Subgroups — C₅₃×#61259 #27268771

Order 13568.

1866 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3492	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3345	C₂	1	4	yes	maximal, complemented
3	C₅₃×#3344	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3337	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3335	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3099	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3092	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3084	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3060	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2952	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#554	—	2	2	no	—
20	C₅₃×#554	C₂²	1	4	yes	—
21	C₅₃×#553	—	2	2	no	—
22	C₅₃×#553	C₂²	1	4	yes	—
23	C₅₃×#552	—	2	2	no	—
24	C₅₃×#552	C₂²	1	4	yes	—
25	C₅₃×#547	—	2	2	no	—
26	C₅₃×#547	C₂²	1	4	yes	—
27	C₅₃×#544	C₂²	1	2	yes	—
28	C₅₃×#539	—	2	4	no	—
29	C₅₃×#537	C₂²	1	2	yes	—
30	C₅₃×#525	—	2	4	no	—
31	C₅₃×#525	C₂²	1	2	yes	—
32	C₅₃×#525	C₂²	1	1	yes	characteristic
33	C₅₃×#520	C₂²	1	2	yes	—
34	C₅₃×#520	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#501	C₂²	1	4	yes	complemented
37	C₅₃×#499	C₂²	1	2	yes	complemented
38	C₅₃×#498	C₂²	1	2	yes	complemented
39	C₅₃×#485	C₂²	1	4	yes	complemented
40	C₅₃×#484	C₂²	1	4	yes	—
41	C₅₃×#483	C₂²	1	4	yes	—
42	C₅₃×#483	C₂²	1	4	yes	complemented
43	C₅₃×#482	C₂²	1	4	yes	complemented
44	C₅₃×#481	C₂²	1	4	yes	—
45	C₅₃×#481	C₂²	1	4	yes	complemented
46	C₅₃×#480	C₂²	1	4	yes	—
47	C₅₃×#479	C₂²	1	4	yes	complemented
48	C₅₃×#478	C₂²	1	4	yes	complemented
49	C₅₃×#476	C₂²	1	4	yes	complemented
50	C₅₃×#475	C₂²	1	4	yes	complemented
51	C₅₃×#470	C₂²	1	2	yes	complemented
52	C₅₃×#469	C₂²	1	2	yes	complemented
53	C₅₃×#467	C₂²	1	2	yes	—
54	C₅₃×#466	C₂²	1	2	yes	—
55	C₅₃×#465	C₂²	1	2	yes	—
56	C₅₃×#465	C₂²	1	2	yes	—
57	C₅₃×#465	C₂²	1	4	yes	complemented
58	C₅₃×#442	C₂²	1	2	yes	complemented
59	C₅₃×#441	C₂²	1	2	yes	complemented
60	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
61	C₅₃×#435	C₂²	1	4	yes	—
62	C₈×D₄×C₅₃	C₂²	1	4	yes	—
63	C₅₃×#428	C₂²	1	2	yes	—
64	C₅₃×#428	C₂²	1	2	yes	—
65	C₅₃×#428	C₂²	1	4	yes	complemented
66	C₅₃×#427	C₂²	1	1	yes	characteristic
67	C₅₃×#427	C₂²	1	2	yes	complemented
68	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
69	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
70	C₂×#106×C₅₃	C₂²	1	4	yes	—
71	C₂×#106×C₅₃	C₂²	1	2	yes	—
72	C₂×#106×C₅₃	C₂²	1	4	yes	—
73	C₂×#106×C₅₃	C₂²	1	2	yes	complemented
74	C₅₃×#424	C₂²	1	1	yes	characteristic
75	C₅₃×#417	C₂²	1	2	yes	—
76	C₅₃×#417	C₂²	1	2	yes	—
77	C₅₃×#416	C₂²	1	2	yes	—
78	C₅₃×#408	C₂²	1	1	yes	characteristic
79	C₅₃×#407	C₂²	1	1	yes	characteristic
80	C₅₃×#405	C₂²	1	1	yes	characteristic
81	C₂×#41×C₅₃	—	2	4	no	—
82	C₂×#41×C₅₃	C₂³	1	2	yes	—
83	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₅₃	—	2	2	no	—
85	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
86	#135×C₅₃	—	2	4	no	—
