Subgroups — C₅₃×#60929 #27268441

Order 13568.

1868 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×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3460	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3263	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3257	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3253	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3252	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3247	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3246	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3084	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#3062	C₂	1	2	yes	maximal, complemented
13	C₅₃×#3060	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3045	C₂	1	2	yes	maximal, complemented
15	C₅₃×#3036	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#3035	C₂	1	1	yes	maximal, characteristic, complemented
17	C₅₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
19	C₂×#135×C₅₃	C₂²	1	2	yes	—
20	C₂×#131×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#556	—	2	1	no	—
22	C₅₃×#555	—	2	1	no	—
23	C₅₃×#554	—	2	1	no	—
24	C₅₃×#553	—	2	2	no	—
25	C₅₃×#548	—	2	2	no	—
26	C₅₃×#547	—	2	1	no	—
27	C₅₃×#544	C₂²	1	1	yes	characteristic
28	C₅₃×#542	—	2	1	no	—
29	C₅₃×#542	C₂²	1	2	yes	—
30	C₅₃×#542	C₂²	1	4	yes	complemented
31	C₅₃×#540	—	2	1	no	—
32	C₅₃×#540	C₂²	1	2	yes	—
33	C₅₃×#538	—	2	2	no	—
34	C₅₃×#537	C₂²	1	1	yes	characteristic
35	C₅₃×#537	C₂²	1	2	yes	—
36	C₅₃×#535	C₂²	1	1	yes	characteristic
37	C₅₃×#535	C₂²	1	1	yes	characteristic
38	C₅₃×#533	C₂²	1	2	yes	complemented
39	C₅₃×#532	—	2	1	no	—
40	C₅₃×#532	C₂²	1	1	yes	characteristic
41	C₅₃×#532	C₂²	1	2	yes	—
42	C₅₃×#532	C₂²	1	2	yes	complemented
43	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#529	—	2	1	no	—
45	C₅₃×#529	C₂²	1	2	yes	—
46	C₅₃×#525	—	2	2	no	—
47	C₅₃×#520	C₂²	1	2	yes	—
48	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
49	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
51	C₅₃×#479	C₂²	1	4	yes	complemented
52	C₅₃×#478	C₂²	1	4	yes	complemented
53	C₅₃×#477	C₂²	1	2	yes	complemented
54	C₅₃×#476	C₂²	1	2	yes	—
55	C₅₃×#476	C₂²	1	2	yes	—
56	C₅₃×#476	C₂²	1	4	yes	—
57	C₅₃×#476	C₂²	1	2	yes	complemented
58	C₅₃×#476	C₂²	1	4	yes	complemented
59	C₅₃×#475	C₂²	1	4	yes	complemented
60	C₅₃×#475	C₂²	1	2	yes	complemented
61	C₅₃×#474	C₂²	1	2	yes	complemented
62	C₅₃×#470	C₂²	1	4	yes	complemented
63	C₅₃×#469	C₂²	1	4	yes	—
64	C₅₃×#469	C₂²	1	4	yes	complemented
65	C₅₃×#468	C₂²	1	4	yes	—
66	C₅₃×#466	C₂²	1	4	yes	—
67	C₅₃×#465	C₂²	1	4	yes	—
68	C₅₃×#465	C₂²	1	4	yes	complemented
69	C₅₃×#465	C₂²	1	4	yes	complemented
70	C₅₃×#464	C₂²	1	4	yes	—
71	C₅₃×#464	C₂²	1	2	yes	complemented
72	C₅₃×#463	C₂²	1	4	yes	—
73	C₅₃×#463	C₂²	1	4	yes	—
74	C₅₃×#463	C₂²	1	2	yes	complemented
75	C₅₃×#462	C₂²	1	1	yes	characteristic, complemented
76	C₅₃×#461	C₂²	1	2	yes	—
77	C₅₃×#461	C₂²	1	1	yes	characteristic, complemented
78	C₅₃×#460	C₂²	1	4	yes	—
79	C₅₃×#460	C₂²	1	1	yes	characteristic, complemented
80	C₅₃×#459	C₂²	1	2	yes	—
81	C₅₃×#459	C₂²	1	4	yes	—
82	C₅₃×#459	C₂²	1	1	yes	characteristic, complemented
83	C₅₃×#428	C₂²	1	2	yes	—
84	C₂×#106×C₅₃	C₂²	1	2	yes	—
85	C₅₃×#424	C₂²	1	2	yes	—
86	C₅₃×#424	C₂²	1	4	yes	complemented
87	C₅₃×#423	C₂²	1	1	yes	characteristic
88	C₅₃×#423	C₂²	1	2	yes	complemented
89	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
90	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
91	C₅₃×#417	C₂²	1	2	yes	—
92	C₅₃×#417	C₂²	1	2	yes	—
93	C₅₃×#416	C₂²	1	2	yes	—
94	C₂×#102×C₅₃	C₂²	1	2	yes	—
95	C₂×#41×C₅₃	—	2	2	no	—
96	C₂×#41×C₅₃	—	2	2	no	—
97	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×#41×C₅₃	C₂³	1	2	yes	—
99	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×Q₈×C₅₃	—	2	1	no	—
101	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂³×C₄×C₅₃	—	2	1	no	—
103	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
104	#135×C₅₃	—	4	4	no	—
105	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
106	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
107	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
