Subgroups — C₅₃×#61324 #27268836

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 880 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3375	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3373	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3371	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3370	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3365	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3361	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3140	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3078	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#3077	C₂	1	2	yes	maximal, complemented
13	C₅₃×#3040	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3038	C₂	1	1	yes	maximal, characteristic
15	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#137×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	2	yes	maximal, complemented
18	C₂×#137×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#559	—	2	1	no	—
21	C₅₃×#559	—	2	2	no	—
22	C₅₃×#559	—	2	1	no	—
23	C₅₃×#559	C₂²	1	2	yes	—
24	C₅₃×#559	C₂²	1	4	yes	—
25	C₅₃×#559	C₂²	1	2	yes	—
26	Q₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₅₃×#557	C₂²	1	1	yes	characteristic, complemented
28	C₅₃×#557	C₂²	1	2	yes	complemented
29	C₅₃×#554	—	2	1	no	—
30	C₅₃×#554	—	2	2	no	—
31	C₅₃×#554	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#554	C₂²	1	2	yes	complemented
33	D₄×Q₈×C₅₃	—	2	1	no	—
34	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#533	—	2	2	no	—
36	C₅₃×#533	—	2	1	no	—
37	C₅₃×#533	C₂²	1	2	yes	—
38	C₅₃×#533	C₂²	1	1	yes	characteristic
39	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#124×C₅₃	—	2	1	no	—
41	C₂×#124×C₅₃	C₂²	1	2	yes	—
42	C₂×#123×C₅₃	—	2	2	no	—
43	C₄×#41×C₅₃	C₂²	1	2	yes	—
44	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
46	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₅₃	—	2	1	no	—
49	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
50	C₅₃×#490	C₂²	1	4	yes	—
51	C₅₃×#489	C₂²	1	4	yes	complemented
52	C₅₃×#489	C₂²	1	4	yes	complemented
53	C₅₃×#488	C₂²	1	4	yes	complemented
54	C₅₃×#488	C₂²	1	4	yes	complemented
55	C₅₃×#486	C₂²	1	2	yes	—
56	C₅₃×#486	C₂²	1	2	yes	—
57	C₅₃×#479	C₂²	1	4	yes	complemented
58	C₅₃×#478	C₂²	1	4	yes	complemented
59	C₅₃×#478	C₂²	1	4	yes	complemented
60	C₅₃×#477	C₂²	1	2	yes	complemented
61	C₅₃×#476	C₂²	1	4	yes	complemented
62	C₅₃×#475	C₂²	1	2	yes	complemented
63	C₅₃×#474	C₂²	1	4	yes	complemented
64	C₅₃×#464	C₂²	1	4	yes	complemented
65	C₅₃×#464	C₂²	1	4	yes	complemented
66	C₅₃×#463	C₂²	1	4	yes	complemented
67	C₅₃×#463	C₂²	1	4	yes	complemented
68	C₅₃×#460	C₂²	1	2	yes	—
69	C₅₃×#460	C₂²	1	2	yes	—
70	C₅₃×#459	C₂²	1	2	yes	—
71	C₅₃×#459	C₂²	1	2	yes	—
72	C₅₃×#446	C₂²	1	4	yes	complemented
73	C₅₃×#446	C₂²	1	2	yes	complemented
74	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
75	C₅₃×#442	C₂²	1	4	yes	complemented
76	C₅₃×#441	C₂²	1	2	yes	complemented
77	C₅₃×#440	C₂²	1	4	yes	complemented
78	C₅₃×#440	C₂²	1	2	yes	complemented
79	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
80	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
81	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
82	C₄×D₈×C₅₃	C₂²	1	4	yes	complemented
83	C₅₃×#428	C₂²	1	2	yes	—
84	C₅₃×#428	C₂²	1	1	yes	characteristic
85	C₅₃×#423	C₂²	1	1	yes	characteristic
86	C₅₃×#423	C₂²	1	2	yes	—
87	C₅₃×#419	C₂²	1	1	yes	characteristic
88	C₅₃×#419	C₂²	1	2	yes	—
89	C₅₃×#418	C₂²	1	2	yes	—
90	C₅₃×#418	C₂²	1	2	yes	—
91	C₅₃×#418	C₂²	1	2	yes	—
92	C₅₃×#417	C₂²	1	1	yes	characteristic
93	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
94	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
95	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×Q₈×C₅₃	—	2	1	no	—
97	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
98	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×D₄×C₅₃	—	2	1	no	—
100	C₂²×D₄×C₅₃	—	2	1	no	—
101	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
102	C₂³×C₄×C₅₃	—	2	2	no	—
103	#137×C₅₃	—	2	4	no	—
104	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
