Subgroups — C₅₃×#61161 #27268673

Order 13568.

2378 conjugacy classes of subgroups.

The subgroup lattice has 992 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3332	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3330	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3308	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3306	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3298	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3294	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#3092	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#3055	C₂	1	2	yes	maximal, complemented
13	C₅₃×#3036	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#3006	C₂	1	2	yes	maximal, complemented
15	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	D₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₅₃×#417	C₂	1	2	yes	maximal, complemented
18	C₂²×#41×C₅₃	—	2	1	no	—
19	C₂²×#41×C₅₃	C₂²	1	2	yes	—
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	complemented
22	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	complemented
23	C₅₃×#550	C₂²	1	1	yes	characteristic, complemented
24	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₅₃×#548	C₂²	1	2	yes	complemented
26	C₅₃×#547	—	2	1	no	—
27	C₅₃×#547	—	2	2	no	—
28	C₅₃×#547	—	2	1	no	—
29	C₅₃×#547	—	2	1	no	—
30	C₅₃×#547	C₂²	1	2	yes	—
31	C₅₃×#547	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#547	C₂²	1	4	yes	complemented
33	C₅₃×#547	C₂²	1	2	yes	complemented
34	C₅₃×#546	—	2	2	no	—
35	C₅₃×#546	C₂²	1	2	yes	complemented
36	D₄²×C₅₃	—	2	1	no	—
37	D₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#532	—	2	1	no	—
39	C₅₃×#532	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#525	—	2	2	no	—
42	C₅₃×#525	C₂²	1	2	yes	—
43	C₂×#123×C₅₃	—	2	2	no	—
44	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
45	C₂×#121×C₅₃	C₂²	1	2	yes	complemented
46	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×D₄×C₅₃	—	2	1	no	—
48	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
51	C₅₃×#489	C₂²	1	4	yes	complemented
52	C₅₃×#487	C₂²	1	2	yes	complemented
53	C₅₃×#486	C₂²	1	2	yes	complemented
54	C₅₃×#484	C₂²	1	4	yes	complemented
55	C₅₃×#483	C₂²	1	4	yes	complemented
56	C₅₃×#481	C₂²	1	4	yes	complemented
57	C₅₃×#480	C₂²	1	4	yes	complemented
58	C₅₃×#477	C₂²	1	2	yes	complemented
59	C₅₃×#476	C₂²	1	2	yes	complemented
60	C₅₃×#475	C₂²	1	2	yes	complemented
61	C₅₃×#474	C₂²	1	2	yes	complemented
62	C₅₃×#469	C₂²	1	4	yes	complemented
63	C₅₃×#468	C₂²	1	4	yes	complemented
64	C₅₃×#466	C₂²	1	4	yes	complemented
65	C₅₃×#465	C₂²	1	4	yes	complemented
66	C₅₃×#463	C₂²	1	4	yes	complemented
67	C₅₃×#461	C₂²	1	2	yes	complemented
68	C₅₃×#459	C₂²	1	2	yes	complemented
69	C₅₃×#449	C₂²	1	4	yes	complemented
70	C₅₃×#449	C₂²	1	4	yes	complemented
71	C₅₃×#448	C₂²	1	4	yes	complemented
72	C₅₃×#447	C₂²	1	4	yes	complemented
73	C₅₃×#442	C₂²	1	2	yes	complemented
74	C₅₃×#440	C₂²	1	2	yes	complemented
75	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
76	C₄×D₈×C₅₃	C₂²	1	2	yes	complemented
77	C₅₃×#436	C₂²	1	2	yes	complemented
78	C₅₃×#435	C₂²	1	4	yes	complemented
79	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
80	C₅₃×#428	C₂²	1	1	yes	characteristic, complemented
81	C₅₃×#428	C₂²	1	2	yes	complemented
82	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
83	C₅₃×#418	C₂²	1	1	yes	characteristic, complemented
84	C₅₃×#417	C₂²	1	1	yes	characteristic
85	C₅₃×#417	C₂²	1	2	yes	—
86	C₅₃×#417	C₂²	1	2	yes	complemented
87	C₅₃×#417	C₂²	1	2	yes	complemented
88	C₅₃×#417	C₂²	1	4	yes	complemented
89	C₅₃×#417	C₂²	1	4	yes	complemented
90	C₅₃×#417	C₂²	1	2	yes	complemented
91	C₂×#103×C₅₃	C₂²	1	4	yes	complemented
92	C₂×#102×C₅₃	C₂²	1	4	yes	complemented
93	C₅₃×#407	C₂²	1	2	yes	complemented
94	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
95	C₂×#41×C₅₃	—	2	2	no	—
96	C₂×#41×C₅₃	—	2	4	no	—
97	C₂×#41×C₅₃	—	2	2	no	—
98	C₂×#41×C₅₃	—	2	2	no	—
99	C₂×#41×C₅₃	—	2	2	no	—
100	C₂×#41×C₅₃	—	2	8	no	—
101	C₂×#41×C₅₃	—	4	4	no	—
102	C₂×#41×C₅₃	—	4	8	no	—
103	C₂×#41×C₅₃	C₂³	1	2	yes	—
104	C₂×#41×C₅₃	C₂³	1	2	yes	—
105	C₂×#41×C₅₃	C₂³	1	2	yes	—
106	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²×Q₈×C₅₃	—	2	1	no	—
108	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
109	C₂²×D₄×C₅₃	—	2	2	no	—
110	C₂²×D₄×C₅₃	—	2	2	no	—
111	C₂²×D₄×C₅₃	—	2	1	no	—
112	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
113	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
115	C₂³×C₄×C₅₃	—	2	2	no	—
116	C₂³×C₄×C₅₃	—	2	2	no	—
117	C₂³×C₄×C₅₃	—	2	1	no	—
118	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
119	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
120	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
121	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
122	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
123	C₂×D₈×C₅₃	C₂³	1	2	yes	—