87	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
89	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
90	#131×C₅₃	—	2	4	no	—
91	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
93	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	—	2	4	no	—
95	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
97	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
98	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
99	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊Q₈×C₅₃	—	2	2	no	—
101	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
102	#126×C₅₃	—	2	4	no	—
103	#126×C₅₃	—	2	4	no	—
104	#125×C₅₃	—	2	4	no	—
105	#125×C₅₃	—	2	4	no	—
106	#125×C₅₃	—	2	2	no	—
107	#125×C₅₃	C₂³	1	4	yes	—
108	#123×C₅₃	—	2	4	no	—
109	#123×C₅₃	—	2	4	no	—
110	#123×C₅₃	—	2	4	no	—
111	#123×C₅₃	—	2	4	no	—
112	#123×C₅₃	—	2	2	no	—
113	#123×C₅₃	D₄	1	2	yes	—
114	C₂²⋊Q₈×C₅₃	—	2	4	no	—
115	C₂²⋊Q₈×C₅₃	—	2	4	no	—
116	C₂²⋊Q₈×C₅₃	—	2	4	no	—
117	C₂²⋊Q₈×C₅₃	—	2	4	no	—
118	C₂²⋊Q₈×C₅₃	—	2	2	no	—
119	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
120	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
121	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
122	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
123	#121×C₅₃	—	2	4	no	—
124	#121×C₅₃	—	2	4	no	—
125	#121×C₅₃	—	2	2	no	—
126	#121×C₅₃	D₄	1	2	yes	—
127	#121×C₅₃	C₂³	1	2	yes	—
128	#121×C₅₃	C₂³	1	2	yes	—
129	#121×C₅₃	C₂³	1	4	yes	—
130	#120×C₅₃	—	2	2	no	—
131	#120×C₅₃	D₄	1	2	yes	—
132	C₄×Q₈×C₅₃	—	2	2	no	—
133	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
134	C₄×D₄×C₅₃	—	2	4	no	—
135	C₄×D₄×C₅₃	—	2	4	no	—
136	C₄×D₄×C₅₃	—	2	4	no	—
137	C₄×D₄×C₅₃	—	2	2	no	—
138	C₄×D₄×C₅₃	—	2	4	no	—
139	C₄×D₄×C₅₃	—	2	2	no	—
140	C₄×D₄×C₅₃	—	2	2	no	—
141	C₄×D₄×C₅₃	C₂³	1	4	yes	—
142	C₄×D₄×C₅₃	C₂³	1	4	yes	—
143	#117×C₅₃	—	2	2	no	—
144	#117×C₅₃	—	2	4	no	—
145	#117×C₅₃	C₂³	1	4	yes	—
146	#117×C₅₃	C₂³	1	2	yes	—
147	#117×C₅₃	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
155	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
156	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
157	#107×C₅₃	C₂³	1	4	yes	—
158	#107×C₅₃	C₂³	1	2	yes	—
159	#107×C₅₃	C₂³	1	4	yes	—
160	#106×C₅₃	C₂³	1	4	yes	—
161	#106×C₅₃	C₂³	1	1	yes	characteristic
162	#106×C₅₃	C₂³	1	2	yes	—
163	#106×C₅₃	C₂³	1	4	yes	—
164	#106×C₅₃	C₂³	1	1	yes	characteristic
165	#106×C₅₃	C₂³	1	2	yes	—
166	#106×C₅₃	C₂³	1	2	yes	—
167	#106×C₅₃	C₂³	1	2	yes	—
168	#106×C₅₃	C₂³	1	4	yes	—
169	#106×C₅₃	C₂³	1	4	yes	—
170	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
171	#103×C₅₃	C₂³	1	2	yes	—
172	#103×C₅₃	C₂³	1	4	yes	—
173	#103×C₅₃	C₂³	1	2	yes	—
174	#103×C₅₃	C₂³	1	4	yes	—
175	#102×C₅₃	C₂³	1	2	yes	—
176	#102×C₅₃	C₂³	1	2	yes	—
177	#102×C₅₃	C₂³	1	4	yes	—
178	#102×C₅₃	C₂³	1	4	yes	—
179	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
180	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
181	#97×C₅₃	C₂³	1	2	yes	—
182	C₄×C₈×C₅₃	C₂³	1	2	yes	—
183	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