105	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
106	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
107	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
108	C₂×D₈×C₅₃	C₂³	1	4	yes	—
109	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
110	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊Q₈×C₅₃	—	2	1	no	—
113	C₄⋊Q₈×C₅₃	—	2	2	no	—
114	C₄⋊Q₈×C₅₃	—	2	1	no	—
115	C₄⋊Q₈×C₅₃	—	2	2	no	—
116	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
119	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
120	#127×C₅₃	—	2	1	no	—
121	#127×C₅₃	—	2	1	no	—
122	#127×C₅₃	C₂³	1	2	yes	—
123	#127×C₅₃	C₂³	1	2	yes	—
124	#125×C₅₃	—	2	2	no	—
125	#125×C₅₃	—	2	2	no	—
126	#125×C₅₃	—	2	2	no	—
127	#125×C₅₃	C₂³	1	2	yes	—
128	#125×C₅₃	C₂³	1	2	yes	—
129	#125×C₅₃	C₂³	1	2	yes	—
130	#124×C₅₃	—	2	2	no	—
131	#124×C₅₃	—	2	2	no	—
132	#124×C₅₃	—	2	2	no	—
133	#124×C₅₃	—	2	2	no	—
134	#124×C₅₃	—	2	4	no	—
135	#124×C₅₃	C₂³	1	4	yes	—
136	#124×C₅₃	C₂³	1	4	yes	—
137	#123×C₅₃	—	2	4	no	—
138	#123×C₅₃	—	2	4	no	—
139	#123×C₅₃	—	2	4	no	—
140	#123×C₅₃	—	2	4	no	—
141	C₂²⋊Q₈×C₅₃	—	2	1	no	—
142	C₂²⋊Q₈×C₅₃	—	2	2	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	—	2	1	no	—
149	C₂²⋊Q₈×C₅₃	—	2	2	no	—
150	C₂²⋊Q₈×C₅₃	—	2	2	no	—
151	C₂²⋊Q₈×C₅₃	—	2	1	no	—
152	C₄×Q₈×C₅₃	—	2	1	no	—
153	C₄×Q₈×C₅₃	—	2	2	no	—
154	C₄×Q₈×C₅₃	—	2	2	no	—
155	C₄×Q₈×C₅₃	—	2	1	no	—
156	C₄×Q₈×C₅₃	—	2	1	no	—
157	C₄×Q₈×C₅₃	—	2	1	no	—
158	C₄×Q₈×C₅₃	—	2	2	no	—
159	C₄×Q₈×C₅₃	—	2	1	no	—
160	C₄×Q₈×C₅₃	—	2	1	no	—
161	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
162	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
163	C₄×Q₈×C₅₃	D₄	1	2	yes	—
164	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
165	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
166	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
167	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
168	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
169	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
170	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
171	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
172	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
173	C₄×D₄×C₅₃	—	2	2	no	—
174	C₄×D₄×C₅₃	—	2	4	no	—
175	C₄×D₄×C₅₃	—	2	2	no	—
176	C₄×D₄×C₅₃	—	2	2	no	—
177	C₄×D₄×C₅₃	—	2	2	no	—
178	C₄×D₄×C₅₃	—	2	1	no	—
179	C₄×D₄×C₅₃	—	2	2	no	—
180	C₄×D₄×C₅₃	—	2	2	no	—
181	C₄×D₄×C₅₃	—	2	1	no	—
182	C₄×D₄×C₅₃	C₂³	1	4	yes	—
183	C₄×D₄×C₅₃	C₂³	1	4	yes	—
184	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
185	C₄×D₄×C₅₃	C₂³	1	2	yes	—
186	C₄×D₄×C₅₃	C₂³	1	2	yes	—
187	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
188	#117×C₅₃	—	2	2	no	—
189	#117×C₅₃	—	2	1	no	—
190	#117×C₅₃	C₂³	1	2	yes	—
191	#117×C₅₃	C₂³	1	1	yes	characteristic
192	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
193	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
194	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
195	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
197	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
198	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
199	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
200	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
201	C₂×C₄²×C₅₃	—	2	2	no	—
202	C₂×C₄²×C₅₃	—	2	1	no	—
203	C₂×C₄²×C₅₃	—	2	1	no	—
204	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
205	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
206	#107×C₅₃	C₂³	1	4	yes	—
207	#107×C₅₃	C₂³	1	2	yes	—
208	#106×C₅₃	C₂³	1	2	yes	—
209	#106×C₅₃	C₂³	1	4	yes	—
210	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
211	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
212	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
213	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
214	#103×C₅₃	C₂³	1	2	yes	—
215	#103×C₅₃	C₂³	1	2	yes	—
216	#103×C₅₃	C₂³	1	4	yes	—
217	#103×C₅₃	C₂³	1	4	yes	—
218	#103×C₅₃	C₂³	1	2	yes	—
219	#102×C₅₃	C₂³	1	4	yes	—
220	#102×C₅₃	C₂³	1	4	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	4	yes	—
225	#102×C₅₃	C₂³	1	2	yes	—
226	#97×C₅₃	C₂³	1	4	yes	—
