Subgroups — C₅₃×#19637 #27227149

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1720	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1719	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1711	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1708	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#498	C₂²	1	2	yes	complemented
18	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#492	C₂²	1	2	yes	—
20	C₅₃×#491	—	2	2	no	—
21	C₅₃×#490	—	2	2	no	—
22	C₅₃×#487	C₂²	1	2	yes	complemented
23	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#487	C₂²	1	2	yes	complemented
25	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
26	C₅₃×#487	C₂²	1	2	yes	complemented
27	C₅₃×#486	—	2	2	no	—
28	C₅₃×#486	C₂²	1	2	yes	—
29	C₅₃×#486	C₂²	1	2	yes	—
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#486	C₂²	1	2	yes	—
32	C₅₃×#486	C₂²	1	2	yes	—
33	C₅₃×#483	—	2	2	no	—
34	C₅₃×#475	—	2	2	no	—
35	C₅₃×#467	—	2	2	no	—
36	C₅₃×#460	—	2	2	no	—
37	C₅₃×#450	—	2	2	no	—
38	C₄×Q₁₆×C₅₃	—	2	2	no	—
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₅₃×#419	—	2	2	no	—
41	C₅₃×#418	—	2	2	no	—
42	C₂×#102×C₅₃	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂²×Q₈×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂×Q₁₆×C₅₃	—	2	2	no	—
52	C₂×SD₁₆×C₅₃	—	2	4	no	—
53	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	D₄	1	2	yes	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	#127×C₅₃	—	2	1	no	—
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	—	2	1	no	—
66	#127×C₅₃	C₂³	1	2	yes	—
67	#125×C₅₃	—	2	1	no	—
68	#125×C₅₃	—	2	1	no	—
69	C₂²⋊Q₈×C₅₃	—	2	1	no	—
70	C₂²⋊Q₈×C₅₃	—	2	1	no	—
71	C₂²⋊Q₈×C₅₃	—	2	1	no	—
72	C₂²⋊Q₈×C₅₃	—	2	1	no	—
73	C₄×Q₈×C₅₃	—	2	1	no	—
74	C₄×Q₈×C₅₃	—	2	1	no	—
75	C₄×Q₈×C₅₃	—	2	1	no	—
76	C₄×Q₈×C₅₃	—	2	1	no	—
77	#117×C₅₃	—	2	1	no	—
78	#117×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	#107×C₅₃	—	2	2	no	—
83	#106×C₅₃	—	2	4	no	—
84	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
85	#103×C₅₃	—	2	4	no	—
86	#103×C₅₃	—	2	4	no	—
87	#103×C₅₃	—	2	2	no	—
88	#103×C₅₃	—	2	2	no	—
89	#103×C₅₃	—	2	4	no	—
90	#102×C₅₃	—	2	4	no	—
91	#102×C₅₃	—	2	4	no	—
92	#102×C₅₃	—	2	4	no	—
93	#102×C₅₃	—	2	4	no	—
94	#102×C₅₃	—	2	4	no	—
95	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
96	#97×C₅₃	D₄	1	2	yes	—
97	C₄×C₈×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂⁴×C₅₃	—	4	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	4	1	no	—
109	C₂×Q₈×C₅₃	—	4	4	no	—
110	C₂×D₄×C₅₃	—	2	2	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	4	4	no	—
114	C₂×D₄×C₅₃	—	4	1	no	—
115	C₂×D₄×C₅₃	—	4	1	no	—
116	C₂×D₄×C₅₃	—	4	1	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	8	2	no	—
119	C₂²×C₄×C₅₃	—	4	1	no	—
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₅₃	—	4	4	no	—
124	SD₁₆×C₅₃	—	8	4	no	—
125	M₁₆×C₅₃	—	2	4	no	—
126	C₂×C₈×C₅₃	—	4	2	no	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	4	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₄⋊C₄×C₅₃	—	4	1	no	—
145	C₄⋊C₄×C₅₃	—	4	1	no	—
146	C₄⋊C₄×C₅₃	—	4	1	no	—
147	C₂²⋊C₄×C₅₃	—	4	1	no	—
148	C₂²⋊C₄×C₅₃	—	4	1	no	—
149	C₄²×C₅₃	—	4	1	no	—
150	C₄²×C₅₃	—	4	1	no	—
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂³×C₅₃	—	4	2	no	—
156	C₂³×C₅₃	—	8	2	no	—
157	C₂³×C₅₃	—	8	2	no	—
158	C₂³×C₅₃	—	8	2	no	—
159	C₂³×C₅₃	#127	1	1	yes	characteristic
160	Q₈×C₅₃	—	4	4	no	—
161	Q₈×C₅₃	—	4	2	no	—
162	Q₈×C₅₃	—	4	2	no	—
163	Q₈×C₅₃	—	4	2	no	—
164	Q₈×C₅₃	—	4	2	no	—
165	Q₈×C₅₃	—	8	2	no	—
166	D₄×C₅₃	—	4	4	no	—
167	D₄×C₅₃	—	4	4	no	—
168	D₄×C₅₃	—	4	4	no	—
169	D₄×C₅₃	—	8	2	no	—
170	D₄×C₅₃	—	8	2	no	—
171	D₄×C₅₃	—	8	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