108	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
109	C₂×D₈×C₅₃	C₂³	1	2	yes	—
110	C₂×D₈×C₅₃	C₂³	1	4	yes	—
111	#131×C₅₃	—	4	4	no	—
112	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
113	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
114	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
115	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
116	C₄⋊Q₈×C₅₃	—	2	1	no	—
117	C₄⋊Q₈×C₅₃	—	2	1	no	—
118	C₄⋊Q₈×C₅₃	—	2	1	no	—
119	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
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	#127×C₅₃	—	2	1	no	—
124	#127×C₅₃	C₂³	1	1	yes	characteristic
125	#127×C₅₃	C₂³	1	2	yes	—
126	#126×C₅₃	—	2	2	no	—
127	#126×C₅₃	—	2	4	no	—
128	#126×C₅₃	—	2	2	no	—
129	#125×C₅₃	—	2	1	no	—
130	#125×C₅₃	—	2	2	no	—
131	#125×C₅₃	C₂³	1	2	yes	—
132	#124×C₅₃	—	2	2	no	—
133	#124×C₅₃	—	2	4	no	—
134	#124×C₅₃	—	2	2	no	—
135	#124×C₅₃	—	2	2	no	—
136	#124×C₅₃	—	2	1	no	—
137	#124×C₅₃	C₂³	1	2	yes	—
138	#124×C₅₃	C₂³	1	2	yes	—
139	#123×C₅₃	—	2	2	no	—
140	#123×C₅₃	—	2	2	no	—
141	#123×C₅₃	—	2	2	no	—
142	#123×C₅₃	—	2	4	no	—
143	#123×C₅₃	—	2	2	no	—
144	C₂²⋊Q₈×C₅₃	—	2	2	no	—
145	C₂²⋊Q₈×C₅₃	—	2	4	no	—
146	C₂²⋊Q₈×C₅₃	—	2	2	no	—
147	C₂²⋊Q₈×C₅₃	—	2	1	no	—
148	C₂²⋊Q₈×C₅₃	—	2	2	no	—
149	C₂²⋊Q₈×C₅₃	—	2	2	no	—
150	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
151	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
152	#121×C₅₃	—	2	1	no	—
153	#121×C₅₃	—	2	1	no	—
154	#121×C₅₃	—	2	2	no	—
155	#121×C₅₃	—	2	2	no	—
156	#121×C₅₃	—	2	2	no	—
157	#121×C₅₃	—	2	1	no	—
158	#121×C₅₃	—	2	2	no	—
159	#121×C₅₃	—	2	4	no	—
160	#121×C₅₃	—	2	2	no	—
161	#121×C₅₃	—	2	1	no	—
162	#121×C₅₃	—	2	1	no	—
163	#121×C₅₃	—	2	2	no	—
164	#121×C₅₃	C₂³	1	2	yes	—
165	#121×C₅₃	C₂³	1	2	yes	—
166	#121×C₅₃	C₂³	1	2	yes	—
167	#121×C₅₃	C₂³	1	4	yes	—
168	#120×C₅₃	—	2	2	no	—
169	#120×C₅₃	—	2	2	no	—
170	C₄×Q₈×C₅₃	—	2	2	no	—
171	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
172	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
173	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
174	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
175	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
176	C₄×D₄×C₅₃	—	2	4	no	—
177	C₄×D₄×C₅₃	—	2	1	no	—
178	C₄×D₄×C₅₃	—	2	1	no	—
179	C₄×D₄×C₅₃	—	2	2	no	—
180	C₄×D₄×C₅₃	—	2	1	no	—
181	C₄×D₄×C₅₃	—	2	1	no	—
182	C₄×D₄×C₅₃	—	2	2	no	—
183	C₄×D₄×C₅₃	—	2	1	no	—
184	C₄×D₄×C₅₃	—	2	1	no	—
185	C₄×D₄×C₅₃	—	2	2	no	—
186	C₄×D₄×C₅₃	D₄	1	2	yes	—
187	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
188	C₄×D₄×C₅₃	D₄	1	2	yes	—
189	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
190	C₄×D₄×C₅₃	C₂³	1	2	yes	—
191	C₄×D₄×C₅₃	C₂³	1	2	yes	—
192	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
193	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
194	C₄×D₄×C₅₃	C₂³	1	4	yes	—
195	#117×C₅₃	—	2	2	no	—
196	#117×C₅₃	—	2	1	no	—
197	#117×C₅₃	—	2	2	no	—
198	#117×C₅₃	—	2	2	no	—
199	#117×C₅₃	C₂³	1	1	yes	characteristic
200	#117×C₅₃	C₂³	1	1	yes	characteristic
201	#117×C₅₃	C₂³	1	2	yes	—
202	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
203	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
204	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
205	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
206	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
207	C₂×C₄²×C₅₃	—	2	1	no	—
208	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
209	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
210	#107×C₅₃	C₂³	1	4	yes	—
211	#106×C₅₃	C₂³	1	4	yes	—
212	#106×C₅₃	C₂³	1	4	yes	—
213	#106×C₅₃	C₂³	1	4	yes	—
214	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
215	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
216	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
217	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
218	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
219	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
220	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
221	#103×C₅₃	C₂³	1	2	yes	—
222	#103×C₅₃	C₂³	1	2	yes	—
223	#103×C₅₃	C₂³	1	4	yes	—