184	#41×C₅₃	—	2	2	no	—
185	#41×C₅₃	—	2	2	no	—
186	#41×C₅₃	—	2	4	no	—
187	#41×C₅₃	—	2	4	no	—
188	#41×C₅₃	—	8	8	no	—
189	C₂×Q₈×C₅₃	—	2	4	no	—
190	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
192	C₂×D₄×C₅₃	—	2	2	no	—
193	C₂×D₄×C₅₃	—	2	2	no	—
194	C₂×D₄×C₅₃	—	2	2	no	—
195	C₂×D₄×C₅₃	—	2	4	no	—
196	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
197	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
198	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
199	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₅₃	—	2	2	no	—
201	C₂²×C₄×C₅₃	—	2	2	no	—
202	C₂²×C₄×C₅₃	—	2	4	no	—
203	C₂²×C₄×C₅₃	—	2	2	no	—
204	C₂²×C₄×C₅₃	—	2	4	no	—
205	C₂²×C₄×C₅₃	—	2	2	no	—
206	C₂²×C₄×C₅₃	—	2	4	no	—
207	C₂²×C₄×C₅₃	—	2	2	no	—
208	C₂²×C₄×C₅₃	—	2	4	no	—
209	C₂²×C₄×C₅₃	—	8	4	no	—
210	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
215	Q₁₆×C₅₃	—	2	2	no	—
216	SD₁₆×C₅₃	—	2	4	no	—
217	D₈×C₅₃	—	2	2	no	—
218	M₁₆×C₅₃	—	2	2	no	—
219	M₁₆×C₅₃	—	2	2	no	—
220	M₁₆×C₅₃	—	2	4	no	—
221	C₂×C₈×C₅₃	—	2	2	no	—
222	C₂×C₈×C₅₃	—	2	4	no	—
223	C₂×C₈×C₅₃	—	2	4	no	—
224	C₂×C₈×C₅₃	—	2	4	no	—
225	C₂×C₈×C₅₃	#41	1	4	yes	complemented
226	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
227	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
228	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
229	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
231	C₄⋊C₄×C₅₃	—	2	2	no	—
232	C₄⋊C₄×C₅₃	—	2	2	no	—
233	C₄⋊C₄×C₅₃	—	2	4	no	—
234	C₄⋊C₄×C₅₃	—	2	4	no	—
235	C₄⋊C₄×C₅₃	—	2	4	no	—
236	C₄⋊C₄×C₅₃	—	2	4	no	—
237	C₄⋊C₄×C₅₃	—	2	4	no	—
238	C₄⋊C₄×C₅₃	—	2	4	no	—
239	C₄⋊C₄×C₅₃	—	2	2	no	—
240	C₄⋊C₄×C₅₃	—	2	4	no	—
241	C₄⋊C₄×C₅₃	—	2	4	no	—
242	C₄⋊C₄×C₅₃	—	2	2	no	—
243	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
245	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
246	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
247	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
248	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
249	C₂²⋊C₄×C₅₃	—	2	2	no	—
250	C₂²⋊C₄×C₅₃	—	2	2	no	—
251	C₂²⋊C₄×C₅₃	—	2	4	no	—
252	C₂²⋊C₄×C₅₃	—	2	2	no	—
253	C₂²⋊C₄×C₅₃	—	2	4	no	—
254	C₂²⋊C₄×C₅₃	—	2	4	no	—
255	C₂²⋊C₄×C₅₃	—	2	4	no	—
256	C₂²⋊C₄×C₅₃	—	2	2	no	—
257	C₂²⋊C₄×C₅₃	—	2	2	no	—
258	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
259	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
260	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
261	C₄²×C₅₃	—	2	4	no	—
262	C₄²×C₅₃	—	2	4	no	—
263	C₄²×C₅₃	—	2	2	no	—
264	C₄²×C₅₃	C₂⁴	1	2	yes	—
265	C₂³×C₅₃	—	2	2	no	—
266	C₂³×C₅₃	—	2	4	no	—
267	C₂³×C₅₃	—	2	2	no	—
268	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
269	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
270	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
271	Q₈×C₅₃	—	2	2	no	—
272	Q₈×C₅₃	—	2	2	no	—
273	Q₈×C₅₃	—	2	2	no	—
274	Q₈×C₅₃	—	8	4	no	—
275	D₄×C₅₃	—	2	4	no	—
276	D₄×C₅₃	—	2	8	no	—
277	D₄×C₅₃	—	2	2	no	—
278	D₄×C₅₃	—	2	2	no	—
279	D₄×C₅₃	—	2	2	no	—