124	C₂×D₈×C₅₃	C₂³	1	4	yes	—
125	C₂×M₁₆×C₅₃	—	4	4	no	—
126	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
128	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
130	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
131	C₄⋊Q₈×C₅₃	—	2	1	no	—
132	C₄⋊Q₈×C₅₃	—	2	1	no	—
133	C₄⋊Q₈×C₅₃	—	2	1	no	—
134	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
135	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
136	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
137	#127×C₅₃	—	2	1	no	—
138	#127×C₅₃	C₂³	1	1	yes	characteristic
139	#124×C₅₃	—	2	2	no	—
140	#124×C₅₃	C₂³	1	2	yes	—
141	#123×C₅₃	—	2	2	no	—
142	#123×C₅₃	—	2	4	no	—
143	#123×C₅₃	—	2	4	no	—
144	#123×C₅₃	—	2	4	no	—
145	#123×C₅₃	—	2	4	no	—
146	#123×C₅₃	—	2	2	no	—
147	#123×C₅₃	—	2	2	no	—
148	C₂²⋊Q₈×C₅₃	—	2	2	no	—
149	C₂²⋊Q₈×C₅₃	—	2	2	no	—
150	C₂²⋊Q₈×C₅₃	—	2	4	no	—
151	C₂²⋊Q₈×C₅₃	—	2	2	no	—
152	C₂²⋊Q₈×C₅₃	—	2	2	no	—
153	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
154	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
155	#121×C₅₃	—	2	1	no	—
156	#121×C₅₃	—	2	2	no	—
157	#121×C₅₃	—	2	2	no	—
158	#121×C₅₃	—	2	2	no	—
159	#121×C₅₃	—	2	1	no	—
160	#121×C₅₃	—	2	2	no	—
161	#121×C₅₃	—	2	2	no	—
162	#121×C₅₃	—	2	1	no	—
163	#121×C₅₃	—	2	1	no	—
164	#121×C₅₃	C₂³	1	4	yes	—
165	#121×C₅₃	C₂³	1	4	yes	—
166	#121×C₅₃	C₂³	1	2	yes	—
167	#121×C₅₃	C₂³	1	2	yes	—
168	#120×C₅₃	—	2	2	no	—
169	#120×C₅₃	—	2	2	no	—
170	#120×C₅₃	—	2	2	no	—
171	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
172	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
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	1	no	—
177	C₄×D₄×C₅₃	—	2	1	no	—
178	C₄×D₄×C₅₃	—	2	1	no	—
179	C₄×D₄×C₅₃	—	2	1	no	—
180	C₄×D₄×C₅₃	—	2	2	no	—
181	C₄×D₄×C₅₃	—	2	1	no	—
182	C₄×D₄×C₅₃	—	2	1	no	—
183	C₄×D₄×C₅₃	—	2	2	no	—
184	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
185	C₄×D₄×C₅₃	D₄	1	2	yes	—
186	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
187	C₄×D₄×C₅₃	C₂³	1	2	yes	—
188	C₄×D₄×C₅₃	C₂³	1	2	yes	—
189	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
190	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
191	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
192	C₄×D₄×C₅₃	C₂³	1	2	yes	—
193	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
194	#117×C₅₃	—	2	1	no	—
195	#117×C₅₃	C₂³	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
197	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
198	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
199	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
200	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
201	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
202	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
203	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
204	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
205	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
206	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
207	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
208	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
209	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
210	C₂×C₄²×C₅₃	—	2	1	no	—
211	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
212	#107×C₅₃	C₂³	1	2	yes	—
213	#107×C₅₃	C₂³	1	4	yes	—
214	#106×C₅₃	C₂³	1	2	yes	—
215	#106×C₅₃	C₂³	1	4	yes	—
216	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
217	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
218	#103×C₅₃	C₂³	1	4	yes	—
219	#103×C₅₃	C₂³	1	4	yes	—
220	#103×C₅₃	C₂³	1	2	yes	—
221	#103×C₅₃	C₂³	1	4	yes	—
222	#103×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	#102×C₅₃	C₂³	1	4	yes	—
227	#102×C₅₃	C₂³	1	4	yes	—
228	#102×C₅₃	C₂³	1	2	yes	—
229	#102×C₅₃	C₂³	1	2	yes	—
230	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
231	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
232	#97×C₅₃	C₂³	1	2	yes	—
233	C₄×C₈×C₅₃	C₂³	1	2	yes	—
234	C₂⁴×C₅₃	—	2	1	no	—
235	C₂⁴×C₅₃	—	2	2	no	—
236	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
237	#41×C₅₃	—	2	2	no	—
238	#41×C₅₃	—	2	4	no	—
239	#41×C₅₃	—	2	4	no	—
240	#41×C₅₃	—	2	4	no	—
241	#41×C₅₃	—	2	4	no	—
242	#41×C₅₃	—	4	8	no	—
243	#41×C₅₃	—	4	2	no	—
244	#41×C₅₃	—	4	2	no	—
245	#41×C₅₃	—	4	8	no	—
246	#41×C₅₃	—	4	16	no	—
247	#41×C₅₃	—	4	8	no	—
248	#41×C₅₃	—	4	8	no	—
249	#41×C₅₃	#41	1	4	yes	complemented
250	C₂×Q₈×C₅₃	—	2	2	no	—
251	C₂×Q₈×C₅₃	—	2	2	no	—
252	C₂×Q₈×C₅₃	—	2	2	no	—
253	C₂×Q₈×C₅₃	—	2	4	no	—
254	C₂×Q₈×C₅₃	—	4	2	no	—
255	C₂×Q₈×C₅₃	—	4	4	no	—
256	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
257	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
258	C₂×D₄×C₅₃	—	2	1	no	—
259	C₂×D₄×C₅₃	—	2	4	no	—
260	C₂×D₄×C₅₃	—	2	4	no	—
261	C₂×D₄×C₅₃	—	2	2	no	—
262	C₂×D₄×C₅₃	—	2	2	no	—
263	C₂×D₄×C₅₃	—	2	2	no	—
264	C₂×D₄×C₅₃	—	2	2	no	—
265	C₂×D₄×C₅₃	—	2	4	no	—
266	C₂×D₄×C₅₃	—	2	4	no	—
267	C₂×D₄×C₅₃	—	2	2	no	—