224	#103×C₅₃	C₂³	1	4	yes	—
225	#102×C₅₃	C₂³	1	4	yes	—
226	#102×C₅₃	C₂³	1	4	yes	—
227	#102×C₅₃	C₂³	1	2	yes	—
228	#102×C₅₃	C₂³	1	2	yes	—
229	#102×C₅₃	C₂³	1	4	yes	—
230	#102×C₅₃	C₂³	1	4	yes	—
231	C₂⁴×C₅₃	—	2	1	no	—
232	#41×C₅₃	—	2	2	no	—
233	#41×C₅₃	—	2	2	no	—
234	#41×C₅₃	—	2	2	no	—
235	#41×C₅₃	—	2	4	no	—
236	#41×C₅₃	—	2	2	no	—
237	#41×C₅₃	—	2	4	no	—
238	#41×C₅₃	—	8	4	no	—
239	#41×C₅₃	—	8	4	no	—
240	C₂×Q₈×C₅₃	—	2	2	no	—
241	C₂×Q₈×C₅₃	—	2	2	no	—
242	C₂×Q₈×C₅₃	—	2	2	no	—
243	C₂×Q₈×C₅₃	—	8	2	no	—
244	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
245	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
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	4	no	—
253	C₂×D₄×C₅₃	—	2	1	no	—
254	C₂×D₄×C₅₃	—	2	1	no	—
255	C₂×D₄×C₅₃	—	2	1	no	—
256	C₂×D₄×C₅₃	—	2	2	no	—
257	C₂×D₄×C₅₃	—	2	2	no	—
258	C₂×D₄×C₅₃	—	2	2	no	—
259	C₂×D₄×C₅₃	—	2	2	no	—
260	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
261	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
262	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
263	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
264	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₅₃	—	2	1	no	—
266	C₂²×C₄×C₅₃	—	2	2	no	—
267	C₂²×C₄×C₅₃	—	2	1	no	—
268	C₂²×C₄×C₅₃	—	2	1	no	—
269	C₂²×C₄×C₅₃	—	2	2	no	—
270	C₂²×C₄×C₅₃	—	2	2	no	—
271	C₂²×C₄×C₅₃	—	2	1	no	—
272	C₂²×C₄×C₅₃	—	2	4	no	—
273	C₂²×C₄×C₅₃	—	2	2	no	—
274	C₂²×C₄×C₅₃	—	8	2	no	—
275	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
276	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
277	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
278	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
279	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
280	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
281	Q₁₆×C₅₃	—	4	2	no	—
282	SD₁₆×C₅₃	—	4	2	no	—
283	SD₁₆×C₅₃	—	4	2	no	—
284	SD₁₆×C₅₃	—	4	4	no	—
285	D₈×C₅₃	—	4	2	no	—
286	D₈×C₅₃	—	4	4	no	—
287	M₁₆×C₅₃	—	2	4	no	—
288	M₁₆×C₅₃	—	4	4	no	—
289	C₂×C₈×C₅₃	—	2	4	no	—
290	C₂×C₈×C₅₃	—	4	4	no	—
291	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
292	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
293	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
294	C₄⋊C₄×C₅₃	—	2	2	no	—
295	C₄⋊C₄×C₅₃	—	2	2	no	—
296	C₄⋊C₄×C₅₃	—	2	4	no	—
297	C₄⋊C₄×C₅₃	—	2	2	no	—
298	C₄⋊C₄×C₅₃	—	2	1	no	—
299	C₄⋊C₄×C₅₃	—	2	1	no	—
300	C₄⋊C₄×C₅₃	—	2	1	no	—
301	C₄⋊C₄×C₅₃	—	2	1	no	—
302	C₄⋊C₄×C₅₃	—	2	2	no	—
303	C₄⋊C₄×C₅₃	—	2	2	no	—
304	C₄⋊C₄×C₅₃	—	2	1	no	—
305	C₄⋊C₄×C₅₃	—	2	2	no	—
306	C₄⋊C₄×C₅₃	—	2	2	no	—
307	C₄⋊C₄×C₅₃	—	2	4	no	—
308	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
309	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
310	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
311	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
312	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
313	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
314	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
315	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
316	C₂²⋊C₄×C₅₃	—	2	4	no	—
317	C₂²⋊C₄×C₅₃	—	2	1	no	—
318	C₂²⋊C₄×C₅₃	—	2	2	no	—
319	C₂²⋊C₄×C₅₃	—	2	2	no	—
320	C₂²⋊C₄×C₅₃	—	2	2	no	—
321	C₂²⋊C₄×C₅₃	—	2	1	no	—
322	C₂²⋊C₄×C₅₃	—	2	1	no	—
323	C₂²⋊C₄×C₅₃	—	2	4	no	—
324	C₂²⋊C₄×C₅₃	—	2	2	no	—
325	C₂²⋊C₄×C₅₃	—	2	2	no	—
326	C₂²⋊C₄×C₅₃	—	2	2	no	—
327	C₂²⋊C₄×C₅₃	—	2	2	no	—
328	C₂²⋊C₄×C₅₃	—	2	4	no	—
329	C₂²⋊C₄×C₅₃	—	2	2	no	—
330	C₂²⋊C₄×C₅₃	—	2	2	no	—
331	C₂²⋊C₄×C₅₃	—	2	2	no	—
332	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
333	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
334	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
335	C₄²×C₅₃	—	2	1	no	—
336	C₄²×C₅₃	—	2	1	no	—
337	C₄²×C₅₃	—	2	1	no	—
338	C₄²×C₅₃	—	2	4	no	—
339	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
340	C₄²×C₅₃	C₂×D₄	1	2	yes	—
341	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
342	C₄²×C₅₃	C₂×D₄	1	2	yes	—
343	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
344	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
345	C₄²×C₅₃	C₂⁴	1	2	yes	—