227	#97×C₅₃	C₂³	1	2	yes	—
228	C₄×C₈×C₅₃	C₂³	1	2	yes	—
229	C₄×C₈×C₅₃	C₂³	1	4	yes	—
230	C₂⁴×C₅₃	—	2	2	no	—
231	#41×C₅₃	—	2	2	no	—
232	#41×C₅₃	#41	1	4	yes	—
233	C₂×Q₈×C₅₃	—	2	2	no	—
234	C₂×Q₈×C₅₃	—	2	1	no	—
235	C₂×Q₈×C₅₃	—	2	1	no	—
236	C₂×Q₈×C₅₃	—	2	2	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	—	4	2	no	—
241	C₂×Q₈×C₅₃	—	4	4	no	—
242	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
243	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
244	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
245	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
246	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×D₄×C₅₃	—	2	1	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	2	no	—
253	C₂×D₄×C₅₃	—	2	4	no	—
254	C₂×D₄×C₅₃	—	2	4	no	—
255	C₂×D₄×C₅₃	—	4	4	no	—
256	C₂×D₄×C₅₃	—	8	2	no	—
257	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
258	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
259	C₂²×C₄×C₅₃	—	2	1	no	—
260	C₂²×C₄×C₅₃	—	2	1	no	—
261	C₂²×C₄×C₅₃	—	2	1	no	—
262	C₂²×C₄×C₅₃	—	2	2	no	—
263	C₂²×C₄×C₅₃	—	2	1	no	—
264	C₂²×C₄×C₅₃	—	2	2	no	—
265	C₂²×C₄×C₅₃	—	2	1	no	—
266	C₂²×C₄×C₅₃	—	2	4	no	—
267	C₂²×C₄×C₅₃	—	2	4	no	—
268	C₂²×C₄×C₅₃	—	8	4	no	—
269	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
270	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
271	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
272	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
273	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
274	Q₁₆×C₅₃	—	2	4	no	—
275	SD₁₆×C₅₃	—	2	8	no	—
276	SD₁₆×C₅₃	—	2	4	no	—
277	D₈×C₅₃	—	2	8	no	—
278	M₁₆×C₅₃	—	2	4	no	—
279	M₁₆×C₅₃	—	2	2	no	—
280	M₁₆×C₅₃	—	2	2	no	—
281	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
282	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
283	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
284	C₄⋊C₄×C₅₃	—	2	2	no	—
285	C₄⋊C₄×C₅₃	—	2	2	no	—
286	C₄⋊C₄×C₅₃	—	2	2	no	—
287	C₄⋊C₄×C₅₃	—	2	2	no	—
288	C₄⋊C₄×C₅₃	—	2	1	no	—
289	C₄⋊C₄×C₅₃	—	2	2	no	—
290	C₄⋊C₄×C₅₃	—	2	2	no	—
291	C₄⋊C₄×C₅₃	—	2	1	no	—
292	C₄⋊C₄×C₅₃	—	2	2	no	—
293	C₄⋊C₄×C₅₃	—	2	1	no	—
294	C₄⋊C₄×C₅₃	—	2	1	no	—
295	C₄⋊C₄×C₅₃	—	2	2	no	—
296	C₄⋊C₄×C₅₃	—	2	2	no	—
297	C₄⋊C₄×C₅₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
302	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
303	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
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	C₂×D₄	1	2	yes	—
308	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
309	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
310	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
311	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
312	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
313	C₂²⋊C₄×C₅₃	—	2	2	no	—
314	C₂²⋊C₄×C₅₃	—	2	1	no	—
315	C₂²⋊C₄×C₅₃	—	2	4	no	—
316	C₂²⋊C₄×C₅₃	—	2	4	no	—
317	C₂²⋊C₄×C₅₃	—	2	4	no	—
318	C₂²⋊C₄×C₅₃	—	2	4	no	—
319	C₂²⋊C₄×C₅₃	—	2	4	no	—
320	C₂²⋊C₄×C₅₃	—	2	4	no	—
321	C₂²⋊C₄×C₅₃	—	2	2	no	—
322	C₂²⋊C₄×C₅₃	—	2	1	no	—
323	C₂²⋊C₄×C₅₃	—	2	1	no	—
324	C₂²⋊C₄×C₅₃	—	2	2	no	—
325	C₄²×C₅₃	—	2	2	no	—
326	C₄²×C₅₃	—	2	1	no	—
327	C₄²×C₅₃	—	2	2	no	—
328	C₄²×C₅₃	—	2	2	no	—
329	C₄²×C₅₃	—	2	2	no	—
330	C₄²×C₅₃	—	2	1	no	—
331	C₄²×C₅₃	—	2	2	no	—
332	C₄²×C₅₃	—	2	1	no	—
333	C₄²×C₅₃	C₂×D₄	1	2	yes	—
334	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
335	C₄²×C₅₃	C₂×D₄	1	2	yes	—
336	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
337	C₄²×C₅₃	C₂⁴	1	2	yes	—
338	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
339	C₄²×C₅₃	C₂⁴	1	2	yes	—
340	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
341	C₂³×C₅₃	—	2	1	no	—
342	C₂³×C₅₃	—	2	4	no	—
343	C₂³×C₅₃	—	4	4	no	—
344	C₂³×C₅₃	—	4	4	no	—
345	C₂³×C₅₃	—	4	4	no	—
346	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
347	Q₈×C₅₃	—	2	2	no	—
348	Q₈×C₅₃	—	2	2	no	—
349	Q₈×C₅₃	—	2	1	no	—
350	Q₈×C₅₃	—	2	2	no	—
351	Q₈×C₅₃	—	2	2	no	—
352	Q₈×C₅₃	—	4	2	no	—
353	Q₈×C₅₃	—	4	4	no	—
354	Q₈×C₅₃	—	4	2	no	—
355	Q₈×C₅₃	—	4	2	no	—
356	Q₈×C₅₃	—	8	2	no	—
357	Q₈×C₅₃	—	8	1	no	—
358	Q₈×C₅₃	C₂×#41	1	2	yes	—
359	D₄×C₅₃	—	2	4	no	—
360	D₄×C₅₃	—	2	4	no	—