182	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#135	1	4	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₈×C₅₃	—	2	4	no	—
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	2	2	no	—
190	C₈×C₅₃	—	2	4	no	—
191	#19637	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	8	2	no	—
195	C₂²×C₅₃	—	8	2	no	—
196	C₂²×C₅₃	—	8	2	no	—
197	C₂²×C₅₃	—	16	4	no	—
198	C₂²×C₅₃	#495	1	2	yes	—
199	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	8	2	no	—
203	C₄×C₅₃	—	8	1	no	—
204	C₄×C₅₃	—	8	1	no	—
205	C₄×C₅₃	—	8	1	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	#495	1	2	yes	—
208	C₄×C₅₃	C₂×#135	1	2	yes	—
209	C₄×C₅₃	#575	1	2	yes	—
210	C₄×C₅₃	#577	1	2	yes	—
211	#3229	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#3208	C₂×C₅₃	1	1	yes	characteristic
213	#3170	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#3139	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#3138	C₂×C₅₃	1	1	yes	characteristic
216	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
217	#1720	C₂×C₅₃	1	2	yes	complemented
218	#1719	C₂×C₅₃	1	2	yes	complemented
219	#1711	C₂×C₅₃	1	2	yes	complemented
220	#1708	C₂×C₅₃	1	2	yes	complemented
221	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×C₅₃	—	2	2	no	—
223	C₂×C₅₃	—	16	2	no	—
224	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
225	C₂×C₅₃	#3153	1	1	yes	characteristic
226	C₂×C₅₃	#3154	1	1	yes	characteristic
227	#533	C₂²×C₅₃	1	1	yes	characteristic
228	#533	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
231	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
232	#498	C₂²×C₅₃	1	2	yes	complemented
233	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
234	#492	C₂²×C₅₃	1	2	yes	—
235	#491	—	2	2	no	—
236	#490	—	2	2	no	—
237	#487	C₂²×C₅₃	1	2	yes	complemented
238	#487	C₂²×C₅₃	1	2	yes	complemented
239	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
240	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
241	#487	C₂²×C₅₃	1	2	yes	complemented
242	#486	—	2	2	no	—
243	#486	C₂²×C₅₃	1	2	yes	—
244	#486	C₂²×C₅₃	1	2	yes	—
245	#486	C₂²×C₅₃	1	2	yes	—
246	#486	C₂²×C₅₃	1	2	yes	—
247	#486	C₂²×C₅₃	1	2	yes	—
248	#483	—	2	2	no	—
249	#475	—	2	2	no	—
250	#467	—	2	2	no	—
251	#460	—	2	2	no	—
252	#450	—	2	2	no	—
253	C₄×Q₁₆	—	2	2	no	—
254	#431	C₂²×C₅₃	1	1	yes	characteristic
255	#419	—	2	2	no	—
256	#418	—	2	2	no	—
257	C₂×#102	—	2	2	no	—
258	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
260	#322	C₂²×C₅₃	1	2	yes	complemented
261	C₈²	C₂²×C₅₃	1	2	yes	complemented
262	C₅₃	#19637	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×Q₈	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂²×D₄	—	2	1	no	—
267	C₂×Q₁₆	—	2	2	no	—
268	C₂×SD₁₆	—	2	4	no	—
269	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
270	C₂²×C₈	D₄×C₅₃	1	2	yes	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
275	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
279	#127	—	2	1	no	—
280	#127	—	2	1	no	—
281	#127	—	2	1	no	—
282	#127	C₂³×C₅₃	1	2	yes	—
283	#125	—	2	1	no	—
284	#125	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	C₂²⋊Q₈	—	2	1	no	—
288	C₂²⋊Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	—	2	1	no	—
292	C₄×Q₈	—	2	1	no	—
293	#117	—	2	1	no	—
294	#117	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
298	#107	—	2	2	no	—
299	#106	—	2	4	no	—
300	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
301	#103	—	2	4	no	—
302	#103	—	2	4	no	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#103	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#102	—	2	4	no	—
310	#102	—	2	4	no	—
311	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