346	C₂³×C₅₃	—	2	1	no	—
347	C₂³×C₅₃	—	2	2	no	—
348	C₂³×C₅₃	—	2	2	no	—
349	C₂³×C₅₃	—	4	2	no	—
350	C₂³×C₅₃	—	8	1	no	—
351	C₂³×C₅₃	—	8	1	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	1	no	—
357	Q₈×C₅₃	—	2	1	no	—
358	Q₈×C₅₃	—	2	2	no	—
359	Q₈×C₅₃	—	4	4	no	—
360	Q₈×C₅₃	—	8	2	no	—
361	Q₈×C₅₃	—	8	2	no	—
362	D₄×C₅₃	—	2	4	no	—
363	D₄×C₅₃	—	2	2	no	—
364	D₄×C₅₃	—	2	2	no	—
365	D₄×C₅₃	—	2	2	no	—
366	D₄×C₅₃	—	2	1	no	—
367	D₄×C₅₃	—	2	2	no	—
368	D₄×C₅₃	—	2	2	no	—
369	D₄×C₅₃	—	2	1	no	—
370	D₄×C₅₃	—	4	2	no	—
371	D₄×C₅₃	—	4	2	no	—
372	D₄×C₅₃	—	4	4	no	—
373	D₄×C₅₃	—	4	2	no	—
374	D₄×C₅₃	—	4	2	no	—
375	D₄×C₅₃	—	8	2	no	—
376	D₄×C₅₃	—	8	2	no	—
377	D₄×C₅₃	—	8	2	no	—
378	D₄×C₅₃	—	8	4	no	—
379	D₄×C₅₃	—	8	2	no	—
380	D₄×C₅₃	—	8	2	no	—
381	C₂×C₄×C₅₃	—	2	2	no	—
382	C₂×C₄×C₅₃	—	2	4	no	—
383	C₂×C₄×C₅₃	—	2	2	no	—
384	C₂×C₄×C₅₃	—	2	2	no	—
385	C₂×C₄×C₅₃	—	2	2	no	—
386	C₂×C₄×C₅₃	—	2	4	no	—
387	C₂×C₄×C₅₃	—	2	1	no	—
388	C₂×C₄×C₅₃	—	2	1	no	—
389	C₂×C₄×C₅₃	—	2	1	no	—
390	C₂×C₄×C₅₃	—	2	2	no	—
391	C₂×C₄×C₅₃	—	4	1	no	—
392	C₂×C₄×C₅₃	—	4	2	no	—
393	C₂×C₄×C₅₃	—	4	2	no	—
394	C₂×C₄×C₅₃	—	4	4	no	—
395	C₂×C₄×C₅₃	—	4	4	no	—
396	C₂×C₄×C₅₃	—	4	2	no	—
397	C₂×C₄×C₅₃	—	8	1	no	—
398	C₂×C₄×C₅₃	—	8	2	no	—
399	C₂×C₄×C₅₃	—	8	4	no	—
400	C₂×C₄×C₅₃	—	8	1	no	—
401	C₂×C₄×C₅₃	—	8	2	no	—
402	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
403	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
404	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
405	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
406	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
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	H(5,𝔽₂)	1	2	yes	—
411	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
412	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
413	C₈×C₅₃	—	2	2	no	—
414	C₈×C₅₃	—	2	2	no	—
415	C₈×C₅₃	—	2	4	no	—
416	#60929	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
417	C₂²×C₅₃	—	2	1	no	—
418	C₂²×C₅₃	—	2	1	no	—
419	C₂²×C₅₃	—	2	2	no	—
420	C₂²×C₅₃	—	2	2	no	—
421	C₂²×C₅₃	—	2	2	no	—
422	C₂²×C₅₃	—	4	2	no	—
423	C₂²×C₅₃	—	4	1	no	—
424	C₂²×C₅₃	—	4	2	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	—	4	2	no	—
431	C₂²×C₅₃	—	8	4	no	—
432	C₂²×C₅₃	#537	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	2	no	—
438	C₄×C₅₃	—	2	1	no	—
439	C₄×C₅₃	—	2	2	no	—
440	C₄×C₅₃	—	4	4	no	—
441	C₄×C₅₃	—	4	1	no	—
442	C₄×C₅₃	—	4	1	no	—
443	C₄×C₅₃	—	4	2	no	—
444	C₄×C₅₃	—	4	2	no	—
445	C₄×C₅₃	—	4	1	no	—
446	C₄×C₅₃	#537	1	2	yes	—
447	C₄×C₅₃	#577	1	2	yes	—
448	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
449	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
450	#3511	C₂×C₅₃	1	1	yes	characteristic, complemented
451	#3460	C₂×C₅₃	1	1	yes	characteristic, complemented
452	#3263	C₂×C₅₃	1	4	yes	complemented
453	#3257	C₂×C₅₃	1	4	yes	complemented
454	#3253	C₂×C₅₃	1	2	yes	complemented
455	#3252	C₂×C₅₃	1	2	yes	complemented
456	#3247	C₂×C₅₃	1	2	yes	complemented
457	#3246	C₂×C₅₃	1	2	yes	complemented
458	#3084	C₂×C₅₃	1	2	yes	complemented
459	#3079	C₂×C₅₃	1	1	yes	characteristic, complemented
460	C₂×#476	C₂×C₅₃	1	1	yes	characteristic, complemented
461	#3062	C₂×C₅₃	1	2	yes	complemented
462	#3060	C₂×C₅₃	1	2	yes	complemented
463	#3045	C₂×C₅₃	1	2	yes	complemented
464	#3036	C₂×C₅₃	1	1	yes	characteristic, complemented
465	#3035	C₂×C₅₃	1	1	yes	characteristic, complemented
466	#2906	C₂×C₅₃	1	1	yes	characteristic, complemented
467	C₂×C₅₃	—	2	2	no	—
468	C₂×C₅₃	—	4	2	no	—
469	C₂×C₅₃	—	8	2	no	—
470	C₂×C₅₃	—	8	1	no	—
471	C₂×C₅₃	—	8	2	no	—
472	C₂×C₅₃	C₂×#537	1	1	yes	characteristic
473	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
474	C₂×C₅₃	#3585	1	1	yes	characteristic
475	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
476	C₂×#135	C₂²×C₅₃	1	2	yes	—
477	C₂×#131	C₂²×C₅₃	1	2	yes	—
478	#556	—	2	1	no	—
479	#555	—	2	1	no	—
480	#554	—	2	1	no	—
481	#553	—	2	2	no	—
482	#548	—	2	2	no	—
483	#547	—	2	1	no	—
484	#544	C₂²×C₅₃	1	1	yes	characteristic
485	#542	—	2	1	no	—
486	#542	C₂²×C₅₃	1	2	yes	—
487	#542	C₂²×C₅₃	1	4	yes	complemented
488	#540	—	2	1	no	—
489	#540	C₂²×C₅₃	1	2	yes	—
490	#538	—	2	2	no	—