361	D₄×C₅₃	—	2	1	no	—
362	D₄×C₅₃	—	2	4	no	—
363	D₄×C₅₃	—	4	4	no	—
364	D₄×C₅₃	—	4	4	no	—
365	D₄×C₅₃	—	4	4	no	—
366	D₄×C₅₃	—	8	2	no	—
367	D₄×C₅₃	—	8	2	no	—
368	D₄×C₅₃	C₂×#41	1	2	yes	—
369	C₂×C₄×C₅₃	—	2	1	no	—
370	C₂×C₄×C₅₃	—	2	4	no	—
371	C₂×C₄×C₅₃	—	2	2	no	—
372	C₂×C₄×C₅₃	—	2	2	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	1	no	—
377	C₂×C₄×C₅₃	—	2	2	no	—
378	C₂×C₄×C₅₃	—	4	2	no	—
379	C₂×C₄×C₅₃	—	4	1	no	—
380	C₂×C₄×C₅₃	—	4	2	no	—
381	C₂×C₄×C₅₃	—	4	4	no	—
382	C₂×C₄×C₅₃	—	4	2	no	—
383	C₂×C₄×C₅₃	—	4	2	no	—
384	C₂×C₄×C₅₃	—	4	2	no	—
385	C₂×C₄×C₅₃	—	8	2	no	—
386	C₂×C₄×C₅₃	—	8	1	no	—
387	C₂×C₄×C₅₃	—	8	4	no	—
388	C₂×C₄×C₅₃	—	8	4	no	—
389	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
390	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
391	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
393	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
394	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
395	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
396	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
397	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
398	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
399	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
400	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
401	C₈×C₅₃	—	2	4	no	—
402	C₈×C₅₃	—	2	2	no	—
403	C₈×C₅₃	—	2	2	no	—
404	#61324	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
405	C₂²×C₅₃	—	2	1	no	—
406	C₂²×C₅₃	—	2	1	no	—
407	C₂²×C₅₃	—	2	2	no	—
408	C₂²×C₅₃	—	4	1	no	—
409	C₂²×C₅₃	—	4	1	no	—
410	C₂²×C₅₃	—	4	4	no	—
411	C₂²×C₅₃	—	4	4	no	—
412	C₂²×C₅₃	—	4	4	no	—
413	C₂²×C₅₃	—	8	8	no	—
414	C₂²×C₅₃	#548	1	2	yes	—
415	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
416	C₄×C₅₃	—	2	2	no	—
417	C₄×C₅₃	—	2	2	no	—
418	C₄×C₅₃	—	2	1	no	—
419	C₄×C₅₃	—	2	1	no	—
420	C₄×C₅₃	—	2	2	no	—
421	C₄×C₅₃	—	2	1	no	—
422	C₄×C₅₃	—	2	2	no	—
423	C₄×C₅₃	—	2	2	no	—
424	C₄×C₅₃	—	4	2	no	—
425	C₄×C₅₃	—	4	2	no	—
426	C₄×C₅₃	—	4	1	no	—
427	C₄×C₅₃	—	4	2	no	—
428	C₄×C₅₃	—	4	2	no	—
429	C₄×C₅₃	—	4	1	no	—
430	C₄×C₅₃	#548	1	2	yes	—
431	C₄×C₅₃	#575	1	2	yes	—
432	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
433	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
434	C₂×#559	C₂×C₅₃	1	1	yes	characteristic, complemented
435	Q₈×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
436	#3375	C₂×C₅₃	1	2	yes	complemented
437	#3373	C₂×C₅₃	1	2	yes	complemented
438	#3371	C₂×C₅₃	1	4	yes	complemented
439	#3370	C₂×C₅₃	1	4	yes	complemented
440	#3365	C₂×C₅₃	1	2	yes	complemented
441	#3361	C₂×C₅₃	1	2	yes	complemented
442	#3140	C₂×C₅₃	1	2	yes	complemented
443	#3138	C₂×C₅₃	1	1	yes	characteristic, complemented
444	#3078	C₂×C₅₃	1	1	yes	characteristic, complemented
445	#3077	C₂×C₅₃	1	2	yes	complemented
446	#3040	C₂×C₅₃	1	2	yes	complemented
447	#3038	C₂×C₅₃	1	1	yes	characteristic
448	Q₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
449	C₄×#137	C₂×C₅₃	1	1	yes	characteristic, complemented
450	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	2	yes	complemented
451	C₂×C₅₃	—	2	2	no	—
452	C₂×C₅₃	—	4	2	no	—
453	C₂×C₅₃	—	8	4	no	—
454	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
455	C₂×C₅₃	C₂×#575	1	1	yes	characteristic
456	C₂×C₅₃	#3585	1	1	yes	characteristic
457	C₂×#137	C₂²×C₅₃	1	1	yes	characteristic
458	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
459	#559	—	2	1	no	—
460	#559	—	2	2	no	—
461	#559	—	2	1	no	—
462	#559	C₂²×C₅₃	1	2	yes	—
463	#559	C₂²×C₅₃	1	4	yes	—
464	#559	C₂²×C₅₃	1	2	yes	—
465	Q₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
466	#557	C₂²×C₅₃	1	1	yes	characteristic, complemented
467	#557	C₂²×C₅₃	1	2	yes	complemented
468	#554	—	2	1	no	—
469	#554	—	2	2	no	—
470	#554	C₂²×C₅₃	1	1	yes	characteristic, complemented
471	#554	C₂²×C₅₃	1	2	yes	complemented
472	D₄×Q₈	—	2	1	no	—
473	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
474	#533	—	2	2	no	—
475	#533	—	2	1	no	—
476	#533	C₂²×C₅₃	1	2	yes	—
477	#533	C₂²×C₅₃	1	1	yes	characteristic
478	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
479	C₂×#124	—	2	1	no	—