268	C₂×D₄×C₅₃	—	2	1	no	—
269	C₂×D₄×C₅₃	—	2	1	no	—
270	C₂×D₄×C₅₃	—	2	2	no	—
271	C₂×D₄×C₅₃	—	2	1	no	—
272	C₂×D₄×C₅₃	—	2	2	no	—
273	C₂×D₄×C₅₃	—	2	2	no	—
274	C₂×D₄×C₅₃	—	4	2	no	—
275	C₂×D₄×C₅₃	—	4	8	no	—
276	C₂×D₄×C₅₃	—	4	8	no	—
277	C₂×D₄×C₅₃	—	4	4	no	—
278	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
279	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
280	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
281	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
282	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
283	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
284	C₂²×C₄×C₅₃	—	2	1	no	—
285	C₂²×C₄×C₅₃	—	2	4	no	—
286	C₂²×C₄×C₅₃	—	2	4	no	—
287	C₂²×C₄×C₅₃	—	2	1	no	—
288	C₂²×C₄×C₅₃	—	2	2	no	—
289	C₂²×C₄×C₅₃	—	2	4	no	—
290	C₂²×C₄×C₅₃	—	2	4	no	—
291	C₂²×C₄×C₅₃	—	2	2	no	—
292	C₂²×C₄×C₅₃	—	2	1	no	—
293	C₂²×C₄×C₅₃	—	2	2	no	—
294	C₂²×C₄×C₅₃	—	2	2	no	—
295	C₂²×C₄×C₅₃	—	2	2	no	—
296	C₂²×C₄×C₅₃	—	2	2	no	—
297	C₂²×C₄×C₅₃	—	2	2	no	—
298	C₂²×C₄×C₅₃	—	2	1	no	—
299	C₂²×C₄×C₅₃	—	4	8	no	—
300	C₂²×C₄×C₅₃	—	4	4	no	—
301	C₂²×C₄×C₅₃	—	4	8	no	—
302	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
303	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
304	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
305	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
306	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
307	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
308	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
309	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
310	SD₁₆×C₅₃	—	2	4	no	—
311	SD₁₆×C₅₃	—	4	4	no	—
312	D₈×C₅₃	—	2	4	no	—
313	D₈×C₅₃	—	4	4	no	—
314	M₁₆×C₅₃	—	2	2	no	—
315	M₁₆×C₅₃	—	2	2	no	—
316	M₁₆×C₅₃	—	2	4	no	—
317	M₁₆×C₅₃	—	4	4	no	—
318	C₂×C₈×C₅₃	—	4	4	no	—
319	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
320	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
321	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
322	C₄⋊C₄×C₅₃	—	2	1	no	—
323	C₄⋊C₄×C₅₃	—	2	2	no	—
324	C₄⋊C₄×C₅₃	—	2	4	no	—
325	C₄⋊C₄×C₅₃	—	2	4	no	—
326	C₄⋊C₄×C₅₃	—	2	1	no	—
327	C₄⋊C₄×C₅₃	—	2	2	no	—
328	C₄⋊C₄×C₅₃	—	2	2	no	—
329	C₄⋊C₄×C₅₃	—	2	2	no	—
330	C₄⋊C₄×C₅₃	—	2	1	no	—
331	C₄⋊C₄×C₅₃	—	2	1	no	—
332	C₄⋊C₄×C₅₃	—	2	1	no	—
333	C₄⋊C₄×C₅₃	—	2	2	no	—
334	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
335	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
336	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
337	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
338	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
339	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
340	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
341	C₂²⋊C₄×C₅₃	—	2	2	no	—
342	C₂²⋊C₄×C₅₃	—	2	1	no	—
343	C₂²⋊C₄×C₅₃	—	2	2	no	—
344	C₂²⋊C₄×C₅₃	—	2	2	no	—
345	C₂²⋊C₄×C₅₃	—	2	2	no	—
346	C₂²⋊C₄×C₅₃	—	2	2	no	—
347	C₂²⋊C₄×C₅₃	—	2	2	no	—
348	C₂²⋊C₄×C₅₃	—	2	4	no	—
349	C₂²⋊C₄×C₅₃	—	2	4	no	—
350	C₂²⋊C₄×C₅₃	—	2	1	no	—
351	C₂²⋊C₄×C₅₃	—	2	1	no	—
352	C₂²⋊C₄×C₅₃	—	2	2	no	—
353	C₂²⋊C₄×C₅₃	—	2	2	no	—
354	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
355	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
356	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
357	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
358	C₄²×C₅₃	—	2	1	no	—
359	C₄²×C₅₃	—	2	1	no	—
360	C₄²×C₅₃	—	2	1	no	—
361	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
362	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
363	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
364	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
365	C₂³×C₅₃	—	2	4	no	—
366	C₂³×C₅₃	—	2	4	no	—
367	C₂³×C₅₃	—	2	1	no	—
368	C₂³×C₅₃	—	2	2	no	—
369	C₂³×C₅₃	—	2	2	no	—
370	C₂³×C₅₃	—	4	2	no	—
371	C₂³×C₅₃	—	4	4	no	—
372	C₂³×C₅₃	—	4	2	no	—
373	C₂³×C₅₃	—	4	2	no	—
374	C₂³×C₅₃	—	4	2	no	—
375	C₂³×C₅₃	—	4	4	no	—
376	C₂³×C₅₃	—	8	2	no	—
377	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
378	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
379	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
380	Q₈×C₅₃	—	2	1	no	—
381	Q₈×C₅₃	—	2	2	no	—
382	Q₈×C₅₃	—	2	2	no	—
383	Q₈×C₅₃	—	4	4	no	—
384	Q₈×C₅₃	—	4	4	no	—
385	Q₈×C₅₃	—	4	4	no	—
386	Q₈×C₅₃	C₂×#41	1	2	yes	complemented
387	D₄×C₅₃	—	2	4	no	—
388	D₄×C₅₃	—	2	4	no	—
389	D₄×C₅₃	—	2	2	no	—
390	D₄×C₅₃	—	2	2	no	—
391	D₄×C₅₃	—	2	2	no	—
392	D₄×C₅₃	—	2	1	no	—
393	D₄×C₅₃	—	2	2	no	—
394	D₄×C₅₃	—	2	2	no	—
395	D₄×C₅₃	—	4	4	no	—
396	D₄×C₅₃	—	4	4	no	—
397	D₄×C₅₃	—	4	4	no	—
398	D₄×C₅₃	—	4	8	no	—
399	D₄×C₅₃	—	4	4	no	—
400	D₄×C₅₃	—	4	4	no	—
401	D₄×C₅₃	—	4	8	no	—
402	D₄×C₅₃	—	4	2	no	—
403	D₄×C₅₃	—	4	2	no	—
404	D₄×C₅₃	—	8	1	no	—
405	D₄×C₅₃	—	8	2	no	—
406	D₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
407	D₄×C₅₃	C₂×#41	1	2	yes	complemented
408	C₂×C₄×C₅₃	—	2	4	no	—
409	C₂×C₄×C₅₃	—	2	1	no	—
410	C₂×C₄×C₅₃	—	2	4	no	—
411	C₂×C₄×C₅₃	—	2	4	no	—
412	C₂×C₄×C₅₃	—	2	4	no	—