491	#537	C₂²×C₅₃	1	1	yes	characteristic
492	#537	C₂²×C₅₃	1	2	yes	—
493	#535	C₂²×C₅₃	1	1	yes	characteristic
494	#535	C₂²×C₅₃	1	1	yes	characteristic
495	#533	C₂²×C₅₃	1	2	yes	complemented
496	#532	—	2	1	no	—
497	#532	C₂²×C₅₃	1	1	yes	characteristic
498	#532	C₂²×C₅₃	1	2	yes	—
499	#532	C₂²×C₅₃	1	2	yes	complemented
500	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
501	#529	—	2	1	no	—
502	#529	C₂²×C₅₃	1	2	yes	—
503	#525	—	2	2	no	—
504	#520	C₂²×C₅₃	1	2	yes	—
505	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
506	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
507	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
508	#479	C₂²×C₅₃	1	4	yes	complemented
509	#478	C₂²×C₅₃	1	4	yes	complemented
510	#477	C₂²×C₅₃	1	2	yes	complemented
511	#476	C₂²×C₅₃	1	2	yes	—
512	#476	C₂²×C₅₃	1	2	yes	—
513	#476	C₂²×C₅₃	1	4	yes	—
514	#476	C₂²×C₅₃	1	2	yes	complemented
515	#476	C₂²×C₅₃	1	4	yes	complemented
516	#475	C₂²×C₅₃	1	2	yes	complemented
517	#475	C₂²×C₅₃	1	4	yes	complemented
518	#474	C₂²×C₅₃	1	2	yes	complemented
519	#470	C₂²×C₅₃	1	4	yes	complemented
520	#469	C₂²×C₅₃	1	4	yes	—
521	#469	C₂²×C₅₃	1	4	yes	complemented
522	#468	C₂²×C₅₃	1	4	yes	—
523	#466	C₂²×C₅₃	1	4	yes	—
524	#465	C₂²×C₅₃	1	4	yes	—
525	#465	C₂²×C₅₃	1	4	yes	complemented
526	#465	C₂²×C₅₃	1	4	yes	complemented
527	#464	C₂²×C₅₃	1	4	yes	—
528	#464	C₂²×C₅₃	1	2	yes	complemented
529	#463	C₂²×C₅₃	1	4	yes	—
530	#463	C₂²×C₅₃	1	4	yes	—
531	#463	C₂²×C₅₃	1	2	yes	complemented
532	#462	C₂²×C₅₃	1	1	yes	characteristic, complemented
533	#461	C₂²×C₅₃	1	2	yes	—
534	#461	C₂²×C₅₃	1	1	yes	characteristic, complemented
535	#460	C₂²×C₅₃	1	4	yes	—
536	#460	C₂²×C₅₃	1	1	yes	characteristic, complemented
537	#459	C₂²×C₅₃	1	2	yes	—
538	#459	C₂²×C₅₃	1	4	yes	—
539	#459	C₂²×C₅₃	1	1	yes	characteristic, complemented
540	#428	C₂²×C₅₃	1	2	yes	—
541	C₂×#106	C₂²×C₅₃	1	2	yes	—
542	#424	C₂²×C₅₃	1	2	yes	—
543	#424	C₂²×C₅₃	1	4	yes	complemented
544	#423	C₂²×C₅₃	1	1	yes	characteristic
545	#423	C₂²×C₅₃	1	2	yes	complemented
546	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
547	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
548	#417	C₂²×C₅₃	1	2	yes	—
549	#417	C₂²×C₅₃	1	2	yes	—
550	#416	C₂²×C₅₃	1	2	yes	—
551	C₂×#102	C₂²×C₅₃	1	2	yes	—
552	C₅₃	#60929	1	1	yes	characteristic, direct product factor, Sylow
553	C₂×#41	—	2	2	no	—
554	C₂×#41	—	2	2	no	—
555	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
556	C₂×#41	C₂³×C₅₃	1	2	yes	—
557	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
558	C₂²×Q₈	—	2	1	no	—
559	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
560	C₂³×C₄	—	2	1	no	—
561	AGL(1,ℤ/8)	—	4	4	no	—
562	#135	—	4	4	no	—
563	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
564	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
565	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
566	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
567	C₂×D₈	C₂³×C₅₃	1	2	yes	—
568	C₂×D₈	C₂³×C₅₃	1	4	yes	—
569	#131	—	4	4	no	—
570	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
571	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
572	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
573	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
574	C₄⋊Q₈	—	2	1	no	—
575	C₄⋊Q₈	—	2	1	no	—
576	C₄⋊Q₈	—	2	1	no	—
577	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
578	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
579	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
580	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
581	#127	—	2	1	no	—
582	#127	C₂³×C₅₃	1	1	yes	characteristic
583	#127	C₂³×C₅₃	1	2	yes	—
584	#126	—	2	2	no	—
585	#126	—	2	4	no	—
586	#126	—	2	2	no	—
587	#125	—	2	1	no	—
588	#125	—	2	2	no	—
589	#125	C₂³×C₅₃	1	2	yes	—
590	#124	—	2	1	no	—
591	#124	—	2	2	no	—
592	#124	—	2	4	no	—
593	#124	—	2	2	no	—
594	#124	—	2	2	no	—
595	#124	C₂³×C₅₃	1	2	yes	—
596	#124	C₂³×C₅₃	1	2	yes	—
597	#123	—	2	2	no	—
598	#123	—	2	2	no	—
599	#123	—	2	2	no	—
600	#123	—	2	2	no	—
601	#123	—	2	4	no	—
602	C₂²⋊Q₈	—	2	1	no	—
603	C₂²⋊Q₈	—	2	2	no	—
604	C₂²⋊Q₈	—	2	2	no	—