280	D₄×C₅₃	—	4	4	no	—
281	D₄×C₅₃	—	4	4	no	—
282	D₄×C₅₃	—	4	4	no	—
283	D₄×C₅₃	—	8	4	no	—
284	C₂×C₄×C₅₃	—	2	2	no	—
285	C₂×C₄×C₅₃	—	2	4	no	—
286	C₂×C₄×C₅₃	—	2	2	no	—
287	C₂×C₄×C₅₃	—	2	2	no	—
288	C₂×C₄×C₅₃	—	2	2	no	—
289	C₂×C₄×C₅₃	—	2	2	no	—
290	C₂×C₄×C₅₃	—	2	4	no	—
291	C₂×C₄×C₅₃	—	2	4	no	—
292	C₂×C₄×C₅₃	—	2	2	no	—
293	C₂×C₄×C₅₃	—	2	2	no	—
294	C₂×C₄×C₅₃	—	2	2	no	—
295	C₂×C₄×C₅₃	—	4	8	no	—
296	C₂×C₄×C₅₃	—	4	4	no	—
297	C₂×C₄×C₅₃	—	8	4	no	—
298	C₂×C₄×C₅₃	—	8	2	no	—
299	C₂×C₄×C₅₃	—	8	4	no	—
300	C₂×C₄×C₅₃	—	8	2	no	—
301	C₂×C₄×C₅₃	—	8	4	no	—
302	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
303	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
307	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
308	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
309	C₈×C₅₃	—	2	2	no	—
310	C₈×C₅₃	—	2	2	no	—
311	C₈×C₅₃	—	2	2	no	—
312	C₈×C₅₃	C₂×#41	1	2	yes	—
313	C₈×C₅₃	C₂×#41	1	2	yes	—
314	#61259	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
315	C₂²×C₅₃	—	2	8	no	—
316	C₂²×C₅₃	—	2	2	no	—
317	C₂²×C₅₃	—	2	2	no	—
318	C₂²×C₅₃	—	2	2	no	—
319	C₂²×C₅₃	—	2	2	no	—
320	C₂²×C₅₃	—	2	4	no	—
321	C₂²×C₅₃	—	4	2	no	—
322	C₂²×C₅₃	—	4	2	no	—
323	C₂²×C₅₃	—	4	2	no	—
324	C₂²×C₅₃	#547	1	2	yes	—
325	C₂²×C₅₃	#577	1	4	yes	—
326	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
327	C₄×C₅₃	—	2	2	no	—
328	C₄×C₅₃	—	2	2	no	—
329	C₄×C₅₃	—	2	4	no	—
330	C₄×C₅₃	—	4	2	no	—
331	C₄×C₅₃	—	4	4	no	—
332	C₄×C₅₃	—	4	4	no	—
333	C₄×C₅₃	—	4	2	no	—
334	C₄×C₅₃	—	4	2	no	—
335	C₄×C₅₃	#547	1	2	yes	—
336	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
337	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
338	#3492	C₂×C₅₃	1	2	yes	complemented
339	#3345	C₂×C₅₃	1	4	yes	complemented
340	#3344	C₂×C₅₃	1	4	yes	complemented
341	#3337	C₂×C₅₃	1	4	yes	complemented
342	#3335	C₂×C₅₃	1	4	yes	complemented
343	#3163	C₂×C₅₃	1	1	yes	characteristic, complemented
344	#3099	C₂×C₅₃	1	2	yes	complemented
345	#3092	C₂×C₅₃	1	2	yes	complemented
346	#3084	C₂×C₅₃	1	2	yes	complemented
347	#3060	C₂×C₅₃	1	1	yes	characteristic, complemented
348	#3050	C₂×C₅₃	1	1	yes	characteristic, complemented
349	#2952	C₂×C₅₃	1	1	yes	characteristic, complemented
350	#2929	C₂×C₅₃	1	1	yes	characteristic, complemented
351	#2912	C₂×C₅₃	1	1	yes	characteristic, complemented
352	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
353	C₂×C₅₃	—	2	2	no	—
354	C₂×C₅₃	—	2	4	no	—
355	C₂×C₅₃	—	4	2	no	—
356	C₂×C₅₃	—	8	2	no	—
357	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
358	C₂×C₅₃	C₂×#577	1	2	yes	—
359	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
360	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
361	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
362	#554	—	2	2	no	—
363	#554	C₂²×C₅₃	1	4	yes	—
364	#553	—	2	2	no	—
365	#553	C₂²×C₅₃	1	4	yes	—
366	#552	—	2	2	no	—
367	#552	C₂²×C₅₃	1	4	yes	—
368	#547	—	2	2	no	—
369	#547	C₂²×C₅₃	1	4	yes	—
370	#544	C₂²×C₅₃	1	2	yes	—
371	#539	—	2	4	no	—
372	#537	C₂²×C₅₃	1	2	yes	—