413	C₂×C₄×C₅₃	—	2	2	no	—
414	C₂×C₄×C₅₃	—	2	2	no	—
415	C₂×C₄×C₅₃	—	2	2	no	—
416	C₂×C₄×C₅₃	—	2	2	no	—
417	C₂×C₄×C₅₃	—	2	2	no	—
418	C₂×C₄×C₅₃	—	2	2	no	—
419	C₂×C₄×C₅₃	—	2	4	no	—
420	C₂×C₄×C₅₃	—	2	2	no	—
421	C₂×C₄×C₅₃	—	2	1	no	—
422	C₂×C₄×C₅₃	—	2	1	no	—
423	C₂×C₄×C₅₃	—	4	1	no	—
424	C₂×C₄×C₅₃	—	4	4	no	—
425	C₂×C₄×C₅₃	—	4	4	no	—
426	C₂×C₄×C₅₃	—	4	4	no	—
427	C₂×C₄×C₅₃	—	4	4	no	—
428	C₂×C₄×C₅₃	—	4	2	no	—
429	C₂×C₄×C₅₃	—	4	4	no	—
430	C₂×C₄×C₅₃	—	4	8	no	—
431	C₂×C₄×C₅₃	—	4	8	no	—
432	C₂×C₄×C₅₃	—	4	4	no	—
433	C₂×C₄×C₅₃	—	4	4	no	—
434	C₂×C₄×C₅₃	—	4	2	no	—
435	C₂×C₄×C₅₃	—	8	1	no	—
436	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
437	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
438	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
439	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
440	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
441	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
442	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
443	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
444	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
445	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
446	C₈×C₅₃	—	2	2	no	—
447	C₈×C₅₃	—	2	2	no	—
448	C₈×C₅₃	—	2	4	no	—
449	#61161	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
450	C₂²×C₅₃	—	2	1	no	—
451	C₂²×C₅₃	—	2	2	no	—
452	C₂²×C₅₃	—	2	2	no	—
453	C₂²×C₅₃	—	2	1	no	—
454	C₂²×C₅₃	—	2	2	no	—
455	C₂²×C₅₃	—	2	2	no	—
456	C₂²×C₅₃	—	2	2	no	—
457	C₂²×C₅₃	—	2	2	no	—
458	C₂²×C₅₃	—	4	8	no	—
459	C₂²×C₅₃	—	4	1	no	—
460	C₂²×C₅₃	—	4	1	no	—
461	C₂²×C₅₃	—	4	2	no	—
462	C₂²×C₅₃	—	4	2	no	—
463	C₂²×C₅₃	—	4	2	no	—
464	C₂²×C₅₃	—	4	2	no	—
465	C₂²×C₅₃	—	4	2	no	—
466	C₂²×C₅₃	—	4	2	no	—
467	C₂²×C₅₃	—	8	8	no	—
468	C₂²×C₅₃	—	8	4	no	—
469	C₂²×C₅₃	#546	1	2	yes	—
470	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
471	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
472	C₄×C₅₃	—	2	1	no	—
473	C₄×C₅₃	—	2	1	no	—
474	C₄×C₅₃	—	2	2	no	—
475	C₄×C₅₃	—	2	2	no	—
476	C₄×C₅₃	—	2	2	no	—
477	C₄×C₅₃	—	2	1	no	—
478	C₄×C₅₃	—	4	1	no	—
479	C₄×C₅₃	—	4	2	no	—
480	C₄×C₅₃	—	4	2	no	—
481	C₄×C₅₃	—	4	4	no	—
482	C₄×C₅₃	—	4	1	no	—
483	C₄×C₅₃	#546	1	2	yes	—
484	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
485	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
486	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
487	D₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
488	C₂×#547	C₂×C₅₃	1	1	yes	characteristic, complemented
489	#3332	C₂×C₅₃	1	2	yes	complemented
490	#3330	C₂×C₅₃	1	2	yes	complemented
491	#3308	C₂×C₅₃	1	4	yes	complemented
492	#3306	C₂×C₅₃	1	4	yes	complemented
493	#3298	C₂×C₅₃	1	2	yes	complemented
494	#3294	C₂×C₅₃	1	2	yes	complemented
495	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
496	#3092	C₂×C₅₃	1	2	yes	complemented
497	#3079	C₂×C₅₃	1	1	yes	characteristic, complemented
498	#3055	C₂×C₅₃	1	2	yes	complemented
499	#3036	C₂×C₅₃	1	1	yes	characteristic, complemented
500	#3006	C₂×C₅₃	1	2	yes	complemented
501	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	1	yes	characteristic, complemented
502	D₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
503	C₂×#417	C₂×C₅₃	1	2	yes	complemented
504	C₂×C₅₃	—	2	2	no	—
505	C₂×C₅₃	—	2	4	no	—
506	C₂×C₅₃	—	4	2	no	—
507	C₂×C₅₃	—	4	2	no	—
508	C₂×C₅₃	—	8	2	no	—
509	C₂×C₅₃	—	8	2	no	—
510	C₂×C₅₃	C₂×#546	1	1	yes	characteristic
511	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
512	C₂×C₅₃	#3586	1	1	yes	characteristic
513	C₂²×#41	—	2	1	no	—
514	C₂²×#41	C₂²×C₅₃	1	2	yes	—
515	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic
516	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	complemented
517	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	complemented
518	#550	C₂²×C₅₃	1	1	yes	characteristic, complemented
519	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
520	#548	C₂²×C₅₃	1	2	yes	complemented
521	#547	—	2	1	no	—
522	#547	—	2	2	no	—
523	#547	—	2	1	no	—
524	#547	—	2	1	no	—
525	#547	C₂²×C₅₃	1	2	yes	—
526	#547	C₂²×C₅₃	1	1	yes	characteristic, complemented
527	#547	C₂²×C₅₃	1	2	yes	complemented
528	#547	C₂²×C₅₃	1	4	yes	complemented
529	#546	—	2	2	no	—
530	#546	C₂²×C₅₃	1	2	yes	complemented
531	D₄²	—	2	1	no	—
532	D₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
533	#532	—	2	1	no	—
534	#532	C₂²×C₅₃	1	1	yes	characteristic
535	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
536	#525	—	2	2	no	—
537	#525	C₂²×C₅₃	1	2	yes	—
538	C₂×#123	—	2	2	no	—
539	C₂×C₂²⋊Q₈	—	2	1	no	—
540	C₂×#121	C₂²×C₅₃	1	2	yes	complemented
541	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
542	C₂×C₄×D₄	—	2	1	no	—
543	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
544	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
545	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
546	#489	C₂²×C₅₃	1	4	yes	complemented