605	C₂²⋊Q₈	—	2	2	no	—
606	C₂²⋊Q₈	—	2	4	no	—
607	C₂²⋊Q₈	—	2	2	no	—
608	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
609	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
610	#121	—	2	2	no	—
611	#121	—	2	1	no	—
612	#121	—	2	2	no	—
613	#121	—	2	4	no	—
614	#121	—	2	2	no	—
615	#121	—	2	1	no	—
616	#121	—	2	1	no	—
617	#121	—	2	2	no	—
618	#121	—	2	1	no	—
619	#121	—	2	1	no	—
620	#121	—	2	2	no	—
621	#121	—	2	2	no	—
622	#121	C₂³×C₅₃	1	2	yes	—
623	#121	C₂³×C₅₃	1	2	yes	—
624	#121	C₂³×C₅₃	1	2	yes	—
625	#121	C₂³×C₅₃	1	4	yes	—
626	#120	—	2	2	no	—
627	#120	—	2	2	no	—
628	C₄×Q₈	—	2	2	no	—
629	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
630	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
631	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
632	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
633	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
634	C₄×D₄	—	2	1	no	—
635	C₄×D₄	—	2	1	no	—
636	C₄×D₄	—	2	2	no	—
637	C₄×D₄	—	2	4	no	—
638	C₄×D₄	—	2	1	no	—
639	C₄×D₄	—	2	1	no	—
640	C₄×D₄	—	2	2	no	—
641	C₄×D₄	—	2	1	no	—
642	C₄×D₄	—	2	1	no	—
643	C₄×D₄	—	2	2	no	—
644	C₄×D₄	D₄×C₅₃	1	2	yes	—
645	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
646	C₄×D₄	D₄×C₅₃	1	2	yes	—
647	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
648	C₄×D₄	C₂³×C₅₃	1	2	yes	—
649	C₄×D₄	C₂³×C₅₃	1	2	yes	—
650	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
651	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
652	C₄×D₄	C₂³×C₅₃	1	4	yes	—
653	#117	—	2	2	no	—
654	#117	—	2	2	no	—
655	#117	—	2	1	no	—
656	#117	—	2	2	no	—
657	#117	C₂³×C₅₃	1	1	yes	characteristic
658	#117	C₂³×C₅₃	1	1	yes	characteristic
659	#117	C₂³×C₅₃	1	2	yes	—
660	C₂×C₄⋊C₄	—	2	1	no	—
661	C₂×C₄⋊C₄	—	2	2	no	—
662	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
663	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
664	C₂×C₂²⋊C₄	—	2	1	no	—
665	C₂×C₄²	—	2	1	no	—
666	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
667	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
668	#107	C₂³×C₅₃	1	4	yes	—
669	#106	C₂³×C₅₃	1	4	yes	—
670	#106	C₂³×C₅₃	1	4	yes	—
671	#106	C₂³×C₅₃	1	4	yes	—
672	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
673	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
674	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
675	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
676	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
677	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
678	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
679	#103	C₂³×C₅₃	1	2	yes	—
680	#103	C₂³×C₅₃	1	2	yes	—
681	#103	C₂³×C₅₃	1	4	yes	—
682	#103	C₂³×C₅₃	1	4	yes	—
683	#102	C₂³×C₅₃	1	4	yes	—
684	#102	C₂³×C₅₃	1	4	yes	—
685	#102	C₂³×C₅₃	1	2	yes	—
686	#102	C₂³×C₅₃	1	2	yes	—
687	#102	C₂³×C₅₃	1	4	yes	—
688	#102	C₂³×C₅₃	1	4	yes	—
689	C₂⁴	—	2	1	no	—
690	#41	—	2	2	no	—
691	#41	—	2	2	no	—
692	#41	—	2	2	no	—
693	#41	—	2	4	no	—
694	#41	—	2	2	no	—
695	#41	—	2	4	no	—
696	#41	—	8	4	no	—
697	#41	—	8	4	no	—
698	C₂×Q₈	—	2	2	no	—
699	C₂×Q₈	—	2	2	no	—
700	C₂×Q₈	—	2	2	no	—
701	C₂×Q₈	—	8	2	no	—
702	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
703	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
704	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
705	C₂×D₄	—	2	1	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	4	no	—
711	C₂×D₄	—	2	1	no	—
712	C₂×D₄	—	2	1	no	—
713	C₂×D₄	—	2	1	no	—
714	C₂×D₄	—	2	2	no	—
715	C₂×D₄	—	2	2	no	—
716	C₂×D₄	—	2	2	no	—
717	C₂×D₄	—	2	2	no	—
718	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
719	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
720	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
721	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
722	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
723	C₂²×C₄	—	2	1	no	—
724	C₂²×C₄	—	2	2	no	—
725	C₂²×C₄	—	2	1	no	—
726	C₂²×C₄	—	2	1	no	—
727	C₂²×C₄	—	2	2	no	—
728	C₂²×C₄	—	2	2	no	—
729	C₂²×C₄	—	2	1	no	—
730	C₂²×C₄	—	2	4	no	—
731	C₂²×C₄	—	2	2	no	—
732	C₂²×C₄	—	8	2	no	—
733	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