373	#525	—	2	4	no	—
374	#525	C₂²×C₅₃	1	2	yes	—
375	#525	C₂²×C₅₃	1	1	yes	characteristic
376	#520	C₂²×C₅₃	1	2	yes	—
377	#520	C₂²×C₅₃	1	2	yes	—
378	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
379	#501	C₂²×C₅₃	1	4	yes	complemented
380	#499	C₂²×C₅₃	1	2	yes	complemented
381	#498	C₂²×C₅₃	1	2	yes	complemented
382	#485	C₂²×C₅₃	1	4	yes	complemented
383	#484	C₂²×C₅₃	1	4	yes	—
384	#483	C₂²×C₅₃	1	4	yes	—
385	#483	C₂²×C₅₃	1	4	yes	complemented
386	#482	C₂²×C₅₃	1	4	yes	complemented
387	#481	C₂²×C₅₃	1	4	yes	—
388	#481	C₂²×C₅₃	1	4	yes	complemented
389	#480	C₂²×C₅₃	1	4	yes	—
390	#479	C₂²×C₅₃	1	4	yes	complemented
391	#478	C₂²×C₅₃	1	4	yes	complemented
392	#476	C₂²×C₅₃	1	4	yes	complemented
393	#475	C₂²×C₅₃	1	4	yes	complemented
394	#470	C₂²×C₅₃	1	2	yes	complemented
395	#469	C₂²×C₅₃	1	2	yes	complemented
396	#467	C₂²×C₅₃	1	2	yes	—
397	#466	C₂²×C₅₃	1	2	yes	—
398	#465	C₂²×C₅₃	1	2	yes	—
399	#465	C₂²×C₅₃	1	2	yes	—
400	#465	C₂²×C₅₃	1	4	yes	complemented
401	#442	C₂²×C₅₃	1	2	yes	complemented
402	#441	C₂²×C₅₃	1	2	yes	complemented
403	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
404	#435	C₂²×C₅₃	1	4	yes	—
405	C₈×D₄	C₂²×C₅₃	1	4	yes	—
406	#428	C₂²×C₅₃	1	2	yes	—
407	#428	C₂²×C₅₃	1	2	yes	—
408	#428	C₂²×C₅₃	1	4	yes	complemented
409	#427	C₂²×C₅₃	1	1	yes	characteristic
410	#427	C₂²×C₅₃	1	2	yes	complemented
411	C₂×#106	C₂²×C₅₃	1	4	yes	—
412	C₂×#106	C₂²×C₅₃	1	2	yes	—
413	C₂×#106	C₂²×C₅₃	1	4	yes	—
414	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
415	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
416	C₂×#106	C₂²×C₅₃	1	2	yes	complemented
417	#424	C₂²×C₅₃	1	1	yes	characteristic
418	#417	C₂²×C₅₃	1	2	yes	—
419	#417	C₂²×C₅₃	1	2	yes	—
420	#416	C₂²×C₅₃	1	2	yes	—
421	#408	C₂²×C₅₃	1	1	yes	characteristic
422	#407	C₂²×C₅₃	1	1	yes	characteristic
423	#405	C₂²×C₅₃	1	1	yes	characteristic
424	C₅₃	#61259	1	1	yes	characteristic, direct product factor, Sylow
425	C₂×#41	—	2	4	no	—
426	C₂×#41	C₂³×C₅₃	1	2	yes	—
427	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
428	C₂³×C₄	—	2	2	no	—
429	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
430	#135	—	2	4	no	—
431	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
432	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
433	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
434	#131	—	2	4	no	—
435	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
436	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
437	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
438	C₂²×C₈	—	2	4	no	—
439	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
440	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
441	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
442	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
443	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
444	C₄⋊Q₈	—	2	2	no	—
445	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
446	#126	—	2	4	no	—
447	#126	—	2	4	no	—