480	C₂×#124	C₂²×C₅₃	1	2	yes	—
481	C₂×#123	—	2	2	no	—
482	C₄×#41	C₂²×C₅₃	1	2	yes	—
483	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
484	C₂×C₄×Q₈	—	2	1	no	—
485	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
486	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
487	C₂×C₄×D₄	—	2	1	no	—
488	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
489	#490	C₂²×C₅₃	1	4	yes	—
490	#489	C₂²×C₅₃	1	4	yes	complemented
491	#489	C₂²×C₅₃	1	4	yes	complemented
492	#488	C₂²×C₅₃	1	4	yes	complemented
493	#488	C₂²×C₅₃	1	4	yes	complemented
494	#486	C₂²×C₅₃	1	2	yes	—
495	#486	C₂²×C₅₃	1	2	yes	—
496	#479	C₂²×C₅₃	1	4	yes	complemented
497	#478	C₂²×C₅₃	1	4	yes	complemented
498	#478	C₂²×C₅₃	1	4	yes	complemented
499	#477	C₂²×C₅₃	1	2	yes	complemented
500	#476	C₂²×C₅₃	1	4	yes	complemented
501	#475	C₂²×C₅₃	1	2	yes	complemented
502	#474	C₂²×C₅₃	1	4	yes	complemented
503	#464	C₂²×C₅₃	1	4	yes	complemented
504	#464	C₂²×C₅₃	1	4	yes	complemented
505	#463	C₂²×C₅₃	1	4	yes	complemented
506	#463	C₂²×C₅₃	1	4	yes	complemented
507	#460	C₂²×C₅₃	1	2	yes	—
508	#460	C₂²×C₅₃	1	2	yes	—
509	#459	C₂²×C₅₃	1	2	yes	—
510	#459	C₂²×C₅₃	1	2	yes	—
511	#446	C₂²×C₅₃	1	4	yes	complemented
512	#446	C₂²×C₅₃	1	2	yes	complemented
513	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
514	#442	C₂²×C₅₃	1	4	yes	complemented
515	#441	C₂²×C₅₃	1	2	yes	complemented
516	#440	C₂²×C₅₃	1	2	yes	complemented
517	#440	C₂²×C₅₃	1	4	yes	complemented
518	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	complemented
519	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
520	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
521	C₄×D₈	C₂²×C₅₃	1	4	yes	complemented
522	#428	C₂²×C₅₃	1	1	yes	characteristic
523	#428	C₂²×C₅₃	1	2	yes	—
524	#423	C₂²×C₅₃	1	2	yes	—
525	#423	C₂²×C₅₃	1	1	yes	characteristic
526	#419	C₂²×C₅₃	1	2	yes	—
527	#419	C₂²×C₅₃	1	1	yes	characteristic
528	#418	C₂²×C₅₃	1	2	yes	—
529	#418	C₂²×C₅₃	1	2	yes	—
530	#418	C₂²×C₅₃	1	2	yes	—
531	#417	C₂²×C₅₃	1	1	yes	characteristic
532	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
533	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
534	C₅₃	#61324	1	1	yes	characteristic, direct product factor, Sylow
535	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
536	C₂²×Q₈	—	2	1	no	—
537	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
538	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
539	C₂²×D₄	—	2	1	no	—
540	C₂²×D₄	—	2	1	no	—
541	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
542	C₂³×C₄	—	2	2	no	—
543	#137	—	2	4	no	—
544	AGL(1,ℤ/8)	—	2	8	no	—
545	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
546	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
547	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
548	C₂×D₈	C₂³×C₅₃	1	4	yes	—
549	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
550	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
551	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
552	C₄⋊Q₈	—	2	1	no	—
553	C₄⋊Q₈	—	2	2	no	—
554	C₄⋊Q₈	—	2	1	no	—
555	C₄⋊Q₈	—	2	2	no	—
556	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
557	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
558	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
559	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
560	#127	—	2	1	no	—
561	#127	—	2	1	no	—
562	#127	C₂³×C₅₃	1	2	yes	—
563	#127	C₂³×C₅₃	1	2	yes	—
564	#125	—	2	2	no	—
565	#125	—	2	2	no	—
566	#125	—	2	2	no	—
567	#125	C₂³×C₅₃	1	2	yes	—
568	#125	C₂³×C₅₃	1	2	yes	—
569	#125	C₂³×C₅₃	1	2	yes	—
570	#124	—	2	2	no	—
571	#124	—	2	2	no	—
572	#124	—	2	4	no	—
573	#124	—	2	2	no	—
574	#124	—	2	2	no	—
575	#124	C₂³×C₅₃	1	4	yes	—
576	#124	C₂³×C₅₃	1	4	yes	—
577	#123	—	2	4	no	—
578	#123	—	2	4	no	—
579	#123	—	2	4	no	—
580	#123	—	2	4	no	—
581	C₂²⋊Q₈	—	2	1	no	—
582	C₂²⋊Q₈	—	2	2	no	—
583	C₂²⋊Q₈	—	2	1	no	—
584	C₂²⋊Q₈	—	2	2	no	—
585	C₂²⋊Q₈	—	2	2	no	—
586	C₂²⋊Q₈	—	2	1	no	—
587	C₂²⋊Q₈	—	2	1	no	—
588	C₂²⋊Q₈	—	2	2	no	—
589	C₂²⋊Q₈	—	2	2	no	—
590	C₂²⋊Q₈	—	2	2	no	—
591	C₂²⋊Q₈	—	2	2	no	—
592	C₄×Q₈	—	2	1	no	—
593	C₄×Q₈	—	2	1	no	—
594	C₄×Q₈	—	2	2	no	—
595	C₄×Q₈	—	2	2	no	—