734	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
735	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
736	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
737	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
738	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
739	Q₁₆	—	4	2	no	—
740	SD₁₆	—	4	2	no	—
741	SD₁₆	—	4	2	no	—
742	SD₁₆	—	4	4	no	—
743	D₈	—	4	2	no	—
744	D₈	—	4	4	no	—
745	M₁₆	—	2	4	no	—
746	M₁₆	—	4	4	no	—
747	C₂×C₈	—	2	4	no	—
748	C₂×C₈	—	4	4	no	—
749	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
750	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
751	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
752	C₄⋊C₄	—	2	2	no	—
753	C₄⋊C₄	—	2	2	no	—
754	C₄⋊C₄	—	2	4	no	—
755	C₄⋊C₄	—	2	2	no	—
756	C₄⋊C₄	—	2	1	no	—
757	C₄⋊C₄	—	2	1	no	—
758	C₄⋊C₄	—	2	1	no	—
759	C₄⋊C₄	—	2	1	no	—
760	C₄⋊C₄	—	2	2	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	1	no	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	2	no	—
765	C₄⋊C₄	—	2	4	no	—
766	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
767	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
768	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
769	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
770	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
771	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
772	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
773	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
774	C₂²⋊C₄	—	2	4	no	—
775	C₂²⋊C₄	—	2	1	no	—
776	C₂²⋊C₄	—	2	2	no	—
777	C₂²⋊C₄	—	2	2	no	—
778	C₂²⋊C₄	—	2	2	no	—
779	C₂²⋊C₄	—	2	1	no	—
780	C₂²⋊C₄	—	2	1	no	—
781	C₂²⋊C₄	—	2	4	no	—
782	C₂²⋊C₄	—	2	2	no	—
783	C₂²⋊C₄	—	2	2	no	—
784	C₂²⋊C₄	—	2	2	no	—
785	C₂²⋊C₄	—	2	2	no	—
786	C₂²⋊C₄	—	2	4	no	—
787	C₂²⋊C₄	—	2	2	no	—
788	C₂²⋊C₄	—	2	2	no	—
789	C₂²⋊C₄	—	2	2	no	—
790	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
791	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
792	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
793	C₄²	—	2	1	no	—
794	C₄²	—	2	1	no	—
795	C₄²	—	2	1	no	—
796	C₄²	—	2	4	no	—
797	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
798	C₄²	C₂×D₄×C₅₃	1	2	yes	—
799	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
800	C₄²	C₂×D₄×C₅₃	1	2	yes	—
801	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
802	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
803	C₄²	C₂⁴×C₅₃	1	2	yes	—
804	C₂³	—	2	2	no	—
805	C₂³	—	2	2	no	—
806	C₂³	—	2	1	no	—
807	C₂³	—	4	2	no	—
808	C₂³	—	8	1	no	—
809	C₂³	—	8	1	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	1	no	—
815	Q₈	—	2	1	no	—
816	Q₈	—	2	2	no	—
817	Q₈	—	4	4	no	—
818	Q₈	—	8	2	no	—
819	Q₈	—	8	2	no	—
820	D₄	—	2	4	no	—
821	D₄	—	2	2	no	—
822	D₄	—	2	2	no	—
823	D₄	—	2	2	no	—
824	D₄	—	2	1	no	—
825	D₄	—	2	2	no	—
826	D₄	—	2	2	no	—
827	D₄	—	2	1	no	—
828	D₄	—	4	2	no	—
829	D₄	—	4	2	no	—
830	D₄	—	4	4	no	—
831	D₄	—	4	2	no	—
832	D₄	—	4	2	no	—
833	D₄	—	8	2	no	—
834	D₄	—	8	2	no	—
835	D₄	—	8	2	no	—
836	D₄	—	8	2	no	—
837	D₄	—	8	2	no	—
838	D₄	—	8	4	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	4	no	—
841	C₂×C₄	—	2	2	no	—
842	C₂×C₄	—	2	2	no	—
843	C₂×C₄	—	2	1	no	—
844	C₂×C₄	—	2	2	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	1	no	—
847	C₂×C₄	—	2	4	no	—
848	C₂×C₄	—	2	2	no	—
849	C₂×C₄	—	4	1	no	—
850	C₂×C₄	—	4	2	no	—
851	C₂×C₄	—	4	2	no	—
852	C₂×C₄	—	4	4	no	—
853	C₂×C₄	—	4	4	no	—
854	C₂×C₄	—	4	2	no	—
855	C₂×C₄	—	8	1	no	—
856	C₂×C₄	—	8	2	no	—
857	C₂×C₄	—	8	4	no	—
858	C₂×C₄	—	8	1	no	—
859	C₂×C₄	—	8	2	no	—
860	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
861	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
862	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
863	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
864	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
865	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
866	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
867	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
868	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