547	#487	C₂²×C₅₃	1	2	yes	complemented
548	#486	C₂²×C₅₃	1	2	yes	complemented
549	#484	C₂²×C₅₃	1	4	yes	complemented
550	#483	C₂²×C₅₃	1	4	yes	complemented
551	#481	C₂²×C₅₃	1	4	yes	complemented
552	#480	C₂²×C₅₃	1	4	yes	complemented
553	#477	C₂²×C₅₃	1	2	yes	complemented
554	#476	C₂²×C₅₃	1	2	yes	complemented
555	#475	C₂²×C₅₃	1	2	yes	complemented
556	#474	C₂²×C₅₃	1	2	yes	complemented
557	#469	C₂²×C₅₃	1	4	yes	complemented
558	#468	C₂²×C₅₃	1	4	yes	complemented
559	#466	C₂²×C₅₃	1	4	yes	complemented
560	#465	C₂²×C₅₃	1	4	yes	complemented
561	#463	C₂²×C₅₃	1	4	yes	complemented
562	#461	C₂²×C₅₃	1	2	yes	complemented
563	#459	C₂²×C₅₃	1	2	yes	complemented
564	#449	C₂²×C₅₃	1	4	yes	complemented
565	#449	C₂²×C₅₃	1	4	yes	complemented
566	#448	C₂²×C₅₃	1	4	yes	complemented
567	#447	C₂²×C₅₃	1	4	yes	complemented
568	#442	C₂²×C₅₃	1	2	yes	complemented
569	#440	C₂²×C₅₃	1	2	yes	complemented
570	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
571	C₄×D₈	C₂²×C₅₃	1	2	yes	complemented
572	#436	C₂²×C₅₃	1	2	yes	complemented
573	#435	C₂²×C₅₃	1	4	yes	complemented
574	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
575	#428	C₂²×C₅₃	1	2	yes	complemented
576	#428	C₂²×C₅₃	1	1	yes	characteristic, complemented
577	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
578	#418	C₂²×C₅₃	1	1	yes	characteristic, complemented
579	#417	C₂²×C₅₃	1	2	yes	—
580	#417	C₂²×C₅₃	1	1	yes	characteristic
581	#417	C₂²×C₅₃	1	4	yes	complemented
582	#417	C₂²×C₅₃	1	4	yes	complemented
583	#417	C₂²×C₅₃	1	2	yes	complemented
584	#417	C₂²×C₅₃	1	2	yes	complemented
585	#417	C₂²×C₅₃	1	2	yes	complemented
586	C₂×#103	C₂²×C₅₃	1	4	yes	complemented
587	C₂×#102	C₂²×C₅₃	1	4	yes	complemented
588	#407	C₂²×C₅₃	1	2	yes	complemented
589	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
590	C₅₃	#61161	1	1	yes	characteristic, direct product factor, Sylow
591	C₂×#41	—	2	2	no	—
592	C₂×#41	—	2	4	no	—
593	C₂×#41	—	2	2	no	—
594	C₂×#41	—	2	2	no	—
595	C₂×#41	—	2	2	no	—
596	C₂×#41	—	2	8	no	—
597	C₂×#41	—	4	4	no	—
598	C₂×#41	—	4	8	no	—
599	C₂×#41	C₂³×C₅₃	1	2	yes	—
600	C₂×#41	C₂³×C₅₃	1	2	yes	—
601	C₂×#41	C₂³×C₅₃	1	2	yes	—
602	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
603	C₂²×Q₈	—	2	1	no	—
604	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
605	C₂²×D₄	—	2	2	no	—
606	C₂²×D₄	—	2	2	no	—
607	C₂²×D₄	—	2	1	no	—
608	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
609	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
610	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
611	C₂³×C₄	—	2	2	no	—
612	C₂³×C₄	—	2	2	no	—
613	C₂³×C₄	—	2	1	no	—
614	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
615	AGL(1,ℤ/8)	—	2	4	no	—
616	AGL(1,ℤ/8)	—	4	4	no	—
617	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
618	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
619	C₂×D₈	C₂³×C₅₃	1	2	yes	—
620	C₂×D₈	C₂³×C₅₃	1	4	yes	—
621	C₂×M₁₆	—	4	4	no	—
622	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
623	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
624	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
625	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
626	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
627	C₄⋊Q₈	—	2	1	no	—
628	C₄⋊Q₈	—	2	1	no	—
629	C₄⋊Q₈	—	2	1	no	—
630	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
631	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
632	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
633	#127	—	2	1	no	—
634	#127	C₂³×C₅₃	1	1	yes	characteristic
635	#124	—	2	2	no	—
636	#124	C₂³×C₅₃	1	2	yes	—
637	#123	—	2	2	no	—
638	#123	—	2	2	no	—
639	#123	—	2	4	no	—
640	#123	—	2	4	no	—
641	#123	—	2	4	no	—
642	#123	—	2	4	no	—
643	#123	—	2	2	no	—
644	C₂²⋊Q₈	—	2	2	no	—
645	C₂²⋊Q₈	—	2	2	no	—
646	C₂²⋊Q₈	—	2	4	no	—
647	C₂²⋊Q₈	—	2	2	no	—
648	C₂²⋊Q₈	—	2	2	no	—
649	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
650	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
651	#121	—	2	2	no	—
652	#121	—	2	1	no	—
653	#121	—	2	2	no	—
654	#121	—	2	2	no	—
655	#121	—	2	1	no	—
656	#121	—	2	1	no	—
657	#121	—	2	1	no	—
658	#121	—	2	2	no	—
659	#121	—	2	2	no	—
660	#121	C₂³×C₅₃	1	4	yes	—
661	#121	C₂³×C₅₃	1	4	yes	—
662	#121	C₂³×C₅₃	1	2	yes	—
663	#121	C₂³×C₅₃	1	2	yes	—
664	#120	—	2	2	no	—
665	#120	—	2	2	no	—
666	#120	—	2	2	no	—
667	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
668	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
669	C₄×D₄	—	2	1	no	—
670	C₄×D₄	—	2	2	no	—
671	C₄×D₄	—	2	1	no	—
672	C₄×D₄	—	2	1	no	—
673	C₄×D₄	—	2	2	no	—
674	C₄×D₄	—	2	1	no	—
675	C₄×D₄	—	2	2	no	—
676	C₄×D₄	—	2	1	no	—
677	C₄×D₄	—	2	1	no	—
678	C₄×D₄	—	2	1	no	—
679	C₄×D₄	—	2	1	no	—
680	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
681	C₄×D₄	D₄×C₅₃	1	2	yes	—
682	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
683	C₄×D₄	C₂³×C₅₃	1	2	yes	—
684	C₄×D₄	C₂³×C₅₃	1	2	yes	—
685	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