596	C₄×Q₈	—	2	1	no	—
597	C₄×Q₈	—	2	1	no	—
598	C₄×Q₈	—	2	1	no	—
599	C₄×Q₈	—	2	2	no	—
600	C₄×Q₈	—	2	1	no	—
601	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
602	C₄×Q₈	D₄×C₅₃	1	2	yes	—
603	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
604	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
605	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
606	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
607	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
608	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
609	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
610	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
611	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
612	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
613	C₄×D₄	—	2	1	no	—
614	C₄×D₄	—	2	2	no	—
615	C₄×D₄	—	2	2	no	—
616	C₄×D₄	—	2	1	no	—
617	C₄×D₄	—	2	2	no	—
618	C₄×D₄	—	2	4	no	—
619	C₄×D₄	—	2	2	no	—
620	C₄×D₄	—	2	2	no	—
621	C₄×D₄	—	2	2	no	—
622	C₄×D₄	C₂³×C₅₃	1	4	yes	—
623	C₄×D₄	C₂³×C₅₃	1	4	yes	—
624	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
625	C₄×D₄	C₂³×C₅₃	1	2	yes	—
626	C₄×D₄	C₂³×C₅₃	1	2	yes	—
627	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
628	#117	—	2	2	no	—
629	#117	—	2	1	no	—
630	#117	C₂³×C₅₃	1	2	yes	—
631	#117	C₂³×C₅₃	1	1	yes	characteristic
632	C₂×C₄⋊C₄	—	2	1	no	—
633	C₂×C₄⋊C₄	—	2	1	no	—
634	C₂×C₄⋊C₄	—	2	2	no	—
635	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
636	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
637	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
638	C₂×C₂²⋊C₄	—	2	2	no	—
639	C₂×C₂²⋊C₄	—	2	2	no	—
640	C₂×C₂²⋊C₄	—	2	2	no	—
641	C₂×C₄²	—	2	1	no	—
642	C₂×C₄²	—	2	2	no	—
643	C₂×C₄²	—	2	1	no	—
644	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
645	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
646	#107	C₂³×C₅₃	1	4	yes	—
647	#107	C₂³×C₅₃	1	2	yes	—
648	#106	C₂³×C₅₃	1	2	yes	—
649	#106	C₂³×C₅₃	1	4	yes	—
650	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
651	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
652	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
653	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
654	#103	C₂³×C₅₃	1	2	yes	—
655	#103	C₂³×C₅₃	1	2	yes	—
656	#103	C₂³×C₅₃	1	2	yes	—
657	#103	C₂³×C₅₃	1	4	yes	—
658	#103	C₂³×C₅₃	1	4	yes	—
659	#102	C₂³×C₅₃	1	2	yes	—
660	#102	C₂³×C₅₃	1	4	yes	—
661	#102	C₂³×C₅₃	1	4	yes	—
662	#102	C₂³×C₅₃	1	4	yes	—
663	#102	C₂³×C₅₃	1	4	yes	—
664	#102	C₂³×C₅₃	1	4	yes	—
665	#102	C₂³×C₅₃	1	4	yes	—
666	#97	C₂³×C₅₃	1	2	yes	—
667	#97	C₂³×C₅₃	1	4	yes	—
668	C₄×C₈	C₂³×C₅₃	1	4	yes	—
669	C₄×C₈	C₂³×C₅₃	1	2	yes	—
670	C₂⁴	—	2	2	no	—
671	#41	—	2	2	no	—
672	#41	#41×C₅₃	1	4	yes	—
673	C₂×Q₈	—	2	2	no	—
674	C₂×Q₈	—	2	1	no	—
675	C₂×Q₈	—	2	1	no	—
676	C₂×Q₈	—	2	2	no	—
677	C₂×Q₈	—	2	1	no	—
678	C₂×Q₈	—	2	2	no	—
679	C₂×Q₈	—	2	2	no	—
680	C₂×Q₈	—	4	2	no	—
681	C₂×Q₈	—	4	4	no	—
682	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
683	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
684	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
685	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
686	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
687	C₂×D₄	—	2	1	no	—
688	C₂×D₄	—	2	2	no	—
689	C₂×D₄	—	2	2	no	—
690	C₂×D₄	—	2	2	no	—
691	C₂×D₄	—	2	2	no	—
692	C₂×D₄	—	2	2	no	—
693	C₂×D₄	—	2	4	no	—
694	C₂×D₄	—	2	4	no	—
695	C₂×D₄	—	4	4	no	—
696	C₂×D₄	—	8	2	no	—
697	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
698	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
699	C₂²×C₄	—	2	1	no	—
700	C₂²×C₄	—	2	1	no	—
701	C₂²×C₄	—	2	1	no	—
702	C₂²×C₄	—	2	2	no	—
703	C₂²×C₄	—	2	1	no	—
704	C₂²×C₄	—	2	2	no	—
705	C₂²×C₄	—	2	1	no	—
706	C₂²×C₄	—	2	4	no	—
707	C₂²×C₄	—	2	4	no	—
708	C₂²×C₄	—	8	4	no	—
709	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
710	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
711	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
712	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
713	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
714	Q₁₆	—	2	4	no	—
715	SD₁₆	—	2	8	no	—