448	#125	—	2	4	no	—
449	#125	—	2	4	no	—
450	#125	—	2	2	no	—
451	#125	C₂³×C₅₃	1	4	yes	—
452	#123	—	2	4	no	—
453	#123	—	2	2	no	—
454	#123	—	2	4	no	—
455	#123	—	2	4	no	—
456	#123	—	2	4	no	—
457	#123	D₄×C₅₃	1	2	yes	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	4	no	—
460	C₂²⋊Q₈	—	2	4	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
464	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
465	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
466	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
467	#121	—	2	4	no	—
468	#121	—	2	4	no	—
469	#121	—	2	2	no	—
470	#121	D₄×C₅₃	1	2	yes	—
471	#121	C₂³×C₅₃	1	2	yes	—
472	#121	C₂³×C₅₃	1	2	yes	—
473	#121	C₂³×C₅₃	1	4	yes	—
474	#120	—	2	2	no	—
475	#120	D₄×C₅₃	1	2	yes	—
476	C₄×Q₈	—	2	2	no	—
477	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
478	C₄×D₄	—	2	4	no	—
479	C₄×D₄	—	2	2	no	—
480	C₄×D₄	—	2	4	no	—
481	C₄×D₄	—	2	2	no	—
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	4	no	—
484	C₄×D₄	—	2	4	no	—
485	C₄×D₄	C₂³×C₅₃	1	4	yes	—
486	C₄×D₄	C₂³×C₅₃	1	4	yes	—
487	#117	—	2	2	no	—
488	#117	—	2	4	no	—
489	#117	C₂³×C₅₃	1	2	yes	—
490	#117	C₂³×C₅₃	1	4	yes	—
491	#117	C₂³×C₅₃	1	1	yes	characteristic
492	C₂×C₄⋊C₄	—	2	2	no	—
493	C₂×C₄⋊C₄	—	2	2	no	—
494	C₂×C₄⋊C₄	—	2	2	no	—
495	C₂×C₄⋊C₄	—	2	2	no	—
496	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
498	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
499	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
500	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
501	#107	C₂³×C₅₃	1	2	yes	—
502	#107	C₂³×C₅₃	1	4	yes	—
503	#107	C₂³×C₅₃	1	4	yes	—
504	#106	C₂³×C₅₃	1	2	yes	—
505	#106	C₂³×C₅₃	1	2	yes	—
506	#106	C₂³×C₅₃	1	2	yes	—
507	#106	C₂³×C₅₃	1	2	yes	—
508	#106	C₂³×C₅₃	1	4	yes	—
509	#106	C₂³×C₅₃	1	4	yes	—
510	#106	C₂³×C₅₃	1	1	yes	characteristic
511	#106	C₂³×C₅₃	1	4	yes	—
512	#106	C₂³×C₅₃	1	1	yes	characteristic
513	#106	C₂³×C₅₃	1	4	yes	—
514	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
515	#103	C₂³×C₅₃	1	2	yes	—
516	#103	C₂³×C₅₃	1	2	yes	—
517	#103	C₂³×C₅₃	1	4	yes	—
518	#103	C₂³×C₅₃	1	4	yes	—
519	#102	C₂³×C₅₃	1	2	yes	—
520	#102	C₂³×C₅₃	1	2	yes	—
521	#102	C₂³×C₅₃	1	4	yes	—
522	#102	C₂³×C₅₃	1	4	yes	—
523	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
524	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
525	#97	C₂³×C₅₃	1	2	yes	—
526	C₄×C₈	C₂³×C₅₃	1	2	yes	—
527	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
528	#41	—	2	2	no	—
529	#41	—	2	2	no	—
530	#41	—	2	4	no	—
531	#41	—	2	4	no	—
532	#41	—	8	8	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
535	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
536	C₂×D₄	—	2	2	no	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	4	no	—
540	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
541	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
542	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