686	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
687	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
688	C₄×D₄	C₂³×C₅₃	1	2	yes	—
689	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
690	#117	—	2	1	no	—
691	#117	C₂³×C₅₃	1	1	yes	characteristic
692	C₂×C₄⋊C₄	—	2	2	no	—
693	C₂×C₄⋊C₄	—	2	2	no	—
694	C₂×C₄⋊C₄	—	2	2	no	—
695	C₂×C₄⋊C₄	—	2	1	no	—
696	C₂×C₄⋊C₄	—	2	1	no	—
697	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
698	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
699	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
700	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
701	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
702	C₂×C₂²⋊C₄	—	2	1	no	—
703	C₂×C₂²⋊C₄	—	2	2	no	—
704	C₂×C₂²⋊C₄	—	2	2	no	—
705	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
706	C₂×C₄²	—	2	1	no	—
707	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
708	#107	C₂³×C₅₃	1	2	yes	—
709	#107	C₂³×C₅₃	1	4	yes	—
710	#106	C₂³×C₅₃	1	2	yes	—
711	#106	C₂³×C₅₃	1	4	yes	—
712	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
713	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
714	#103	C₂³×C₅₃	1	2	yes	—
715	#103	C₂³×C₅₃	1	4	yes	—
716	#103	C₂³×C₅₃	1	4	yes	—
717	#103	C₂³×C₅₃	1	4	yes	—
718	#103	C₂³×C₅₃	1	4	yes	—
719	#102	C₂³×C₅₃	1	2	yes	—
720	#102	C₂³×C₅₃	1	4	yes	—
721	#102	C₂³×C₅₃	1	4	yes	—
722	#102	C₂³×C₅₃	1	4	yes	—
723	#102	C₂³×C₅₃	1	4	yes	—
724	#102	C₂³×C₅₃	1	2	yes	—
725	#102	C₂³×C₅₃	1	2	yes	—
726	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
727	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
728	#97	C₂³×C₅₃	1	2	yes	—
729	C₄×C₈	C₂³×C₅₃	1	2	yes	—
730	C₂⁴	—	2	1	no	—
731	C₂⁴	—	2	2	no	—
732	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
733	#41	—	2	2	no	—
734	#41	—	2	4	no	—
735	#41	—	2	4	no	—
736	#41	—	2	4	no	—
737	#41	—	2	4	no	—
738	#41	—	4	8	no	—
739	#41	—	4	2	no	—
740	#41	—	4	2	no	—
741	#41	—	4	8	no	—
742	#41	—	4	16	no	—
743	#41	—	4	8	no	—
744	#41	—	4	8	no	—
745	#41	#41×C₅₃	1	4	yes	complemented
746	C₂×Q₈	—	2	2	no	—
747	C₂×Q₈	—	2	2	no	—
748	C₂×Q₈	—	2	2	no	—
749	C₂×Q₈	—	2	4	no	—
750	C₂×Q₈	—	4	2	no	—
751	C₂×Q₈	—	4	4	no	—
752	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
753	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
754	C₂×D₄	—	2	1	no	—
755	C₂×D₄	—	2	4	no	—
756	C₂×D₄	—	2	4	no	—
757	C₂×D₄	—	2	2	no	—
758	C₂×D₄	—	2	2	no	—
759	C₂×D₄	—	2	2	no	—
760	C₂×D₄	—	2	2	no	—
761	C₂×D₄	—	2	4	no	—
762	C₂×D₄	—	2	4	no	—
763	C₂×D₄	—	2	2	no	—
764	C₂×D₄	—	2	1	no	—
765	C₂×D₄	—	2	1	no	—
766	C₂×D₄	—	2	2	no	—
767	C₂×D₄	—	2	1	no	—
768	C₂×D₄	—	2	2	no	—
769	C₂×D₄	—	2	2	no	—
770	C₂×D₄	—	4	2	no	—
771	C₂×D₄	—	4	8	no	—
772	C₂×D₄	—	4	8	no	—
773	C₂×D₄	—	4	4	no	—
774	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
775	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
776	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
777	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
778	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
779	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
780	C₂²×C₄	—	2	1	no	—
781	C₂²×C₄	—	2	4	no	—
782	C₂²×C₄	—	2	4	no	—
783	C₂²×C₄	—	2	1	no	—
784	C₂²×C₄	—	2	2	no	—
785	C₂²×C₄	—	2	4	no	—
786	C₂²×C₄	—	2	4	no	—
787	C₂²×C₄	—	2	2	no	—
788	C₂²×C₄	—	2	1	no	—
789	C₂²×C₄	—	2	2	no	—
790	C₂²×C₄	—	2	2	no	—
791	C₂²×C₄	—	2	2	no	—
792	C₂²×C₄	—	2	2	no	—
793	C₂²×C₄	—	2	2	no	—
794	C₂²×C₄	—	2	1	no	—
795	C₂²×C₄	—	4	8	no	—
796	C₂²×C₄	—	4	4	no	—
797	C₂²×C₄	—	4	8	no	—
798	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
799	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
800	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
801	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
802	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
803	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
804	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
805	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
806	SD₁₆	—	2	4	no	—
807	SD₁₆	—	4	4	no	—
808	D₈	—	2	4	no	—
809	D₈	—	4	4	no	—
810	M₁₆	—	2	2	no	—
811	M₁₆	—	2	2	no	—
812	M₁₆	—	2	4	no	—
813	M₁₆	—	4	4	no	—
814	C₂×C₈	—	4	4	no	—
815	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
816	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
817	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
818	C₄⋊C₄	—	2	1	no	—
819	C₄⋊C₄	—	2	2	no	—
820	C₄⋊C₄	—	2	4	no	—
821	C₄⋊C₄	—	2	4	no	—
822	C₄⋊C₄	—	2	1	no	—
823	C₄⋊C₄	—	2	2	no	—
824	C₄⋊C₄	—	2	2	no	—
825	C₄⋊C₄	—	2	2	no	—
826	C₄⋊C₄	—	2	1	no	—
827	C₄⋊C₄	—	2	1	no	—
828	C₄⋊C₄	—	2	1	no	—
829	C₄⋊C₄	—	2	2	no	—
830	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
831	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
832	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
833	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
834	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