716	SD₁₆	—	2	4	no	—
717	D₈	—	2	8	no	—
718	M₁₆	—	2	4	no	—
719	M₁₆	—	2	2	no	—
720	M₁₆	—	2	2	no	—
721	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
722	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
723	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
724	C₄⋊C₄	—	2	2	no	—
725	C₄⋊C₄	—	2	2	no	—
726	C₄⋊C₄	—	2	2	no	—
727	C₄⋊C₄	—	2	2	no	—
728	C₄⋊C₄	—	2	1	no	—
729	C₄⋊C₄	—	2	2	no	—
730	C₄⋊C₄	—	2	2	no	—
731	C₄⋊C₄	—	2	1	no	—
732	C₄⋊C₄	—	2	1	no	—
733	C₄⋊C₄	—	2	2	no	—
734	C₄⋊C₄	—	2	2	no	—
735	C₄⋊C₄	—	2	1	no	—
736	C₄⋊C₄	—	2	2	no	—
737	C₄⋊C₄	—	2	1	no	—
738	C₄⋊C₄	—	2	2	no	—
739	C₄⋊C₄	—	2	2	no	—
740	C₄⋊C₄	—	2	2	no	—
741	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
742	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
743	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
744	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
745	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
746	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
747	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
748	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
749	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
750	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
751	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
752	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₂²⋊C₄	—	2	1	no	—
755	C₂²⋊C₄	—	2	4	no	—
756	C₂²⋊C₄	—	2	4	no	—
757	C₂²⋊C₄	—	2	4	no	—
758	C₂²⋊C₄	—	2	4	no	—
759	C₂²⋊C₄	—	2	4	no	—
760	C₂²⋊C₄	—	2	4	no	—
761	C₂²⋊C₄	—	2	1	no	—
762	C₂²⋊C₄	—	2	2	no	—
763	C₂²⋊C₄	—	2	2	no	—
764	C₂²⋊C₄	—	2	1	no	—
765	C₄²	—	2	2	no	—
766	C₄²	—	2	1	no	—
767	C₄²	—	2	2	no	—
768	C₄²	—	2	2	no	—
769	C₄²	—	2	2	no	—
770	C₄²	—	2	1	no	—
771	C₄²	—	2	2	no	—
772	C₄²	—	2	1	no	—
773	C₄²	C₂×D₄×C₅₃	1	2	yes	—
774	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
775	C₄²	C₂×D₄×C₅₃	1	2	yes	—
776	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
777	C₄²	C₂⁴×C₅₃	1	2	yes	—
778	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
779	C₄²	C₂⁴×C₅₃	1	2	yes	—
780	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
781	C₂³	—	2	1	no	—
782	C₂³	—	2	4	no	—
783	C₂³	—	4	4	no	—
784	C₂³	—	4	4	no	—
785	C₂³	—	4	4	no	—
786	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
787	Q₈	—	2	2	no	—
788	Q₈	—	2	2	no	—
789	Q₈	—	2	1	no	—
790	Q₈	—	2	2	no	—
791	Q₈	—	2	2	no	—
792	Q₈	—	4	2	no	—
793	Q₈	—	4	4	no	—
794	Q₈	—	4	2	no	—
795	Q₈	—	4	2	no	—
796	Q₈	—	8	2	no	—
797	Q₈	—	8	1	no	—
798	Q₈	C₂×#41×C₅₃	1	2	yes	—
799	D₄	—	2	4	no	—
800	D₄	—	2	4	no	—
801	D₄	—	2	1	no	—
802	D₄	—	2	4	no	—
803	D₄	—	4	4	no	—
804	D₄	—	4	4	no	—
805	D₄	—	4	4	no	—
806	D₄	—	8	2	no	—
807	D₄	—	8	2	no	—
808	D₄	C₂×#41×C₅₃	1	2	yes	—
809	C₂×C₄	—	2	1	no	—
810	C₂×C₄	—	2	4	no	—
811	C₂×C₄	—	2	2	no	—
812	C₂×C₄	—	2	2	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	1	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	1	no	—
818	C₂×C₄	—	4	2	no	—
819	C₂×C₄	—	4	1	no	—
820	C₂×C₄	—	4	2	no	—
821	C₂×C₄	—	4	4	no	—
822	C₂×C₄	—	4	2	no	—
823	C₂×C₄	—	4	2	no	—
824	C₂×C₄	—	4	2	no	—
825	C₂×C₄	—	8	2	no	—
826	C₂×C₄	—	8	1	no	—
827	C₂×C₄	—	8	4	no	—
828	C₂×C₄	—	8	4	no	—
829	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
830	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
831	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
832	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
833	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
834	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
835	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
836	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
837	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
838	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
839	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
840	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
841	C₈	—	2	4	no	—
842	C₈	—	2	2	no	—
843	C₈	—	2	2	no	—
844	C₂²	—	2	1	no	—
845	C₂²	—	2	1	no	—
846	C₂²	—	2	2	no	—
847	C₂²	—	4	1	no	—
848	C₂²	—	4	1	no	—
849	C₂²	—	4	4	no	—
850	C₂²	—	4	4	no	—
851	C₂²	—	4	4	no	—
852	C₂²	—	8	8	no	—
853	C₂²	C₅₃×#548	1	2	yes	—
