×#557	C₂²	semidirect
22. C₅₃×#556	C₂²	semidirect
24. C₅₃×#552	C₂²	semidirect
36. C₅₃×#533	C₂²	semidirect
37. C₅₃×#533	C₂²	semidirect
38. C₅₃×#533	C₂²	semidirect
41. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
48. C₅₃×#501	C₂²	semidirect
49. C₅₃×#500	C₂²	semidirect
50. C₅₃×#490	C₂²	semidirect
51. C₅₃×#486	C₂²	semidirect
52. C₅₃×#484	C₂²	semidirect
53. C₅₃×#483	C₂²	semidirect
54. C₅₃×#482	C₂²	semidirect
55. C₅₃×#482	C₂²	semidirect
56. C₅₃×#480	C₂²	semidirect
58. C₅₃×#478	C₂²	semidirect
59. C₅₃×#477	C₂²	semidirect
61. C₅₃×#476	C₂²	semidirect
62. C₅₃×#476	C₂²	semidirect
63. C₅₃×#475	C₂²	semidirect
66. C₅₃×#474	C₂²	semidirect
67. C₅₃×#474	C₂²	semidirect
68. C₅₃×#474	C₂²	semidirect
69. C₅₃×#474	C₂²	semidirect
70. C₅₃×#468	C₂²	semidirect
71. C₅₃×#468	C₂²	semidirect
72. C₅₃×#465	C₂²	semidirect
74. C₅₃×#461	C₂²	semidirect
75. C₅₃×#461	C₂²	semidirect
76. C₅₃×#460	C₂²	semidirect
77. C₅₃×#460	C₂²	semidirect
80. C₅₃×#446	C₂²	semidirect
82. C₅₃×#440	C₂²	semidirect
84. C₄×SD₁₆×C₅₃	C₂²	semidirect
85. C₅₃×#436	C₂²	semidirect
86. C₅₃×#435	C₂²	semidirect
87. C₅₃×#432	C₂²	semidirect
88. C₅₃×#431	C₂²	semidirect
89. C₅₃×#428	C₂²	semidirect
90. C₂×#107×C₅₃	C₂²	semidirect
91. C₅₃×#423	C₂²	semidirect
92. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
415. #62109	C₅₃	direct (N × H)
450. #3509	C₂×C₅₃	semidirect
452. #3386	C₂×C₅₃	semidirect
453. #3368	C₂×C₅₃	semidirect
454. #3353	C₂×C₅₃	semidirect
455. #3343	C₂×C₅₃	semidirect
456. #3324	C₂×C₅₃	semidirect
457. #3310	C₂×C₅₃	semidirect
458. #3254	C₂×C₅₃	semidirect
459. #3246	C₂×C₅₃	semidirect
460. #3235	C₂×C₅₃	semidirect
461. #3226	C₂×C₅₃	semidirect
462. #3110	C₂×C₅₃	semidirect
463. #3101	C₂×C₅₃	semidirect
464. #3077	C₂×C₅₃	semidirect
465. C₂×#474	C₂×C₅₃	semidirect
466. #2989	C₂×C₅₃	semidirect
477. #557	C₂²×C₅₃	semidirect
479. #556	C₂²×C₅₃	semidirect
481. #552	C₂²×C₅₃	semidirect
493. #533	C₂²×C₅₃	semidirect
494. #533	C₂²×C₅₃	semidirect
495. #533	C₂²×C₅₃	semidirect
498. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
505. #501	C₂²×C₅₃	semidirect
506. #500	C₂²×C₅₃	semidirect
507. #490	C₂²×C₅₃	semidirect
508. #486	C₂²×C₅₃	semidirect
509. #484	C₂²×C₅₃	semidirect
510. #483	C₂²×C₅₃	semidirect
511. #482	C₂²×C₅₃	semidirect
512. #482	C₂²×C₅₃	semidirect
513. #480	C₂²×C₅₃	semidirect
515. #478	C₂²×C₅₃	semidirect
516. #477	C₂²×C₅₃	semidirect
518. #476	C₂²×C₅₃	semidirect
519. #476	C₂²×C₅₃	semidirect
520. #475	C₂²×C₅₃	semidirect
523. #474	C₂²×C₅₃	semidirect
524. #474	C₂²×C₅₃	semidirect
525. #474	C₂²×C₅₃	semidirect
526. #474	C₂²×C₅₃	semidirect
527. #468	C₂²×C₅₃	semidirect
528. #468	C₂²×C₅₃	semidirect
529. #465	C₂²×C₅₃	semidirect
531. #461	C₂²×C₅₃	semidirect
532. #461	C₂²×C₅₃	semidirect
533. #460	C₂²×C₅₃	semidirect
534. #460	C₂²×C₅₃	semidirect
537. #446	C₂²×C₅₃	semidirect
539. #440	C₂²×C₅₃	semidirect
541. C₄×SD₁₆	C₂²×C₅₃	semidirect
542. #436	C₂²×C₅₃	semidirect
543. #435	C₂²×C₅₃	semidirect
544. #432	C₂²×C₅₃	semidirect
545. #431	C₂²×C₅₃	semidirect
546. #428	C₂²×C₅₃	semidirect
547. C₂×#107	C₂²×C₅₃	semidirect
548. #423	C₂²×C₅₃	semidirect
549. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
552. C₅₃	#62109	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×#3509	2
D₄×#41×C₅₃	2
C₅₃×#3386	2
C₅₃×#3368	2
C₅₃×#3353	2
C₅₃×#3343	2
C₅₃×#3324	2
C₅₃×#3310	2
C₅₃×#3254	2
C₅₃×#3246	2
C₅₃×#3235	2
C₅₃×#3226	2
C₅₃×#3110	2
C₅₃×#3101	2
C₅₃×#3077	2
C₂×C₅₃×#474	2
C₅₃×#2989	2
#62109	53

Subgroups — C53×#62109 #27269621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62109 #27269621