543	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	2	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	4	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	8	4	no	—
554	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
555	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
556	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
558	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
559	Q₁₆	—	2	2	no	—
560	SD₁₆	—	2	4	no	—
561	D₈	—	2	2	no	—
562	M₁₆	—	2	2	no	—
563	M₁₆	—	2	2	no	—
564	M₁₆	—	2	4	no	—
565	C₂×C₈	—	2	2	no	—
566	C₂×C₈	—	2	4	no	—
567	C₂×C₈	—	2	4	no	—
568	C₂×C₈	—	2	4	no	—
569	C₂×C₈	#41×C₅₃	1	4	yes	complemented
570	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
571	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
572	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
573	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
574	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
588	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
589	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
590	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
591	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
603	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
604	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
605	C₄²	—	2	4	no	—
606	C₄²	—	2	4	no	—
607	C₄²	—	2	2	no	—
608	C₄²	C₂⁴×C₅₃	1	2	yes	—
609	C₂³	—	2	2	no	—
610	C₂³	—	2	4	no	—
611	C₂³	—	2	2	no	—
612	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
613	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
614	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
615	Q₈	—	2	2	no	—
616	Q₈	—	2	2	no	—
617	Q₈	—	2	2	no	—
618	Q₈	—	8	4	no	—
619	D₄	—	2	4	no	—
620	D₄	—	2	8	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	2	no	—
623	D₄	—	2	2	no	—
624	D₄	—	4	4	no	—
625	D₄	—	4	4	no	—
626	D₄	—	4	4	no	—
627	D₄	—	8	4	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	4	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	4	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	4	8	no	—
640	C₂×C₄	—	4	4	no	—
641	C₂×C₄	—	8	4	no	—
642	C₂×C₄	—	8	2	no	—
643	C₂×C₄	—	8	4	no	—
644	C₂×C₄	—	8	2	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
647	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
648	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
650	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
651	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
652	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
653	C₈	—	2	2	no	—
654	C₈	—	2	2	no	—
655	C₈	—	2	2	no	—
656	C₈	C₂×#41×C₅₃	1	2	yes	—
657	C₈	C₂×#41×C₅₃	1	2	yes	—
658	C₂²	—	2	8	no	—
659	C₂²	—	2	2	no	—
660	C₂²	—	2	2	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	4	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	C₅₃×#547	1	2	yes	—
668	C₂²	C₅₃×#577	1	4	yes	—
669	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	4	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	4	no	—
675	C₄	—	4	4	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	2	no	—