854	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
855	C₄	—	2	2	no	—
856	C₄	—	2	1	no	—
857	C₄	—	2	2	no	—
858	C₄	—	2	2	no	—
859	C₄	—	2	2	no	—
860	C₄	—	2	2	no	—
861	C₄	—	2	1	no	—
862	C₄	—	2	1	no	—
863	C₄	—	4	2	no	—
864	C₄	—	4	2	no	—
865	C₄	—	4	1	no	—
866	C₄	—	4	2	no	—
867	C₄	—	4	2	no	—
868	C₄	—	4	1	no	—
869	C₄	C₅₃×#548	1	2	yes	—
870	C₄	C₅₃×#575	1	2	yes	—
871	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
872	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
873	C₂	—	2	2	no	—
874	C₂	—	4	2	no	—
875	C₂	—	8	4	no	—
876	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
877	C₂	C₂×C₅₃×#575	1	1	yes	characteristic
878	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×#3038	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#559	C₂	semidirect
2. Q₈×#41×C₅₃	C₂	semidirect
3. C₅₃×#3375	C₂	semidirect
4. C₅₃×#3373	C₂	semidirect
5. C₅₃×#3371	C₂	semidirect
6. C₅₃×#3370	C₂	semidirect
7. C₅₃×#3365	C₂	semidirect
8. C₅₃×#3361	C₂	semidirect
9. C₅₃×#3140	C₂	semidirect
10. C₅₃×#3138	C₂	semidirect
11. C₅₃×#3078	C₂	semidirect
12. C₅₃×#3077	C₂	semidirect
13. C₅₃×#3040	C₂	semidirect
15. Q₈×M₁₆×C₅₃	C₂	semidirect
16. C₄×#137×C₅₃	C₂	semidirect
17. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
26. Q₈²×C₅₃	C₂²	semidirect
27. C₅₃×#557	C₂²	semidirect
28. C₅₃×#557	C₂²	semidirect
31. C₅₃×#554	C₂²	semidirect
32. C₅₃×#554	C₂²	semidirect
34. D₄×Q₈×C₅₃	C₂²	semidirect
51. C₅₃×#489	C₂²	semidirect
52. C₅₃×#489	C₂²	semidirect
53. C₅₃×#488	C₂²	semidirect
54. C₅₃×#488	C₂²	semidirect
57. C₅₃×#479	C₂²	semidirect
58. C₅₃×#478	C₂²	semidirect
59. C₅₃×#478	C₂²	semidirect
60. C₅₃×#477	C₂²	semidirect
61. C₅₃×#476	C₂²	semidirect
62. C₅₃×#475	C₂²	semidirect
63. C₅₃×#474	C₂²	semidirect
64. C₅₃×#464	C₂²	semidirect
65. C₅₃×#464	C₂²	semidirect
66. C₅₃×#463	C₂²	semidirect
67. C₅₃×#463	C₂²	semidirect
72. C₅₃×#446	C₂²	semidirect
73. C₅₃×#446	C₂²	semidirect
74. C₈×Q₈×C₅₃	C₂²	semidirect
75. C₅₃×#442	C₂²	semidirect
76. C₅₃×#441	C₂²	semidirect
77. C₅₃×#440	C₂²	semidirect
78. C₅₃×#440	C₂²	semidirect
79. C₄×Q₁₆×C₅₃	C₂²	semidirect
80. C₄×SD₁₆×C₅₃	C₂²	semidirect
81. C₄×SD₁₆×C₅₃	C₂²	semidirect
82. C₄×D₈×C₅₃	C₂²	semidirect
404. #61324	C₅₃	direct (N × H)
434. C₂×#559	C₂×C₅₃	semidirect
435. Q₈×#41	C₂×C₅₃	semidirect
436. #3375	C₂×C₅₃	semidirect
437. #3373	C₂×C₅₃	semidirect
438. #3371	C₂×C₅₃	semidirect
439. #3370	C₂×C₅₃	semidirect
440. #3365	C₂×C₅₃	semidirect
441. #3361	C₂×C₅₃	semidirect
442. #3140	C₂×C₅₃	semidirect
443. #3138	C₂×C₅₃	semidirect
444. #3078	C₂×C₅₃	semidirect
445. #3077	C₂×C₅₃	semidirect
446. #3040	C₂×C₅₃	semidirect
448. Q₈×M₁₆	C₂×C₅₃	semidirect
449. C₄×#137	C₂×C₅₃	semidirect
450. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
465. Q₈²	C₂²×C₅₃	semidirect
466. #557	C₂²×C₅₃	semidirect
467. #557	C₂²×C₅₃	semidirect
470. #554	C₂²×C₅₃	semidirect
471. #554	C₂²×C₅₃	semidirect
473. D₄×Q₈	C₂²×C₅₃	semidirect
490. #489	C₂²×C₅₃	semidirect
491. #489	C₂²×C₅₃	semidirect
492. #488	C₂²×C₅₃	semidirect
493. #488	C₂²×C₅₃	semidirect
496. #479	C₂²×C₅₃	semidirect
497. #478	C₂²×C₅₃	semidirect
498. #478	C₂²×C₅₃	semidirect
499. #477	C₂²×C₅₃	semidirect
500. #476	C₂²×C₅₃	semidirect
501. #475	C₂²×C₅₃	semidirect
502. #474	C₂²×C₅₃	semidirect
503. #464	C₂²×C₅₃	semidirect
504. #464	C₂²×C₅₃	semidirect
505. #463	C₂²×C₅₃	semidirect
506. #463	C₂²×C₅₃	semidirect
511. #446	C₂²×C₅₃	semidirect
512. #446	C₂²×C₅₃	semidirect
513. C₈×Q₈	C₂²×C₅₃	semidirect
514. #442	C₂²×C₅₃	semidirect
515. #441	C₂²×C₅₃	semidirect
516. #440	C₂²×C₅₃	semidirect
517. #440	C₂²×C₅₃	semidirect
518. C₄×Q₁₆	C₂²×C₅₃	semidirect
519. C₄×SD₁₆	C₂²×C₅₃	semidirect
520. C₄×SD₁₆	C₂²×C₅₃	semidirect
521. C₄×D₈	C₂²×C₅₃	semidirect
534. C₅₃	#61324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#559	2
Q₈×#41×C₅₃	2
C₅₃×#3375	2
C₅₃×#3373	2
C₅₃×#3371	2
C₅₃×#3370	2
C₅₃×#3365	2
C₅₃×#3361	2
C₅₃×#3140	2
C₅₃×#3138	2
C₅₃×#3078	2
C₅₃×#3077	2
C₅₃×#3040	2
C₅₃×#3038	2
Q₈×M₁₆×C₅₃	2
C₄×#137×C₅₃	2
C₄×AGL(1,ℤ/8)×C₅₃	2
#61324	53

Subgroups — C53×#61324 #27268836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61324 #27268836