835	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
836	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
837	C₂²⋊C₄	—	2	2	no	—
838	C₂²⋊C₄	—	2	1	no	—
839	C₂²⋊C₄	—	2	2	no	—
840	C₂²⋊C₄	—	2	2	no	—
841	C₂²⋊C₄	—	2	2	no	—
842	C₂²⋊C₄	—	2	2	no	—
843	C₂²⋊C₄	—	2	2	no	—
844	C₂²⋊C₄	—	2	4	no	—
845	C₂²⋊C₄	—	2	4	no	—
846	C₂²⋊C₄	—	2	1	no	—
847	C₂²⋊C₄	—	2	1	no	—
848	C₂²⋊C₄	—	2	2	no	—
849	C₂²⋊C₄	—	2	2	no	—
850	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
851	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
852	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
853	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
854	C₄²	—	2	1	no	—
855	C₄²	—	2	1	no	—
856	C₄²	—	2	1	no	—
857	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
858	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
859	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
860	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
861	C₂³	—	2	4	no	—
862	C₂³	—	2	4	no	—
863	C₂³	—	2	1	no	—
864	C₂³	—	2	2	no	—
865	C₂³	—	2	2	no	—
866	C₂³	—	4	2	no	—
867	C₂³	—	4	4	no	—
868	C₂³	—	4	2	no	—
869	C₂³	—	4	2	no	—
870	C₂³	—	4	2	no	—
871	C₂³	—	4	4	no	—
872	C₂³	—	8	2	no	—
873	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
874	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
875	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
876	Q₈	—	2	1	no	—
877	Q₈	—	2	2	no	—
878	Q₈	—	2	2	no	—
879	Q₈	—	4	4	no	—
880	Q₈	—	4	4	no	—
881	Q₈	—	4	4	no	—
882	Q₈	C₂×#41×C₅₃	1	2	yes	complemented
883	D₄	—	2	4	no	—
884	D₄	—	2	4	no	—
885	D₄	—	2	2	no	—
886	D₄	—	2	2	no	—
887	D₄	—	2	2	no	—
888	D₄	—	2	1	no	—
889	D₄	—	2	2	no	—
890	D₄	—	2	2	no	—
891	D₄	—	4	4	no	—
892	D₄	—	4	4	no	—
893	D₄	—	4	4	no	—
894	D₄	—	4	8	no	—
895	D₄	—	4	4	no	—
896	D₄	—	4	4	no	—
897	D₄	—	4	8	no	—
898	D₄	—	4	2	no	—
899	D₄	—	4	2	no	—
900	D₄	—	8	1	no	—
901	D₄	—	8	2	no	—
902	D₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
903	D₄	C₂×#41×C₅₃	1	2	yes	complemented
904	C₂×C₄	—	2	4	no	—
905	C₂×C₄	—	2	1	no	—
906	C₂×C₄	—	2	4	no	—
907	C₂×C₄	—	2	4	no	—
908	C₂×C₄	—	2	4	no	—
909	C₂×C₄	—	2	2	no	—
910	C₂×C₄	—	2	2	no	—
911	C₂×C₄	—	2	2	no	—
912	C₂×C₄	—	2	2	no	—
913	C₂×C₄	—	2	2	no	—
914	C₂×C₄	—	2	1	no	—
915	C₂×C₄	—	2	1	no	—
916	C₂×C₄	—	2	2	no	—
917	C₂×C₄	—	2	2	no	—
918	C₂×C₄	—	2	4	no	—
919	C₂×C₄	—	4	1	no	—
920	C₂×C₄	—	4	4	no	—
921	C₂×C₄	—	4	4	no	—
922	C₂×C₄	—	4	4	no	—
923	C₂×C₄	—	4	4	no	—
924	C₂×C₄	—	4	2	no	—
925	C₂×C₄	—	4	4	no	—
926	C₂×C₄	—	4	8	no	—
927	C₂×C₄	—	4	8	no	—
928	C₂×C₄	—	4	4	no	—
929	C₂×C₄	—	4	4	no	—
930	C₂×C₄	—	4	2	no	—
931	C₂×C₄	—	8	1	no	—
932	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
933	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
934	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
935	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
936	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
937	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
938	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
939	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
940	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
941	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
942	C₈	—	2	2	no	—
943	C₈	—	2	2	no	—
944	C₈	—	2	4	no	—
945	C₂²	—	2	2	no	—
946	C₂²	—	2	1	no	—
947	C₂²	—	2	1	no	—
948	C₂²	—	2	2	no	—
949	C₂²	—	2	2	no	—
950	C₂²	—	2	2	no	—
951	C₂²	—	2	2	no	—
952	C₂²	—	2	2	no	—
953	C₂²	—	4	8	no	—
954	C₂²	—	4	1	no	—
955	C₂²	—	4	1	no	—
956	C₂²	—	4	2	no	—
957	C₂²	—	4	2	no	—
958	C₂²	—	4	2	no	—
959	C₂²	—	4	2	no	—
960	C₂²	—	4	2	no	—
961	C₂²	—	4	2	no	—
962	C₂²	—	8	8	no	—
963	C₂²	—	8	4	no	—
964	C₂²	C₅₃×#546	1	2	yes	—
965	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
966	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
967	C₄	—	2	2	no	—
968	C₄	—	2	2	no	—
969	C₄	—	2	2	no	—
970	C₄	—	2	1	no	—
971	C₄	—	2	1	no	—
972	C₄	—	2	1	no	—
973	C₄	—	4	2	no	—
974	C₄	—	4	2	no	—
975	C₄	—	4	4	no	—
976	C₄	—	4	1	no	—
977	C₄	—	4	1	no	—
978	C₄	C₅₃×#546	1	2	yes	—
979	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
980	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
981	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
982	C₂	—	2	2	no	—
983	C₂	—	2	4	no	—
984	C₂	—	4	2	no	—
985	C₂	—	4	2	no	—
986	C₂	—	8	2	no	—
987	C₂	—	8	2	no	—
988	C₂	C₂×C₅₃×#546	1	1	yes	characteristic
989	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
990	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 ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. D₄×#41×C₅₃	C₂	semidirect
2. C₂×C₅₃×#547	C₂	semidirect
3. C₅₃×#3332	C₂	semidirect
4. C₅₃×#3330	C₂	semidirect
5. C₅₃×#3308	C₂	semidirect
6. C₅₃×#3306	C₂	semidirect
7. C₅₃×#3298	C₂	semidirect
8. C₅₃×#3294	C₂	semidirect
9. C₅₃×#3141	C₂	semidirect
10. C₅₃×#3092	C₂	semidirect
11. C₅₃×#3079	C₂	semidirect
12. C₅₃×#3055	C₂	semidirect
13. C₅₃×#3036	C₂	semidirect
14. C₅₃×#3006	C₂	semidirect
15. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
16. D₄×M₁₆×C₅₃	C₂	semidirect
17. C₂×C₅₃×#417	C₂	semidirect
21. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
22. C₂²×M₁₆×C₅₃	C₂²	semidirect
23. C₅₃×#550	C₂²	semidirect
24. D₄×Q₈×C₅₃	C₂²	semidirect
25. C₅₃×#548	C₂²	semidirect
31. C₅₃×#547	C₂²	semidirect
32. C₅₃×#547	C₂²	semidirect
33. C₅₃×#547	C₂²	semidirect
35. C₅₃×#546	C₂²	semidirect
37. D₄²×C₅₃	C₂²	semidirect
40. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
45. C₂×#121×C₅₃	C₂²	semidirect
49. C₂×C₄×D₄×C₅₃	C₂²	semidirect
50. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
51. C₅₃×#489	C₂²	semidirect
52. C₅₃×#487	C₂²	semidirect
53. C₅₃×#486	C₂²	semidirect
54. C₅₃×#484	C₂²	semidirect
55. C₅₃×#483	C₂²	semidirect
56. C₅₃×#481	C₂²	semidirect
57. C₅₃×#480	C₂²	semidirect
58. C₅₃×#477	C₂²	semidirect
59. C₅₃×#476	C₂²	semidirect
60. C₅₃×#475	C₂²	semidirect
61. C₅₃×#474	C₂²	semidirect
62. C₅₃×#469	C₂²	semidirect
63. C₅₃×#468	C₂²	semidirect
64. C₅₃×#466	C₂²	semidirect
65. C₅₃×#465	C₂²	semidirect
66. C₅₃×#463	C₂²	semidirect
67. C₅₃×#461	C₂²	semidirect
68. C₅₃×#459	C₂²	semidirect
69. C₅₃×#449	C₂²	semidirect
70. C₅₃×#449	C₂²	semidirect
71. C₅₃×#448	C₂²	semidirect
72. C₅₃×#447	C₂²	semidirect
73. C₅₃×#442	C₂²	semidirect
74. C₅₃×#440	C₂²	semidirect
75. C₄×SD₁₆×C₅₃	C₂²	semidirect
76. C₄×D₈×C₅₃	C₂²	semidirect
77. C₅₃×#436	C₂²	semidirect
78. C₅₃×#435	C₂²	semidirect
79. C₈×D₄×C₅₃	C₂²	semidirect
80. C₅₃×#428	C₂²	semidirect
81. C₅₃×#428	C₂²	semidirect
82. C₅₃×#423	C₂²	semidirect
83. C₅₃×#418	C₂²	semidirect
86. C₅₃×#417	C₂²	semidirect
87. C₅₃×#417	C₂²	semidirect
88. C₅₃×#417	C₂²	semidirect
89. C₅₃×#417	C₂²	semidirect
90. C₅₃×#417	C₂²	semidirect
91. C₂×#103×C₅₃	C₂²	semidirect
92. C₂×#102×C₅₃	C₂²	semidirect
93. C₅₃×#407	C₂²	semidirect
94. C₄×M₁₆×C₅₃	C₂²	semidirect
249. #41×C₅₃	#41	semidirect
386. Q₈×C₅₃	C₂×#41	semidirect
406. D₄×C₅₃	AGL(1,ℤ/8)	semidirect
407. D₄×C₅₃	C₂×#41	semidirect
449. #61161	C₅₃	direct (N × H)
487. D₄×#41	C₂×C₅₃	semidirect
488. C₂×#547	C₂×C₅₃	semidirect
489. #3332	C₂×C₅₃	semidirect
490. #3330	C₂×C₅₃	semidirect
491. #3308	C₂×C₅₃	semidirect
492. #3306	C₂×C₅₃	semidirect
493. #3298	C₂×C₅₃	semidirect
494. #3294	C₂×C₅₃	semidirect
495. #3141	C₂×C₅₃	semidirect
496. #3092	C₂×C₅₃	semidirect
497. #3079	C₂×C₅₃	semidirect
498. #3055	C₂×C₅₃	semidirect
499. #3036	C₂×C₅₃	semidirect
500. #3006	C₂×C₅₃	semidirect
501. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
502. D₄×M₁₆	C₂×C₅₃	semidirect
503. C₂×#417	C₂×C₅₃	semidirect
516. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
517. C₂²×M₁₆	C₂²×C₅₃	semidirect
518. #550	C₂²×C₅₃	semidirect
519. D₄×Q₈	C₂²×C₅₃	semidirect
520. #548	C₂²×C₅₃	semidirect
526. #547	C₂²×C₅₃	semidirect
527. #547	C₂²×C₅₃	semidirect
528. #547	C₂²×C₅₃	semidirect
530. #546	C₂²×C₅₃	semidirect
532. D₄²	C₂²×C₅₃	semidirect
535. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
540. C₂×#121	C₂²×C₅₃	semidirect
544. C₂×C₄×D₄	C₂²×C₅₃	semidirect
545. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
546. #489	C₂²×C₅₃	semidirect
547. #487	C₂²×C₅₃	semidirect
548. #486	C₂²×C₅₃	semidirect
549. #484	C₂²×C₅₃	semidirect
550. #483	C₂²×C₅₃	semidirect
551. #481	C₂²×C₅₃	semidirect
552. #480	C₂²×C₅₃	semidirect
553. #477	C₂²×C₅₃	semidirect
554. #476	C₂²×C₅₃	semidirect
555. #475	C₂²×C₅₃	semidirect
556. #474	C₂²×C₅₃	semidirect
557. #469	C₂²×C₅₃	semidirect
558. #468	C₂²×C₅₃	semidirect
559. #466	C₂²×C₅₃	semidirect
560. #465	C₂²×C₅₃	semidirect
561. #463	C₂²×C₅₃	semidirect
562. #461	C₂²×C₅₃	semidirect
563. #459	C₂²×C₅₃	semidirect
564. #449	C₂²×C₅₃	semidirect
565. #449	C₂²×C₅₃	semidirect
566. #448	C₂²×C₅₃	semidirect
567. #447	C₂²×C₅₃	semidirect
568. #442	C₂²×C₅₃	semidirect
569. #440	C₂²×C₅₃	semidirect
570. C₄×SD₁₆	C₂²×C₅₃	semidirect
571. C₄×D₈	C₂²×C₅₃	semidirect
572. #436	C₂²×C₅₃	semidirect
573. #435	C₂²×C₅₃	semidirect
574. C₈×D₄	C₂²×C₅₃	semidirect
575. #428	C₂²×C₅₃	semidirect
576. #428	C₂²×C₅₃	semidirect
577. #423	C₂²×C₅₃	semidirect
578. #418	C₂²×C₅₃	semidirect
581. #417	C₂²×C₅₃	semidirect
582. #417	C₂²×C₅₃	semidirect
583. #417	C₂²×C₅₃	semidirect
584. #417	C₂²×C₅₃	semidirect
585. #417	C₂²×C₅₃	semidirect
586. C₂×#103	C₂²×C₅₃	semidirect
587. C₂×#102	C₂²×C₅₃	semidirect
588. #407	C₂²×C₅₃	semidirect
589. C₄×M₁₆	C₂²×C₅₃	semidirect
590. C₅₃	#61161	direct (N × H)
745. #41	#41×C₅₃	semidirect
882. Q₈	C₂×#41×C₅₃	semidirect
902. D₄	AGL(1,ℤ/8)×C₅₃	semidirect
903. D₄	C₂×#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
D₄×#41×C₅₃	2
C₂×C₅₃×#547	2
C₅₃×#3332	2
C₅₃×#3330	2
C₅₃×#3308	2
C₅₃×#3306	2
C₅₃×#3298	2
C₅₃×#3294	2
C₅₃×#3141	2
C₅₃×#3092	2
C₅₃×#3079	2
C₅₃×#3055	2
C₅₃×#3036	2
C₅₃×#3006	2
C₄×AGL(1,ℤ/8)×C₅₃	2
D₄×M₁₆×C₅₃	2
C₂×C₅₃×#417	2
#61161	53

Subgroups — C53×#61161 #27268673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61161 #27268673