678	C₄	C₅₃×#547	1	2	yes	—
679	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
680	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
681	C₂	—	2	2	no	—
682	C₂	—	2	4	no	—
683	C₂	—	4	2	no	—
684	C₂	—	8	2	no	—
685	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
686	C₂	C₂×C₅₃×#577	1	2	yes	—

	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 ◁ C₅₃×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3492	C₂	semidirect
2. C₅₃×#3345	C₂	semidirect
3. C₅₃×#3344	C₂	semidirect
4. C₅₃×#3337	C₂	semidirect
5. C₅₃×#3335	C₂	semidirect
6. C₅₃×#3163	C₂	semidirect
7. C₅₃×#3099	C₂	semidirect
8. C₅₃×#3092	C₂	semidirect
9. C₅₃×#3084	C₂	semidirect
10. C₅₃×#3060	C₂	semidirect
11. C₅₃×#3050	C₂	semidirect
12. C₅₃×#2952	C₂	semidirect
13. C₅₃×#2929	C₂	semidirect
14. C₅₃×#2912	C₂	semidirect
15. C₂²×#106×C₅₃	C₂	semidirect
36. C₅₃×#501	C₂²	semidirect
37. C₅₃×#499	C₂²	semidirect
38. C₅₃×#498	C₂²	semidirect
39. C₅₃×#485	C₂²	semidirect
42. C₅₃×#483	C₂²	semidirect
43. C₅₃×#482	C₂²	semidirect
45. C₅₃×#481	C₂²	semidirect
47. C₅₃×#479	C₂²	semidirect
48. C₅₃×#478	C₂²	semidirect
49. C₅₃×#476	C₂²	semidirect
50. C₅₃×#475	C₂²	semidirect
51. C₅₃×#470	C₂²	semidirect
52. C₅₃×#469	C₂²	semidirect
57. C₅₃×#465	C₂²	semidirect
58. C₅₃×#442	C₂²	semidirect
59. C₅₃×#441	C₂²	semidirect
60. C₄×SD₁₆×C₅₃	C₂²	semidirect
65. C₅₃×#428	C₂²	semidirect
67. C₅₃×#427	C₂²	semidirect
73. C₂×#106×C₅₃	C₂²	semidirect
225. C₂×C₈×C₅₃	#41	semidirect
314. #61259	C₅₃	direct (N × H)
338. #3492	C₂×C₅₃	semidirect
339. #3345	C₂×C₅₃	semidirect
340. #3344	C₂×C₅₃	semidirect
341. #3337	C₂×C₅₃	semidirect
342. #3335	C₂×C₅₃	semidirect
343. #3163	C₂×C₅₃	semidirect
344. #3099	C₂×C₅₃	semidirect
345. #3092	C₂×C₅₃	semidirect
346. #3084	C₂×C₅₃	semidirect
347. #3060	C₂×C₅₃	semidirect
348. #3050	C₂×C₅₃	semidirect
349. #2952	C₂×C₅₃	semidirect
350. #2929	C₂×C₅₃	semidirect
351. #2912	C₂×C₅₃	semidirect
352. C₂²×#106	C₂×C₅₃	semidirect
379. #501	C₂²×C₅₃	semidirect
380. #499	C₂²×C₅₃	semidirect
381. #498	C₂²×C₅₃	semidirect
382. #485	C₂²×C₅₃	semidirect
385. #483	C₂²×C₅₃	semidirect
386. #482	C₂²×C₅₃	semidirect
388. #481	C₂²×C₅₃	semidirect
390. #479	C₂²×C₅₃	semidirect
391. #478	C₂²×C₅₃	semidirect
392. #476	C₂²×C₅₃	semidirect
393. #475	C₂²×C₅₃	semidirect
394. #470	C₂²×C₅₃	semidirect
395. #469	C₂²×C₅₃	semidirect
400. #465	C₂²×C₅₃	semidirect
401. #442	C₂²×C₅₃	semidirect
402. #441	C₂²×C₅₃	semidirect
403. C₄×SD₁₆	C₂²×C₅₃	semidirect
408. #428	C₂²×C₅₃	semidirect
410. #427	C₂²×C₅₃	semidirect
416. C₂×#106	C₂²×C₅₃	semidirect
424. C₅₃	#61259	direct (N × H)
569. C₂×C₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3492	2
C₅₃×#3345	2
C₅₃×#3344	2
C₅₃×#3337	2
C₅₃×#3335	2
C₅₃×#3163	2
C₅₃×#3099	2
C₅₃×#3092	2
C₅₃×#3084	2
C₅₃×#3060	2
C₅₃×#3050	2
C₅₃×#2952	2
C₅₃×#2929	2
C₅₃×#2912	2
C₂²×#106×C₅₃	2
#61259	53

Subgroups — C53×#61259 #27268771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61259 #27268771