312	#97	D₄×C₅₃	1	2	yes	—
313	C₄×C₈	D₄×C₅₃	1	2	yes	—
314	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₅₃	1	2	yes	—
316	C₄×C₈	C₂³×C₅₃	1	2	yes	—
317	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
318	C₂⁴	—	4	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	—	4	4	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	4	4	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂²×C₄	—	4	1	no	—
336	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
339	Q₁₆	—	4	4	no	—
340	SD₁₆	—	8	4	no	—
341	M₁₆	—	2	4	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
345	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
346	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
370	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
371	C₂³	—	4	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	—	8	2	no	—
375	C₂³	#127×C₅₃	1	1	yes	characteristic
376	Q₈	—	4	4	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	2	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	8	2	no	—
382	D₄	—	4	4	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	4	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	2	no	—
387	D₄	—	8	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	2	no	—
396	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
397	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
398	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
399	C₂×C₄	#135×C₅₃	1	4	yes	—
400	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
403	C₈	—	2	4	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	2	no	—
406	C₈	—	2	4	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	8	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	16	4	no	—
413	C₂²	C₅₃×#495	1	2	yes	—
414	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	8	2	no	—
418	C₄	—	8	1	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	1	no	—
421	C₄	—	8	1	no	—
422	C₄	C₅₃×#495	1	2	yes	—
423	C₄	C₂×#135×C₅₃	1	2	yes	—
424	C₄	C₅₃×#575	1	2	yes	—
425	C₄	C₅₃×#577	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	16	2	no	—
428	C₂	C₂×C₅₃×#495	1	1	yes	characteristic
429	C₂	C₅₃×#3153	1	1	yes	characteristic
430	C₂	C₅₃×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3229	C₂	semidirect
3. C₅₃×#3170	C₂	semidirect
4. C₅₃×#3139	C₂	semidirect
7. C₅₃×#1720	C₂	semidirect
8. C₅₃×#1719	C₂	semidirect
9. C₅₃×#1711	C₂	semidirect
10. C₅₃×#1708	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
16. C₅₃×#500	C₂²	semidirect
17. C₅₃×#498	C₂²	semidirect
18. C₅₃×#494	C₂²	semidirect
22. C₅₃×#487	C₂²	semidirect
23. C₅₃×#487	C₂²	semidirect
24. C₅₃×#487	C₂²	semidirect
25. C₅₃×#487	C₂²	semidirect
26. C₅₃×#487	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
191. #19637	C₅₃	direct (N × H)
211. #3229	C₂×C₅₃	semidirect
213. #3170	C₂×C₅₃	semidirect
214. #3139	C₂×C₅₃	semidirect
217. #1720	C₂×C₅₃	semidirect
218. #1719	C₂×C₅₃	semidirect
219. #1711	C₂×C₅₃	semidirect
220. #1708	C₂×C₅₃	semidirect
221. C₈×M₁₆	C₂×C₅₃	semidirect
231. #500	C₂²×C₅₃	semidirect
232. #498	C₂²×C₅₃	semidirect
233. #494	C₂²×C₅₃	semidirect
237. #487	C₂²×C₅₃	semidirect
238. #487	C₂²×C₅₃	semidirect
239. #487	C₂²×C₅₃	semidirect
240. #487	C₂²×C₅₃	semidirect
241. #487	C₂²×C₅₃	semidirect
260. #322	C₂²×C₅₃	semidirect
261. C₈²	C₂²×C₅₃	semidirect
262. C₅₃	#19637	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×#3229	2
C₅₃×#3208	2
C₅₃×#3170	2
C₅₃×#3139	2
C₅₃×#3138	2
C₂×C₅₃×#486	2
C₅₃×#1720	2
C₅₃×#1719	2
C₅₃×#1711	2
C₅₃×#1708	2
C₈×M₁₆×C₅₃	2
#19637	53

Subgroups — C53×#19637 #27227149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19637 #27227149