869	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
870	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
871	C₈	—	2	2	no	—
872	C₈	—	2	2	no	—
873	C₈	—	2	4	no	—
874	C₂²	—	2	1	no	—
875	C₂²	—	2	1	no	—
876	C₂²	—	2	2	no	—
877	C₂²	—	2	2	no	—
878	C₂²	—	2	2	no	—
879	C₂²	—	4	2	no	—
880	C₂²	—	4	1	no	—
881	C₂²	—	4	2	no	—
882	C₂²	—	4	2	no	—
883	C₂²	—	4	1	no	—
884	C₂²	—	4	2	no	—
885	C₂²	—	4	2	no	—
886	C₂²	—	4	1	no	—
887	C₂²	—	4	2	no	—
888	C₂²	—	8	4	no	—
889	C₂²	C₅₃×#537	1	2	yes	—
890	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
891	C₄	—	2	2	no	—
892	C₄	—	2	1	no	—
893	C₄	—	2	2	no	—
894	C₄	—	2	1	no	—
895	C₄	—	2	2	no	—
896	C₄	—	2	1	no	—
897	C₄	—	4	1	no	—
898	C₄	—	4	2	no	—
899	C₄	—	4	2	no	—
900	C₄	—	4	1	no	—
901	C₄	—	4	4	no	—
902	C₄	—	4	1	no	—
903	C₄	C₅₃×#537	1	2	yes	—
904	C₄	C₅₃×#577	1	2	yes	—
905	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
906	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
907	C₂	—	2	2	no	—
908	C₂	—	4	2	no	—
909	C₂	—	8	2	no	—
910	C₂	—	8	1	no	—
911	C₂	—	8	2	no	—
912	C₂	C₂×C₅₃×#537	1	1	yes	characteristic
913	C₂	C₂×C₅₃×#577	1	1	yes	characteristic
914	C₂	C₅₃×#3585	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×#3511	C₂	semidirect
2. C₅₃×#3460	C₂	semidirect
3. C₅₃×#3263	C₂	semidirect
4. C₅₃×#3257	C₂	semidirect
5. C₅₃×#3253	C₂	semidirect
6. C₅₃×#3252	C₂	semidirect
7. C₅₃×#3247	C₂	semidirect
8. C₅₃×#3246	C₂	semidirect
9. C₅₃×#3084	C₂	semidirect
10. C₅₃×#3079	C₂	semidirect
11. C₂×C₅₃×#476	C₂	semidirect
12. C₅₃×#3062	C₂	semidirect
13. C₅₃×#3060	C₂	semidirect
14. C₅₃×#3045	C₂	semidirect
15. C₅₃×#3036	C₂	semidirect
16. C₅₃×#3035	C₂	semidirect
17. C₅₃×#2906	C₂	semidirect
30. C₅₃×#542	C₂²	semidirect
38. C₅₃×#533	C₂²	semidirect
42. C₅₃×#532	C₂²	semidirect
51. C₅₃×#479	C₂²	semidirect
52. C₅₃×#478	C₂²	semidirect
53. C₅₃×#477	C₂²	semidirect
57. C₅₃×#476	C₂²	semidirect
58. C₅₃×#476	C₂²	semidirect
59. C₅₃×#475	C₂²	semidirect
60. C₅₃×#475	C₂²	semidirect
61. C₅₃×#474	C₂²	semidirect
62. C₅₃×#470	C₂²	semidirect
64. C₅₃×#469	C₂²	semidirect
68. C₅₃×#465	C₂²	semidirect
69. C₅₃×#465	C₂²	semidirect
71. C₅₃×#464	C₂²	semidirect
74. C₅₃×#463	C₂²	semidirect
75. C₅₃×#462	C₂²	semidirect
77. C₅₃×#461	C₂²	semidirect
79. C₅₃×#460	C₂²	semidirect
82. C₅₃×#459	C₂²	semidirect
86. C₅₃×#424	C₂²	semidirect
88. C₅₃×#423	C₂²	semidirect
90. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
416. #60929	C₅₃	direct (N × H)
450. #3511	C₂×C₅₃	semidirect
451. #3460	C₂×C₅₃	semidirect
452. #3263	C₂×C₅₃	semidirect
453. #3257	C₂×C₅₃	semidirect
454. #3253	C₂×C₅₃	semidirect
455. #3252	C₂×C₅₃	semidirect
456. #3247	C₂×C₅₃	semidirect
457. #3246	C₂×C₅₃	semidirect
458. #3084	C₂×C₅₃	semidirect
459. #3079	C₂×C₅₃	semidirect
460. C₂×#476	C₂×C₅₃	semidirect
461. #3062	C₂×C₅₃	semidirect
462. #3060	C₂×C₅₃	semidirect
463. #3045	C₂×C₅₃	semidirect
464. #3036	C₂×C₅₃	semidirect
465. #3035	C₂×C₅₃	semidirect
466. #2906	C₂×C₅₃	semidirect
487. #542	C₂²×C₅₃	semidirect
495. #533	C₂²×C₅₃	semidirect
499. #532	C₂²×C₅₃	semidirect
508. #479	C₂²×C₅₃	semidirect
509. #478	C₂²×C₅₃	semidirect
510. #477	C₂²×C₅₃	semidirect
514. #476	C₂²×C₅₃	semidirect
515. #476	C₂²×C₅₃	semidirect
516. #475	C₂²×C₅₃	semidirect
517. #475	C₂²×C₅₃	semidirect
518. #474	C₂²×C₅₃	semidirect
519. #470	C₂²×C₅₃	semidirect
521. #469	C₂²×C₅₃	semidirect
525. #465	C₂²×C₅₃	semidirect
526. #465	C₂²×C₅₃	semidirect
528. #464	C₂²×C₅₃	semidirect
531. #463	C₂²×C₅₃	semidirect
532. #462	C₂²×C₅₃	semidirect
534. #461	C₂²×C₅₃	semidirect
536. #460	C₂²×C₅₃	semidirect
539. #459	C₂²×C₅₃	semidirect
543. #424	C₂²×C₅₃	semidirect
545. #423	C₂²×C₅₃	semidirect
547. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
552. C₅₃	#60929	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×#3511	2
C₅₃×#3460	2
C₅₃×#3263	2
C₅₃×#3257	2
C₅₃×#3253	2
C₅₃×#3252	2
C₅₃×#3247	2
C₅₃×#3246	2
C₅₃×#3084	2
C₅₃×#3079	2
C₂×C₅₃×#476	2
C₅₃×#3062	2
C₅₃×#3060	2
C₅₃×#3045	2
C₅₃×#3036	2
C₅₃×#3035	2
C₅₃×#2906	2
#60929	53

Subgroups — C53×#60929 #27268441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60929 #27268441